Autorské dílo v rámci projektu Rozvoj jazykových kompetencí pracovníků VŠB-TUO: InterDV, registrační číslo projektu: CZ.1.07/2.2.00/15.

Transcription

1 MODAL TESTING Autoské dílo v ámci pojektu Rozvoj jazykových kompetencí pacovníků VŠB-TUO: InteDV, egistační číslo pojektu: CZ..7/../5.3 Ostava Alena Bilošová

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3 CONTENTS Foewod...7 List of Symbols...8 List of Abbeviations.... Intoduction.... Application of Modal Tests.... Summay of Theoy Vaious Types of Fequency Response Functions Summay of Measuement Methods Summay of Modal Analysis...7. Dual-channel Analysis...8. Autospectum...8. Coss Spectum Coheence....4 System desciptos Fequency Response Function (FRF) Impulse Response Function - IRF Effect of Noise on the FRF Digital Signal Pocessing Aliasing Leakage Fequency Zoom Aveaging Theoetical Basis of Modal Analysis Single Degee-of-feedom System (SDOF) Undamped Single Degee-of-feedom System Viscously damped SDOF system Fee vibation Foced vibation Detemination of Resonance Tuning Detemination of Damping fom FRF Plots Single Degee-of-feedom System with Hysteetic (Stuctual) Damping Foced Vibation

4 3..3. Detemination of Damping fom FRF Plots Vaious Foms of FRF fo Single Degee-of-feedom System Multi Degee-of-feedom System (MDOF) Undamped MDOF System Fee vibation Othogonal Popeties of Eigenvectos Nomalization of Mode Shapes Foced Response Analysis of Multi Degee-of-feedom System Chaacteistics and Pesentation of Multi Degee-of-Feedom FRF Data Damped Multi Degee-of-Feedom System Popotional Viscous Damping Popotional Hysteetic Damping Hysteetic Damping - Geneal Case MDOF System with Geneal Hysteetic Damping - Foced Response Solution MDOF Systems - Summay fo Vaious Types of Damping Excitation by a Geneal Foce Vecto Excitation by a Vecto of Mono-Phased Foces Modal Test Pepaation Pepaation of the Measued Stuctue Fee Suppot Gounded (Fixed) Suppot Suppot in-situ Pepaation of Expeimental Model Measuement Techniques Basic Measuement Setup Excitation Techniques Tansduces Used fo Excitation Foce and Response Measuements Analyze Pepaation of Measuement Refeence Point Measuement Expeimental Modal Analysis - Modal Paametes Identification Single Degee-of-Feedom Appoach Peak-picking method Cicle-fit Method

5 4.3. MDOF System Appoximation Methods Modal Model Pesentation of the Obtained Modal Model Veification of the Obtained Modal Model Compaison of the Expeimentally and Computationally Deived Models Compaison of Natual Fequencies Gaphical Compaison of Modal Shapes Numeical Compaison of Mode Shapes Opeational Deflection Shapes (ODS) Opeational Modal Analysis (OMA) Identification Methods Pesentation of Results...8 Refeences

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7 Foewod The aea of modal testing is quite extensive and to maste it pefectly, it is necessay to integate knowledge fom diffeent fields: vibation measuements, signal pocessing, postpocessing, mathematical backgound, issues of vibating multi degee-of-feedom systems with diffeent models of damping, etc. This text does not aim to discuss in detail all aspects of modal testing, but only to familiaize students with this issue enough to be able to pefom a modal test independently and be awae of poblems that may occu duing measuements and data pocessing. Most of the eades will be students of Applied Mechanics specialization. My main goal is to ceate awaeness in thei minds of what a modal test is about, what it is used fo and when they should conside its implementation in thei engineeing pactice. I have noticed duing the yeas of woking with students that thei confidence of the esults obtained using finite element method is sometimes too high and that they often do not ealize that even the esults obtained using vey sophisticated finite element pogams can be fa fom eality - usually due to the fact that when using these pogams, they neglected o ovely simplified something. Theefoe, veifying and updating the computational model with the use of expeimentally deived data is highly advisable and in some banches (e.g. aicaft) even mandatoy. The basic souce fo witing this text was the book "Modal Analysis theoy, pactice and application" witten by pof. David J. Ewins fom Impeial College of Science, Technology and Medicine in London, who can be egaded as a leading figue in the modal analysis in Euope. I hope he would appeciate that I left his wods unchanged wheeve it was appopiate. The second elevant souce I used wee mateials povided by Büel&Kjæ company - aticles, application notes, pesentations and pictues. This company poduces all the equipment used fo modal testing and also povides a technical suppot to thei customes by publishing a lot of technical papes and oganizing technical taining. Of couse, I added my own expeience and tied to ceate a text that would be useful and well eadable fo students. I wish to all of you that the time spent by eading this text will not be a loss but contibute to you effot to become a competent and well educated mechanical enginee. autho - 7 -

8 List of Symbols * complex conjugate [ ] T matix tanspose a(t) time ecod in channel A A(f) instantaneous spectum of channel A A(ω) inetance (acceleance) [kg - ] b viscous damping constant [kg s - ] b(t) time ecod in channel B B(f) instantaneous spectum of channel B [B] viscous damping matix [kg s - ] F b damping foce [kg m s - ] {F} vecto of complex magnitudes of excitation foces [kg m s - ] {f(t)} excitation foces vecto [kg m s - ] f natual fequency of damped vibation [Hz] f f s G AA (f) G BB (f) G AB (f) G BA (f) h(t) natual fequency of undamped vibation [Hz] sampling fequency [Hz] autospectum of channel A of an analyze (one-sided) autospectum of channel B of an analyze (one-sided) coss spectum fom channel A to channel B of an analyze (one-sided) coss spectum fom channel B to channel A of an analyze (one-sided) impulse esponse function [H] hysteetic damping matix [kg s - ] H(ω) fequency esponse function (eceptance, mobility o inetance) H(s) tansfe function H (f) estimate of fequency esponse function H (f) estimate of fequency esponse function i imaginay unit [I] unity matix (diagonal) k stiffness [kg s - ] [K] stiffness matix [kg s - ] m mass [kg] [M] mass matix [kg] N numbe of degees of feedom - 8 -

9 p pole [s - ] {p} vecto of modal coodinates R esiduum [kg - ] S AA (f) autospectum of channel A of an analyze (two-sided) S BB (f) autospectum of channel B of an analyze (two-sided) S AB (f) coss spectum fom channel A to channel B of an analyze (two-sided) S BA (f) coss spectum fom channel B to channel A of an analyze (two-sided) t time [s] T peiod [s] T(f) tansmissibility [-] {X} vecto of complex magnitudes of displacements [m] { ( t) } x vecto of displacements [m] { x& ( t) }, { ( t) } {& x& ( t) }, { ( t) } v velocity vecto [m s - ] a acceleation vecto [m s - ] Y(ω) mobility [kg - s ] α(ω) eceptance [kg - s ] δ decay constant [s - ] Φ j j th element of the th eigenvecto [-] {Φ} th eigenvecto [-] [Φ] modal matix (mass-nomalized) [-] γ stuctual damping loss facto [-] η tuning coefficient [-] λ eigenvalue of the th mode [s - ] τ time constant of exponential weighting window [s] υ logaithmic decement [-] ω angula fequency of excitation [s - ] Ω angula natual fequency of damped vibation [s - ] Ω angula natual fequency of undamped vibation [s - ] [Ψ] modal matix (geneal fom) [-] ζ damping atio [-] ( f ) γ coheence function [-] - 9 -

10 List of Abbeviations COP DFT DOF FDD FRF FFT IRF MAC MDOF MIMO MSF ODS OMA PSD SDOF SISO SIMO SSI SVD Coheence Output Powe Discete Fouie Tansfom Degee of Feedom Fequency Domain Decomposition Fequency Response Function Fast Fouie Tansfom Impulse Response Function Modal Assuance Citeion Multi Degee of Feedom System Multiple Input Multiple Output Modal Scale Facto Opeational Deflection Shape Opeational Modal Analysis Powe Spectal Density Single Degee of Feedom System Single Input Single Output Single Input Multiple Output Stochastic Subspace Identification Singula Value Decomposition - -

11 . Intoduction Befoe speaking about the aea of modal testing itself, it is woth to know something about diffeent appoaches to vibation measuements. Fom both methodical and pactical points of view, it is useful to distinguish the two expeimental appoaches dealing with vibations: ) Assessment of the natue and level of vibation esponses - signal analysis ) Deiving theoetical models and pesumptions and thei evaluation - system analysis Two types of measuements coespond to these two appoaches: ad ) Vibation esponses of the machine o the stuctue unde investigation ae measued duing opeation conditions. Vibation diagnostics deals with this aea. ad ) Stuctue o a machine pat is put into vibation by means of known excitation foces, often out of its woking envionment. This pocess is substantial fo modal tests. It is obvious that we ae able to get moe accuate and detailed infomation about the measued system unde contolled conditions athe than by simple esponse measuements. This mateial deals with the latte appoach thooughly. By pefoming a modal test, we ae able to detemine modal paametes of the system, thus having a base fo solving many poblems caused by stuctual vibations. Poblems with stuctual vibations pose significant isks and limitations fo design of a wide ange of machiney poducts. They could be a cause of a stuctual integity failue (fo instance beaking of a tubine blade) o they could educe machiney pefomance. At least, excessive vibations always cause excessive noise and discomfot duing opeation. Modal test : "Pocesses applied to the tested pats o stuctues with the aim to get mathematical desciption of thei dynamic behaviou.". Application of Modal Tests Thee ae seveal easons fo pefoming modal tests. Hee they ae soted by accuacy equiements and by the degee of elationship to theoetical analysis: a) Identification of modal paametes (natual fequencies, modal shapes and modal damping espectively) without elationship to theoetical model. Doing this, we can discove e.g. whethe excessive vibations duing opeation ae caused by esonance and what the excited modal shape looks like. - -

12 b) Identification of modal paametes to compae the expeimentally obtained data with coesponding data obtained by FEM o othe theoetical methods. The goal is to veify the theoetical model befoe othe calculations such as esponses to diffeent loads ae caied out. Fo this we need: - pecise detemination of natual fequencies - identification of modal shapes with such pecision that thei compaison with computed modal shapes is possible - matching the coesponding modal shapes togethe c) The same as b) plus coection of the theoetical model so that it bette matches the measued values. This is done by tial-and-eo method usually, e.g. by a slight change in mateial paametes o inclusion of modal damping in the theoetical model. d) Coelation of expeimental and theoetical esults - two sets of data ae numeically compaed in ode to pecisely identify the causes of discepancy between the calculated and measued popeties. This equies a much moe accuate measuement of modal shapes than when we only want to animate them (as it is in the pevious cases). e) Using modal testing to obtain a mathematical model of a component that can be included in a complex one. This appoach is often used fo theoetical analysis of complex stuctues. It equies accuate values of natual fequencies, modal damping and modal shapes. All modes must be included, it is not possible to fit the model to a few individual natual fequencies. Excluded modes affect dynamic behaviou of the entie stuctue in the obseved fequency ange. This application is moe demanding than all pevious ones. f) Ceating a model that can be used to pedict the impact of stuctual modifications to the oiginal test stuctue. It is a smalle change than in the case of the substuctues, so hee ae a little lowe accuacy equiements than in the pevious case. Yet, in both of these cases complications usually occu with non-measued otational degees of feedom. g) Using the model obtained though modal testing to detemine the excitation foces. It is possible to compae esponses caused by excitation foces with the mathematical desciption of tansfe functions of the stuctue and, on the base of this compaison, to estimate the excitation foces. Successful modal testing equies a combination of the thee skills and knowledge: - theoetical famewok - accuate measuement of vibation - ealistic and detailed data pocessing In this intoductoy chapte, only the basis of these thee equiements ae pesented. They will futhe be explained in details. - -

13 . Summay of Theoy The system unde investigation can be descibed by thee diffeent types of models. Each of them is defined by system matices. spatial model - [M[... mass matix - [K]... stiffness matix - [B] o [H]... viscous damping o stuctual (hysteetic) damping matix Matices ae of dimensions N N. (N numbe of degees of feedom numbe of equations of motion) modal model - [λ ]... spectal matix, diagonal, eigenvalues ae on the diagonal - [Φ]... modal matix, columns ae modal shapes 3 d mode nn mode st mode esponse model - [H(ω)]... FRF matix (matix of FRFs - fequency esponse functions, e.g. mobilities Y(ω) o IRFs - impulse esponse functions, symmetical) Pefoming theoetical vibation analysis, we advance fom a spatial model to a esponse model in the following steps:. establish equations of motion spatial model. fee vibation analysis modal model 3. analysis of foced vibation using hamonic excitation esponse model Pefoming expeimental vibation analysis, we advance in opposite diection in the following steps:. measuements of the appopiate set of FRFs esponse model. cuve-fitting of the measued data modal model 3. futhe calculations spatial model (it is not common to pefom this step) Fequency Response Function - FRF, which is the basis of the esponse model, can be expessed as: output movement H ( ω ) input foce esponse excitation - 3 -

14 Thee ae thee basic types of FRFs accoding to the type of esponse paamete, which can be eithe displacement, velocity o acceleation - see Table.. One element of the eceptance matix α jk (ω) epesents hamonic esponse x j at point j caused by a single hamonic foce F k acting at the diffeent point k. Pecise definition of one element of fequency esponse function (fo eceptance matix [α(ω)]): α jk x j ( ω) F k Φ Φ N j k λ ω whee: λ - eigenvalue of the th mode (natual fequency + modal damping) Φ j - the j th element of the th natual shapes' vecto {Φ}, i.e. elative displacement at the j th point as vibating with the th shape N - numbe of modes (.) Note: With the expeimental pocedue, the numbe of extacted modes N is usually smalle than the numbe of degees of feedom (DOF), which is caused by limited fequency ange of measuements. An expeimentally obtained model is a so called incomplete model, in contast to a complete model obtained by computation; we can theoetically obtain a numbe of modes of vibation equal to a numbe of DOFs in a computational model. The expession (.) is the basis of modal tests - it eflects a diect connection between modal popeties of the system and its esponse chaacteistics. Fom a puely theoetical point of view it povides an effective means to calculate esponses, wheeas fom a pactical point of view it allows to detemine modal popeties fom mobility measuements. If we apply the theoetical knowledge of the elationship between eceptance functions and modal paametes, it is possible to pove that the "appopiate" set of measued eceptances α (ω). In pactice, this must only contain one ow o one column of the mobility matix [ ] means that we eithe excite the stuctue at one point and measue esponses at all points o we measue the esponse at one point and excite the stuctue at all points. The fist option applies when a dynamic excite is used, the second when an impact hamme o othe contactless device is used. dynamic excite one column of FRF matix is measued impact hamme excitation one ow of FRF matix is measued.3 Vaious Types of Fequency Response Functions When a fequency esponse function is efeed to without esponse paamete specification, it is usually denoted as H(ω). When a esponse paamete is specified, individual FRFs have thei own denotation (see Table.)

15 Fequency esponse function Response paamete Standad F Invese F displacement X velocity V acceleation A eceptance admittance dynamic compliance dynamic flexibility α(ω) mobility Y(ω) inetance acceleance A(ω) dynamic stiffness mechanical impedance appaent mass Tab. Vaious Types of FRFs Accoding to Response Paamete Displacement as a function of time is in complex notation expessed as: x(t) iωt Xe (.) Expessions fo velocity and acceleation can be obtain by simple deivative: v(t) iωt x(t) & iωxe (.3) t Xe i ω & (.4) a(t) x(t) & ω FRF of type eceptance with displacement as a esponse paamete is defined: X α ( ω) F (.5) And again, using deivatives we obtain anothe types of FRF: V X Y( ω ) iω iωα( ω) F F... mobility (.6) A A( ω ) ω α( ω) F... inetance (.7).4 Summay of Measuement Methods The following aspects demand special attention in ode to ensue acquisition of highquality data: a) mechanical aspects of suppoting and coectly exciting the stuctue b) coect tansduction of the quantities to be measued - foce input and motion esponse c) signal pocessing which is appopiate to the type of the test used - 5 -

16 Vaious Ways of Suppoting the Testpiece Befoe a modal test, we have to conside vaious ways of suppoting the measued stuctue. Geneally, we choose one of the following thee options of suppot coesponding to the aim of the modal test and to the limitations caused by opeating conditions, espectively: - fee (unestained) - It is the simplest way how to suppot the testpiece and it is pefeably used each time we want to coelate expeimental model with a theoetical one. It is usually implemented by suspending the testpiece on vey soft spings (ubbe opes o foam pad). - gounded (clamped) - It equies igid clamping of the testpiece at cetain points. It is moe complicated, because ideal fixation is impossible in eal. Then, discepancies between an expeimental and theoetical model could lagely oiginate fom unequal bounday conditions. Howeve, this suppot needs to be used occasionally (e.g. fo detemination modal paametes of tubine blades). - in situ (unde opeational conditions) - It is used when modal paametes unde eal opeational conditions ae needed and no coelation with a theoetical model is equied. Excitation of the Stuctue The way of how to excite vibations of the measued stuctue is again given mostly by the aim of the modal test, pecision equiements and fequency ange in which the modal paametes ae to be detemined. Thee ae basically two ways of excitation: - excitation by means of dynamic excite by hamonic signal by andom signal by othe types of signals (see chapte 4...3) - impulse excitation by means of impact hamme step elease (fom defomed position) Tansduces Tansduces used fo sensing foce and esponse should affect the measued stuctue as little as possible and thei effectiveness should be adequate to measuement fequency ange and to displacements unde consideation. Nowadays, piezoelectic tansduces of both foce and acceleation ae used; esponse is mostly picked in the fom of acceleation. Chapte deals with tansduces in moe detail

17 .5 Summay of Modal Analysis An analysis of measued data is a pocess in which the measued fequency esponse functions ae analyzed in ode to find a theoetical model that most closely esembles the dynamic behaviou of the stuctue unde test. This pat of the modal test is called expeimental modal analysis, although this tem is often incoectly used fo the entie modal test. The pocess of data analysis poceeds in two stages:. Identifying the appopiate type of model (with viscous o stuctual damping). This choice is often in pactice limited by softwae used fo the modal analysis. Most of softwae packages wok with one type of damping and give no choice to the use.. Detemining appopiate paametes of the chosen model. This stage, also called modal paametes extaction, is done by cuve-fitting of the measued fequency esponse functions to the theoetical expessions. This stage is discussed in detail in chapte

18 . Dual-channel Analysis In this chapte we will intoduce some tems that concen modal tests and that fall within signal pocessing aea. Whilst we use a single-channel analysis fo vibation diagnostics tasks (even if thee could be moe channels pocessed simultaneously), system analysis tasks wok on the pinciple of a dual-channel analysis. The basic scheme of dual-channel FFT (Fast Fouie Tansfom) analyze is in Fig... Duing a simultaneous analysis of signals in at least two channels, the signals themselves ae not in the foefont any moe, but athe popeties of the physical system esponsible fo the diffeences between those signals. The methods can be theoetically expanded to any numbe of channels, but basically two of them ae pocessed simultaneously each time. In the following text, individual functions that occu in system analysis will be descibed. channel A time a (t) Fouie spectum A (f) autospectum G AA (f) fequency esponse - - autocoelation R aa (τ) impulse esponse coss spectum H (f) - h(τ) coss coelation G AB (f) R ab (τ) coheence coheent output powe γ (f) γ (f) G BB (f) channel B time Fouie spectum autospectum - autocoelation b (t) B (f) G BB (f) R bb (τ) ecoding, sampling Fouie tansfom Fig.. aveaging Scheme of Dual-channel Analyze pocessing. Autospectum Autospectum is a function commonly exploed both in signal and system analysis. It is computed fom the instantaneous (Fouie) spectum as: * S ( f ) E[ A( f ) A ( f )] E [ a t *a t ] [ R ( τ) ] (.) Im AA ( ) ( ) +φ A φ A A A S AA Re ( f ) ( ) ( ) e i φa A f A f A S aa (.) iφ ( f ) ( f ) A( f ) e (.3) * A i [ e ] E A ( f ) * ( f ) E A( f ) A ( f ) AA [ ] (.4) - 8 -

19 Thee is a new, fundamental function - coss spectum - in the dual channel pocessing. It is computed fom instantaneous specta of both channels. All othe functions in the scheme in Fig.. ae computed duing post-pocessing fom the coss spectum and the two autospecta. Of couse, all functions ae the functions of fequency.. Coss Spectum Based on complex instantaneous specta A(f) and B(f), the coss spectum S AB (fom A to B) is defined as: * S ( f ) E[ A ( f ) B( f )] E [ a t * b t ] [ R ( τ) ] (.5) A(f) A*(f) B(f) φ A AB ( ) ( ) Im φ B φ B - φ A B(f) Re ( f ) ( ) ( ) e i φa A f A f ab (.6) ( f ) ( ) ( ) e i φb B f B f S AB (.7) ( ) [ ] i φb ( f ) φa ( f ) ( f ) E A( f ) B( f ) e (.8) Amplitude of the coss spectum S AB is the poduct of amplitudes, its phase is the diffeence of both phases (fom A to B). Coss spectum S BA (fom B to A) has the same amplitude, but opposite phase. The phase of the coss spectum is the phase of the system as well. Both autospecta and coss spectum can be defined eithe as two-sided (notation S AA, S BB, S AB, S BA ) o as one-sided (notation G AA, G BB, G AB, G BA ). One-sided spectum is obtained fom the two-sided one as: S AB (f) G AB (f) fo f < G AB (f) S AB (f) fo f (.9) S AB (f) fo f > f The coss spectum itself has little impotance, but it is used to compute othe functions. Its amplitude G AB indicates the extent to which the two signals coelate as the function of fequency, phase angle G AB indicates the phase shift between the two signals as the function of fequency. The advantage of the coss spectum is that influence of noise can be educed by aveaging. That is because the phase angle of the noise spectum takes andom values so that the sum of those seveal andom specta tends to zeo (see Fig..). It can be seen that the measued autospectum is a sum of the tue autospectum and autospectum of noise, whilst the measued coss spectum is equal to the tue coss spectum

20 U(f) tue input H(f) tue output V(f) M(f) A(f) N(f) noise at input measued input noise at output B(f) measued output AA channel A autospectum: * * [( U + M) ( U + M) ] Ε ( U U) * * * [ ] + Ε[ ( U M) ] + Ε[ ( M U) ] + Ε[ ( M M) ] SUU SMM S Ε + BB channel B autospectum: * * [( V + N) ( V + N) ] Ε ( V V) * * * [ ] + Ε[ ( V N) ] + Ε[ ( N V) ] + Ε[ ( N N) ] SVV SNN S Ε + AB coss spectum: * * [( U + M) ( V + N) ] Ε ( U V) * * * [ ] + Ε[ ( U N) ] + Ε[ ( M V) ] + Ε[ ( M N) ] SUV S Ε Im U * i M i U i * M i Re Fig.. Reduction of Noise at the Coss Spectum by Aveaging.3 Coheence Coheence function indicates the degee of linea elationship between two signals as a function of fequency. It is defined by two autospecta and a coss spectum as: AB ( f ) ( f ) G ( f ) G γ ( f ) (.) G AA BB At each fequency coheence can be taken as a coelation coefficient (squaed) which expesses the degee of linea elationship between two vaiables, whee the magnitudes of autospecta coespond to vaiances of those two vaiables and the magnitude of coss spectum coesponds to covaiance. Coheence value vaies fom zeo to one. Zeo means no elationship between the input A and output B, whilst one means a pefectly linea elationship (see Fig..3). ( f ) γ (.) Thee ae fou possible elationships between input A and output B in Fig..3 : a) pefectly linea elationship b) sufficiently linea elationship with a slight scatte caused by noise c) non-linea elationship d) no elationship - -

21 B B a γ AB b γ < AB A A B B c γ < AB d γ AB Fig..3 A Analogy Between Coheence and Coelation Coefficient A Coheence function povides useful infomation only when specta G AA (f), G BB (f) a G AB (f) ae estimates, i.e. specta aveaged fom moe ecods. Fo only one sample (without aveaging) applies: G AB ( f ) A( f ) B( f ) G ( f ) G ( f ) ( f ) AA BB γ (.) In the case of no aveaging, coheence is always equal to. In the case of aveaging and samples G AB influenced by noise, deviations in the phase angles cause that the esulting magnitude G AB is lowe than it would be without pesence of noise (see Fig..4). Pesence of non-lineaities has simila influence. If signals ae andom o if they include some noise, a moe eliable estimate could be obtained with the help of aveaging. Geneally, the esult may be loaded with two types of eos: - systematic (bias) eos - andom eos andom eo systematic eo eal value Fo linea systems, a systematic eo doesn't occu in the coss spectum if the analysis is pefomed with sufficient esolution (see chapte.6.). with noise Σ G ABi < Σ G ABi γ < without noise Σ G ABi Σ G ABi γ Im Im G AB G AB Σ G ABi G AB3 G AB4 Σ G ABi G AB G AB G AB3 G AB4 Re G AB G AB4 G AB3 Σ G ABi G AB Σ G ABi G AB G AB G AB3 G AB4 Re Fig..4 Influence of Noise to the Coheence - -

22 The most impotant application of the coheence function is veification of othe functions and detemination whethe they ae affected by noise o by the pesence of nonlineaities. Low coheence does not automatically mean that the measuement was invalid, but sometimes it is a sign that a lot of aveages should be pefomed to get a valid esult. The easons fo educed coheence may be: - difficult measuements: noise in measued output signal noise in measued input signal othe inputs not coelated with measued input signal system nonlineaities - bad measuements: leakage time vaying systems DOF jitte (while impact excitation, when we do not hit exactly the same position in all of the hits) Coheence is also used to obtain some of the deived functions that have vaious applications. One of these functions is Coheent Output Powe: ( f ) COP γ G (.3) BB COP gives a measue of what pat of the measued output autospectum, G BB (f), is fully coheent with a paticula input signal epesented by autospectum G AA (f). COP can be used when low coheence is caused by noise in the measued output signal. It has no sense when thee is noise in the input signal o when thee ae nonlineaities in the system. Anothe function deived fom the coheence function is Signal-to-Noise Ratio: S/ N γ (.4) γ Hee, noise in the measued output signal is consideed as the only facto affecting the coheence. Then, coheent output (popotional to γ ) gives the measue of signal contained in the output and the non-coheent output (popotional to -γ ) gives the measue of noise in the output..4 System desciptos When the signals A and B epesent input and output of the physical system, fequency esponse function H(f) in the fequency domain and impulse esponse function h(τ) in the time domain ae used to descibe the elationship between these two signals (see Fig..5). Fequency esponse function and impulse esponse function ae so-called system desciptos. They ae independent of the signals involved. - -

23 excitation a(t) esponse b(t) testpiece h(τ) H(f) time fequency a(t) h(τ) b(t) A(f) H(f) B(f) Convolution: b (t) h ( τ) a( t τ) dτ h( t) * a( t) Fig..5 Multiplication: B System Desciptos ( f ) H( f ) A( f ).4. Fequency Response Function (FRF) The main eason fo using FRFs is the simplicity with which the esponse of the eal system can be descibed. A detailed deivation of FRF of a single degee of feedom system (SDOF, see Fig..6) will be caied out in chapte 3.. Hee, we will mention only the fact that fo an ideal physical system, the popeties of which could be descibed by system of linea diffeential nd ode equations, using Laplace tansfom leads to convesion of these diffeential equations to algebaic equations of the Laplace vaiable s. Solution of these equations can be expessed in the fom of tansfe functions H ij (s) that epesent the atio of the esponse in the point i to the input in the point j. The typical tansfe function of the n degee-of-feedom system (Multi-Degee-of-Feedom system - MDOF) can be expessed as: n * R ( ) ijk R ijk H ij s + (.5) * k s p k s p k whee: p k poles - global popety fo all the tansfe functions of a system R ijk esidues - specific fo each of the tansfe functions Each membe in the sum epesents the esponse of single degee of feedom system with the pole p k -δ k + iω k (.6) The eal pat epesents damping and the imaginay pat epesents natual angula fequency of the damped vibation of the k th mode

24 Ideal physical system: F(t) - mass m is a poínt mass - movement only in one diection (x) - dampe b and sping k ae massless - dampe b and sping k ae linea - m, k a b constant in time k m b x(t), v(t), a(t) Fig..6 Ideal SDOF System A tansfe function is thee-dimensional; fo an SDOF system it is shown in Fig..7. If iω is put fo s (i.e. tansfe function is evaluated along the imaginay axis), fequency esponse function H ij (iω) is obtained, that is in fact a slice of the tansfe function along the imaginay axis. The same as with the tansfe function, FRF also could be teated as a sum of components, each of which coesponds to the esponse of an SDOF system. Global popeties δ k a Ω k could be basically obtained fom any of the measued functions H ij, wheeas esidues R ijk define the eigenvecto Φ k and ae specific fo each of the H ij functions. eal pat Re (H(s)) δ iω magnitude (H(s)) δ iω R s p * R s p ( ) + * H s imaginay pat Im (H(s)) iω phase iω p -δ + iω p* -δ - iω δ δ R imω Fig..7 Tansfe Function FRF fo a single degee of feedom system fom Fig..6 is shown in Fig..8. Vaious foms of displaying FRF ae discussed in detail in chaptes 3... and An SDOF system (o one mode of MDOF system) is descibed by means of 3 paametes: - undamped natual fequency - damping atio - esiduum k Ω (.7) m b δ ζ (.8) km Ω R (.9) imω

25 R iω p * R iω p ( ω ) + * H i Η(iω) k m b p -δ + iω Η(iω) -9-8 Ω k / m ω ω Fig..8 FRF of Single Degee-of-Feedom System.4. Impulse Response Function - IRF An impulse esponse of a system is an output signal when the Diac impulse (unit impulse, delta function) is applied at the input. It is an invese Fouie tansfom of the fequency esponse function, and this is the pocedue used to calculated it in an FFT analyze: h(t) - {H(f)} (.) An impulse esponse of an SDOF system is one-sided damped sinusoid (see Fig..9) given by the fomula: h(t) R e δt sin ( Ωt) (.) Just like the FRF, also IRF of an MDOF system is a sum of n IRFs of n SDOF systems. Summing all the n modes, moe geneal fomula is obtained: n δk t h ij (t) R ijk e sin( Ω k t) (.) k The aveage decay constant of this summaized impulse esponse can be used to estimate aveage damping popeties of the system. h(t) R R e -δt t T π Ω Fig..9 IRF of Single Degee-of-Feedom System - 5 -

26 If the magnitude of IRF is displayed in logaithmic scale, the envelope of the function is a line, and its slope indicates damping of the system. Come out of the definition of logaithmic decement υ: x( t) ( t + T) υ ln δt (.3) x Magnitude deceases e-times in the time τ (τ is so-called systems's time constant): ln e δ τ δ τ δ /τ Witten in db: log e 8.7 db If thee is logaithmic scale fo the magnitude of IRF on the vetical axis, time τ, duing which magnitude dops by 8.7 db can be ead out and decay constant δ can be calculated (see Fig..). This pocedue is the same fo both estimating the decay constant of the SDOF system and estimating the aveage decay constant of the MDOF system. h(t) [db] R 8,7 db τ t Fig.. Estimating Damping fom Impulse Response Function.5 Effect of Noise on the FRF A fequency esponse function could be also defined as a slope of the line which, fo linea systems, defines output as a function of input. If the system is not linea, its linea appoximation is obtained using Fouie tansfom. The influence of andom noise is eliminated by linea appoximation as well (see Fig..). A fequency esponse function is defined as the atio of output to input. Thee altenative estimates ae at disposal using a dual channel analyze. They ae defined using autospecta and coss spectum: ( f ) ( f ) G AB H ( f ) (.4) G AA ( f ) ( f ) G BB H ( f ) (.5) G BA - 6 -

27 ( f ) ( f ) ( f ) ( f ) G BB G AB H 3 ( f ) H( f ) H ( f ) (.6) G G AA Coheence function could be defined as: AB AB ( f ) ( f ) G ( f ) ( f ) ( f ) G H γ ( f ) (.7) G H AA BB common state - system with noise nonlinea system B(f) H(f) - the best linea appoximation B(f) H(f) - the best linea appoximation A(f) A(f) Fig.. Lineaization Which of these thee estimates is bette to use depends on whethe thee is noise on the input o output. When FRF is measued using impact excitation, the input signal is clean, without noise, whilst the output signal is modified by system esponse and deteioated by noise, paticulaly in antiesonances. On the contay, when FRF is measued using dynamic excite, the input signal is deteioated by noise in the vicinity of esonances, paticulaly fo slightly damped stuctues. The stuctue behaves as a shot cicuit in the vicinity of esonances and the input powe spectum has low values even if the signal enteing the excite is white noise. The output signal is elatively clean. If thee is a need fo having as accuate values of FRF magnitudes as possible, the best solution is to ead magnitudes of esonant peaks fom H function and the est fom H function. G AA G AA G AB G AV G AB G AV H H G G H G AA G AA + G H (f ) G H + ( ) ( ) V f A f G BB VV NN NN BA G VA G BA G VA G VV BB VV NN BB G VV G NN NN H 3 H G AA G AA G VV G + H a(t) h( τ ) H(f) n(t) G v(t) b(t) G + G γ po G NN G BB H + G NN / G VV G + γ γ po G NN Fig.. Influence of Noise at the Output - 7 -

28 The influence of noise at the output is shown in detail in Fig.., the influence of noise at the input in Fig..3 and the influence of noise at both input and output, which is a common case, is shown in Fig..4. m(t) u(t) a(t) h( τ ) H(f) b(t) H (f ) ( ) ( ) B f U f G G + G AA UU MM G AB G UV H G G AB AA G UU G UB + G MM H + G MM / G UU G BB G BB G BA G BU H H G G BA BU G BB G BB H 3 G G BB AA G UU G BB + G MM H + G MM / G UU H γ MM AA H G MM + G UU γ po G G γ po G MM Fig..3 Influence of Noise at the Input u(t) h( τ ) v(t) H(f) m(t) a(t) n(t) b(t) H (f ) ( ) ( ) V f U f ε IN G G MM UU ε OUT G G NN VV H G G AB AA H + ε IN γ H H ( + ε ) ( + ε ) IN OUT H G BB H H H H G BA + ε OUT H 3 G G BB AA H + ε + ε OUT IN Fig..4 Influence of Noise at Both Input and Output - 8 -

29 .6 Digital Signal Pocessing The main function of a spectal analyze is to pefom Fouie tansfom of signals that ae coming to the input. It is useful to ecall the elationship between two majo vesions of basic Fouie tansfom, between time and fequency domain. In its simplest fom it means that the function x(t), peiodic in time T, can be expessed as an infinite sequence: a πnt πnt x (t) + a n cos + bn sin (.8) n T T whee a n and b n can be computed fom x(t) using fomulas (.9) a (.3): T πnt a n x(t) cos dt T T T πnt bn x(t) sin dt T T (.9) (.3) When x(t) is discetized and takes finite time, so it is only defined on the set of N individual time moments t k (k, N), we can wite the finite Fouie seies: x N / a k n πnt T πnt T k k ( x( t )) + a cos + b sin ; k, N k n n (.3) Coefficients a n and b n ae Fouie o spectal coefficients of the function x(t) and they ae often denoted in the fom of amplitude c n and phase φ n : ( X ) a b c + a n n n n n n a n φ b actg (.3) This is the fom of Fouie tansfom concened in pactical applications of theoy used in modal test aea. Due to discetization of the input signals (fom foce tansduces and acceleometes) it is called discete Fouie tansfom (DFT). The input signal is then digitized by an A/D convete and ecoded as a set of N discete values with egula spacing in the time inteval T duing which the measuement was made. Then, assuming that the sample is peiodic in time T, finite Fouie seies (tansfom) is calculated accoding to the elation (.3), as an estimate of the desied Fouie tansfom. Thee is a basic elationship between the length of the sample T, numbe of discete values N, sample (o captue) fequency f s and ange and esolution of the fequency spectum. The ange of the spectum is -f max, whee f max is Nyquist fequency and the esolution between fequency lines is f, whee: f s N f max (.33) T f f S (.34) N T Since the tansfomation size (N) is usually fixed fo the given type of analyze, and it is usually (even though not always) the powe of, i.e. 5, 4 etc., the fequency ange and spectal esolution is detemined only by duation of each sample

30 The fundamental equation, that is solved fo detemination of spectal content is deived fom the equation (.35): x x x M x 3 N cos cos cos cos ( π / T) ( 4π / T) ( 6π / T) M K a K a K b K M ( Nπ / T) K M o { x } [ C] { } (.35) k a n To solve unknown spectal o Fouie coefficients contained in { a n } following equation is used: { a } [ C] { } (.36) n x k An optimized algoithm of solving the equation (.36) was deived, that is called Fast Fouie Tansfom (FFT). This algoithm equies N to be an integal powe of. Usually, values between 56 and 496 ae used. Digital Fouie analysis has many featues which, if not popely teated, can lead to eoneous esults. Geneally speaking, they esult fom discetization and fom the need to educe the length of time signal. In the following sections, specific featues of aliasing, leakage, weighting windows, fequency zoom and aveaging will be discussed..6. Aliasing Thee is a poblem called "aliasing", which is associated with a digital spectal analysis and esults fom discetisation of the oiginally continuous time signal. If the sampling fequency in elation to the fequency content of the signal is too small, the pesence of high fequencies in the oiginal signal could be misintepeted in the discetisation pocess. In fact, such high fequencies will appea as low fequencies, o, they will be athe indistinguishable fom genuine low fequency components. Fig..5 shows that digitising a low fequency signal (above) poduces exactly the same set of discete values as a esult fom the same pocess applied to a highe fequency signal (below). low-fequency signal high-fequency signal Fig..5 Aliasing - High Fequency Manifested as Low Fequency If the sampling fequency is f s, then the signal of fequency f and signal of fequency (f s -f) ae indistinguishable afte discetization, and this causes distotion of the measued specta using DFT, although the calculation is pefomed accuately. In the desciption of the DFT, it - 3 -

31 was stated that the highest fequency that can be included in the spectum (tansfom) is f s /, and the spectum should stop at this fequency, egadless of the numbe of discete values. The signal, which has the actual fequency content displayed in Fig..6, appeas in DFT as a distoted fom. Distotion towads the uppe end of the applicable fequency ange can be explained by the fact that the potion of the signal which has fequency components above f s / will be eflected in the ange -f s /. These high fequency components then put on the appeaance of being low fequency ones and ceate an indistinguishable mixtue with the eal low fequency components. tue spectum of signal f s indicated spectum fom DFT eflecting of high-fequency components f s / Fig..6 f s Alias Distotion of Spectum by DFT oiginal spectum anti-aliasing filte filteed spectum f s / f s Fig..7 Anti-aliasing Filte Pocess The solution to the poblem is to use an anti-aliasing filte which subjects the oiginal time signal to a low-pass, shap cut-off filte with a chaacteistic of the fom shown in Fig..7. This esults in submitting a modified time histoy to the analyze. Because the filtes used ae inevitably less than pefect and have a finite cut-off ate, it emains necessay to eject the spectal measuements in a fequency ange appoaching the Nyquist fequency, f s /. Typically, the ange fom.8 f s / to f s / is ejected. It is fo this eason that a 48-point - 3 -

32 tansfom does not esult in a complete 4-line spectum being given on the analyze display. Typically, only the fist 8 lines will be shown because the highe ones ae liable to be contaminated by impefect anti-aliasing. It can be concluded that time signal should be submitted to an anti-aliasing filte pio it entes an A/D convete, and theefoe these filtes ae an integal pat of each analyze..6. Leakage Leakage is a poblem which is a diect consequence of the need to take only a finite length of time histoy coupled with the assumption of peiodicity. The poblem is best illustated by the two examples shown in Fig..8, whee two sinusoidal signals of slightly diffeent fequencies ae subjected to the same analysis pocess. On the left side, the signal is pefectly peiodic in the time window T, and the esulting spectum is simply a single line at the fequency of the sine wave. On the ight side, the peiodicity assumption is not satisfied and thee is a discontinuity at the end of the sample. As a esult, the spectum does not indicate the single fequency which the oiginal time signal possessed, and this fequency is not even pevailing in the spectal lines. The enegy has "leaked" into a numbe of spectal lines close to the tue fequency and the spectum is spead ove seveal lines. The two examples epesent the best case and the wost case. The poblem is moe seious fo low fequency signals. Leakage is a seious poblem in many application of digital signal pocessing, including FRF measuements. Thee ae seveal ways of avoiding o at least minimizing its effects: - Changing the duation of the measuement sample length to match any undelying peiodicity in the signal, e.g. by changing measuement time T so that to captue an exact numbe of cycles of the signal. Although such a solution can emove the leakage effect altogethe, it can only do so if the signal being analyzed is peiodic - which is not always the case - and if the peiod of that signal can be detemined - which is often difficult and it could be the fist objective of the analysis. Moeove, measuement time T can not be changed fully abitay in FFT analyzes, but only by some steps accoding to fequency ange of measuement (see fomula.33). - Inceasing the duation of measuement time T, so that the sepaation between the spectal lines - the fequency esolution - is fine (see fomula.34). This does not emove but does educe the seveity of the leakage effect. - Modifying the signal sample obtained in such a way as to educe the seveity of the leakage effect. This pocess is efeed to as windowing o window tansfomation and is widely employed in signal pocessing and modal testing. Windowing involves the imposition of a pescibed pofile w(t) on the time signal pio to pefoming Fouie tansfom. The analyzed signal is then poduct of oiginal signal and window pofile (see Fig..9). The influence of often used Hanning window to Fouie tansfom of a signal is shown in Fig..8 (below). Othe types of windows often used in modal testing (tansient and exponential) ae discussed in the chapte 4... Othe well known type is flat-top window, which is used fo tansduces' calibation

33 a(t) peiodic signal b(t) non-peiodic signal T time T time ectangula window (no window) A(f) B(f) fequency fequency Hanning window πt cos T A(f) B(f) fequency fequency Fig..8 Influence of Weighting Windows and Peiodicity of the Signal on Leakage Eo Fig..9 shows what influence has tuncation of time signal on fequency spectum. Fo digital signal pocessing, spectal fequency esolution f is equal to the invese time length of the sample T (accoding to fomula.34). It means that the bette fequency esolution is equied (with the same fequency ange), the longe the measuement time should be. And vice vesa: the longe measuement time is, the moe time the tansient signal has to decay to zeo and, consequently, the less is the leakage eo (and bette fequency esolution at the same time). Thus, the leakage eo can be eliminated by extending the measuement time. Similaly to spectum, a leakage eo also occus in fequency esponse function as esolution bias eo. Due to this eo, magnitude of the measued FRF could be, compaed with the tue value, lowe in esonances and highe in antiesonances (see Fig..). This eo occus if the fequency esolution of measuement f is much coase than the fequency esolution of the system f s, i.e. than the fequency esolution than would captue the function accuately. It coesponds to time tuncation of the signal, i.e. to the fact that the measuement time T is much shote than the tue esponse of the system T s : T << T s f >> f s

34 a(t) continuous signal A(f) w(t) a(t) w(t) time * W(f) time A(f)*W(f) fequency fequency a(t) time digital signal - measued data time time limitation DFT D A(f) leakage f T fequency fequency Fig..9 Relation Between Time Limitation of the Signal and Leakage Eo in Spectum Measuement time can be extended by inceasing the numbe of fequency lines of Fouie tansfom when using FFT analyze (see chapte.6). This way, the fequency esolution inceases without changing the fequency ange of the measuement. When using an analyze with a fixed numbe of fequency lines, thee is anothe way how to impove fequency esolution - to educe the fequency ange of the measuement. When measuements ae done in baseband (fom to f max Hz), it means to limit the fequency ange fom above. If esonances of inteest ae of highe fequencies, anothe appoach - fequency zoom - should be applied. The fequency ange is then fom f min to f max (see chapte.6.3). esolution bias measued values f f f tue FRF Fig.. Influence of Leakage in FRF - Resolution Bias Eo The elationship between the tue FRF ant its altenative estimates H and H when leakage occus is shown in Fig... The magnitude of the tue FRF is always highe in esonances

35 and always lowe in antiesonances than both estimates. Nevetheless, close to the tue values is H in esonances and H in antiesonances. H tue FRF esonance: H > H > H H antiesonance: H < H < H Fig.. Influence of Leakage in Altenative FRF Estimates.6.3 Fequency Zoom The common solution to the need fo a fine fequency esolution is to "zoom in" on the fequency ange of inteest and to concentate all the spectal lines to the naow band between f min and f max (instead of between and f max ). Thee ae vaious ways of achieving this esult but pehaps the one which is easiest to undestood physically is that which uses a fequency shifting pocess coupled with a contolled aliasing device. Suppose the signal to be analysed, x(t), has a spectum, X(ω), of the type shown in Fig.. and that a detailed (zoom) analysis aound the second peak, between f and f, is of inteest. If a band-pass filte to the signal is applied (see Fig.. below), and DFT is pefomed between and f -f, then because of the aliasing phenomenon descibed in chapte.7, the fequency components between f and f will appea aliased in the analysis ange to f -f with the advantage of a fine esolution. X(ω) f f Fig.. f f Fequency Zoom Realized by Band-Pass Filte This is not the only way of achieving a zoom measuement, but it seves to illustate the concept. Othe methods ae based on effectively shifting the fequency oigin of the spectum

36 by multiplying the oiginal time histoy by a cos(f t) function and then filteing out the highe of the two components thus poduced. Fo example, suppose the signal to be analysed is: x ( t) A sin ( πft) Multiplying this by cos(πf t) yields: A ( t) A sin ( πf ) cos( πf t) ( sin π( f f) t + sin π( f + f ) t) (.37) x and if the second component is then filteed out, the oiginal signal tanslated down the fequency ange by f is left. The modified signal is then analysed in the ange to f -f, yielding a zoom measuement of the oiginal signal between f and f. In this method, sample times ae multiplied by the zoom magnification facto (, 4 etc.) but the sampling is caied out at a slowe ate (also, 4, etc.) dictated by the new effective fequency ange. When using a fequency zoom fo measuing FRF in a naow fequency band, it is impotant to ensue that as low vibational enegy as possible is out of the fequency band of inteest. It means that wheneve possible, excitation of the stuctue should be esticted to the fequency band of analysis. This poblem is discussed in moe detail in chapte Aveaging This chapte deals with anothe featue of a digital spectal analysis that concens paticula equiements fo pocessing andom signals. When analyzing andom vibation signals, it is not sufficient to compute the Fouie tansfom (stictly, it does not exist fo a andom pocess), and instead estimates fo spectal densities and coelation functions which ae used to chaacteize this type of signal must be obtained. Although these popeties ae computed fom the Fouie tansfom, thee ae additional consideations concening thei accuacy and statistical eliability which must be given due attention. Geneally, it is necessay to pefom an aveaging pocess, involving seveal individual time ecods (samples) befoe a esult which can be used with confidence is obtained. The two majo consideations which detemine the numbe of aveages equied ae the statistical eliability and the emoval of spuious andom noise fom the signals. Thee ae seveal possibilities o aveaging modes povided by analyzes - common ae: - peak hold - it is used mostly in vibation diagnostics with displacement tansduces - exponential - latest samples ae weighting moe than olde signals - linea - all samples ae weighting equally In modal testing, linea aveaging is used, eithe with o without ovelap. When aveaging without ovelap is used, it means fo m samples each of duation T that the oveall measuement time would be m T (see Fig..3 below). Nowadays, analyzes compute DFT in extemely shot times, which enables to compute a new tansfomation pio to captuing a complete new data sample. In this case it is often bette to pefom a new tansfomation as soon as possible and use the last N data points, even if some of them could have aleady been used in the pevious tansfom. This pocess is called ovelapping (see Fig..3 below). It is clea that aveages pefomed with ovelapping do not have the same statistical paametes as if completely independent samples ae aveaged. Nevetheless, the pocess

37 with ovelapping is moe efficient than without ovelapping and povides smoothe specta. This is pehaps because of windowing - when using Hanning window, samples ae suppessed to zeo in thei edges and consequently, when using aveaging without ovelap, pats of the signal ae not utilized. aveaging without ovelap aveaging with ovelap pocessing time Fig..3 Types of Aveaging

38 3. Theoetical Basis of Modal Analysis A complex stuctue can be consideed as a numbe of masses inteconnected by spings and damping elements. Since the damping foces in a eal stuctue cannot be estimated with anything like the same accuacy as the elastic and inetia foces, a igoous mathematic simulation of the damping effects is futile. Nevetheless, to account the dissipative foces in the stuctue, assumptions of the fom of damping have to be made, to get as good estimate of the damping foces in pactice as possible. The fom has to be conductive to easy mathematical manipulation, specifically adaptable to linea equations of motion - implying that the damping foces ae hamonic when excitation is hamonic. Two such suitable foms of damping ae: - viscous damping - damping effect is popotional to velocity b v - hysteetic (stuctual) damping - damping coefficient is invesely popotional to k γ angula velocity F b v ω 3. Single Degee-of-feedom System (SDOF) Although vey few pactical stuctues could ealistically be modelled by a single degeeof-feedom system, popeties of such a system ae vey impotant because those fo a moe complex multi degee-of-feedom system can always be epesented as a linea supeposition of a numbe of SDOF chaacteistics. Basic spatial model of SDOF system (see Fig. 3.) consists of mass m and a sping k and, in the case of damped system, of eithe viscous dashpot b o hysteetic dampe h. In this model, f(t) is geneal time vaying foce and x(t) is esponse quantity. F b f(t) In this chapte, thee types of SDOF model will be descibed: - undamped - viscously damped - hysteetically (o stuctually) damped k k m b x(t), v(t), a(t) Fig. 3. Spatial model of SDOF system 3.. Undamped Single Degee-of-feedom System Spatial model of this system consists of mass m and sping k. Fo modal model, popeties of the system without extenal foce, i.e. f(t) will be consideed. In this case, equation of motion is: ma + kx (3.) when substitute fo a & x m & x + kx (3.)

39 Expected solution of this equation is: x(t) iωt Xe (3.3) Putting into equation of motion leads to the equiement that k ω m (3.4) Modal model consists of a single solution (mode of vibation) with a natual fequency given by k Ω (3.5) m Fo fequency esponse analysis, excitation is consideed of the fom f (t) iωt Fe (3.6) and solution is assumed of the fom x(t) iωt Xe (3.7) whee X and F ae complex to accommodate both the amplitude and phase infomation. Now the equation of motion is (k i t iωt ω m)xe ω Fe (3.8) fom which the equied esponse model in the fom of a fequency esponse function is extacted: X F α( ω) (3.9) k ω m This paticula fom of fequency esponse function, α(ω), with esponse paamete displacement, is called eceptance. This function, along with othe vesions of FRF, is independent of the excitation. 3.. Viscously damped SDOF system 3... Fee vibation Adding a viscous dashpot b, the equation of motion fo fee vibation becomes m a + b v + k x (3.) m & x + b x& + k x (3.) Expected solution is of a moe geneal fom (s is complex, athe than imaginay, as with undamped system) st x ( t) Xe Deivatives: st x &( t) Xse st & x ( t) Xs e Having put these into equation of motion, chaacteistic equation is obtained: ms + bs + k (3.)

40 Solution of chaacteistic equation: s s s, b ± b 4km (3.3) m, b b k ± (3.4) m m m, δ ± i Ω δ δ ± iω ζ (3.5) s, δ ± iω (3.6) Whee: k Ω undamped natual fequency m b δ decay constant (3.7) m δ b b ζ... damping atio (3.8) Ω km b k Ω Ω δ Ω ζ... damped natual fequency (3.9) Roots of chaacteistic equation (poles) depend on the value of damping atio ζ. Fo so called positive damping (ζ ), thee may be 3 cases, and theefoe 3 diffeent types of motion (see Fig.3.): - ζ undamped vibation s and s ae imaginay - ζ < damped vibation s and s ae complex conjugates (see Fig. 3.4) - ζ apeiodic movement s and s ae eal (fo ζ: s s -δ) If the eal pat of the pole is positive, which means that ζ < (negative damping), selfexcited vibations occu (see Fig. 3.3., ight side). Fig. 3. Position of Poles Accoding to Damping Ratio Values - 4 -

41 e ( δ + iω )t vaious values of fequency e ( +δ + iω )t +iω i e + Ω t e ( +δ + iω )t e ( +δ + iω )t e ( δ + iω )t e ( δ + iω )t i e + Ω t e ( +δ + iω )t e ( +δ + iω )t e δ t t e δ t e e + δ t e + δ t vaious values of damping δ Fig. 3.3 Fequency Response as a Function of Natual Fequency and Damping Values iω p -δ + iω Ω... undamped natual fequency Ω... damped natual fequency ϑ δ... decay constant δ p* -δ - iω Fig. 3.4 δ ζ cos ϑ... damping atio Ω Complex Conjugates - Poles - in Laplace Plane 3... Foced vibation If movement is caused by acting of hamonic foce, the equation of motion of viscously damped system has the fom: ( t) + bx& ( t) + kx( t) f (t) m & x (3.) iωt whee f ( t) Fe x x& & x hamonic excitation foce iωt ( t) Xe expected solution and its deivatives: iωt ( t) iωxe t ( t) Xe i ω ω - 4 -

42 Dividing the equation (3.) by mass m and putting the expected solution togethe with equations (3.5) a (3.7) in it gives: F ω X + iω ζω X + Ω X Ω (3.) k Then, complex displacement amplitude is: X Ω F k X st Ω ω + iζωω k F static displacement F k ω X η tuning facto ω ω Ω + iζ Ω Ω X X st (3.) η + iζη X X... amplitude of displacement (3.3) st ( η ) + ( ζη) Now, steady state solution of equation of motion will be deived: F X amplitude of displacement (complex) η + iζη k x(t) + iζη F k iωt iωt Xe e displacement time histoy η Displacement is popotional to the acting foce, popotionality constant is: H( η) (3.4) η + iζη what is so-called fequency esponse function eceptance (dimensionless). As the displacement is a complex numbe, it can be divided to the eal pat and imaginay pat (by multiplying both numeato and denominato by denominato's complex conjugate): x ( t) η ζη iωt i e (3.5) ( η ) + ( ζη) ( η ) + ( ζη) It can be seen, that the displacement has one pat Re ( x) η iωt e (3.6) ( η ) + ( ζη) F k F k that is in-phase with the excitation foce and the second pat ( ) Im x ζη iωt e (3.7) ( η ) + ( ζη) F k - 4 -

43 that lags the excitation foce at 9. In Fig. 3.5, vectos OA and OB epesent eal and imaginay pat of displacement. Vecto OC epesents the amplitude of displacement given by Re ( x) Im ( x) x +, thus: ( t) iωt e (3.8) ( η ) + ( ζη) Displacement lags the excitation foce at an angle θ, defined as: F k ζη θ actg (3.9) η Steady-state solution of the equation of motion can thus be expessed in the fom: ( ) F i( ωt θ) x t e (3.3) ( ) ( ) η + ζη k The expession in squae backets is the absolute value of complex fequency esponse. It is often called as an amplification facto and it expesses a dimensionless atio between the amplitude of displacement X and static displacement F/k. H η (3.3) ( ) ( η ) + ( ζη) Im(x) O θ Re ( x) Fe iωt A Re(x) η ( η ) + ( ζη) F e k iωt ( ) Im x ζη ( η ) + ( ξη) F e k iωt x ( t) ( η ) + ( ξη) F e k iωt Fig. 3.5 B Relationship Between the Complex Displacement and the Excitation Foce C Fequency esponse function (3.4) is complex and it is a function of fequency (o of tuning facto, espectively) at the same time. It means that it can not be displayed in a single two-dimensional plot. Its 3D plot is in Fig The ed cuve epesents a damped system, the geen cuve epesents an undamped system (in that case, the plot lies in the plane η-re(h(η)) and is two-dimensional)

44 Im (H(η)) ζ Re (H(η)) η η ( η) η + iζη H Fig D plot of the Fequency Response Function - Undamped and Damped Pojection of this 3D plot to the individual planes is in Fig and in Fig. 3.8 (left). In Fig. 3.8, two of the possible ways of displaying FRF ae shown - simultaneous displaying of eal and imaginay pat of FRF as functions of fequency o of tuning facto (top left), o so called Nyquist diagam which is plot in the plane [Re(H(η)); Im(H(η))] - bottom left. In Nyquist plot, infomation about the fequency is hidden - the plot is dawn fom the initial fequency to the final fequency clockwise; the majo pat of the cicle epesents esonance and its vicinity (moe detailed in chapte 4.3..). Diffeent colous of plots ae fo vaious levels of damping - fom geen fo an undamped system to magenta fo a citically damped system (ζ). Fig D Plot of the Fequency Response Function (Receptance)

45 3 3 Re (H(η)) 6 6 (H(η)) ReH( η, ξ ) ReH( η, ξ ) ReH( η, ξ3 ) eálná eal pat složka magh( η, ξ ) η magh( η, ξ ) magh( η, ξ3 ) amplituda magnitude ReH( η, ξ4 ) ReH( η, ξ5 ) magh( η, ξ4 ) magh( η, ξ5 ) 3 ImH( η, ξ ) ImH( η, ξ ) ImH( η, ξ3 ) ImH( η, ξ4 ) ImH( η, ξ5 ) η Im (H(η)) imaginání imaginay pat složka η (H(η)) η Bodeho plot gaf η η phase fáze η Re (H(η)) ImH( η, ξ ) ImH( η, ξ3 ) ImH( η, ξ4 ) ImH( η, ξ5 ) 3 4 Nyquistův plot gaf Im (H(η)) ReH( η, ξ ), ReH( η, ξ3 ), ReH( η, ξ4 ), ReH( η, ξ5 ) Fig. 3.8 Vaious Foms of Displaying FRF - Viscously Damped System FRF is fequently displayed as Bode plot, which is simultaneously display of amplitude and phase of FRF, both as a function of fequency (o tuning facto) - see Fig. 3.8 (ight)

46 3...3 Detemination of Resonance Tuning Resonance can be defined as a state when magnitude of FRF is maximal. The plot of FRF magnitude as a function of tuning facto shows, that esonant peak fo undamped system occus when η and is shifted to the left when damping inceases. To detemine the esonant tuning facto, equation (3.3) should be deived with espect to the tuning facto and then put the deivative equal to zeo. H ( η ) ( η) dh dη ( η ) + ( ζη) η es ζ (3.3) Then, esonant excitation fequency is: ω es Ω ζ (3.33) Magnitude of FRF and displacement in esonance ae: H( ω es ) (3.34) ζ ζ X es X st (3.35) ζ ζ Fo light damping (ζ <.5) the cuves ae nealy symmetical along the vetical axis passing though η. Peak value of H(ω) at the immediate vicinity of η is given by H( ω es ) & Q Q quality facto (3.36) ζ Detemination of Damping fom FRF Plots a) Detemination of damping fom plot of eal pat H(η) as a function of η The following pocedue will deive fo which tuning factos η and η (and coesponding excitation fequencies ω and ω ) occu local extemes in the gaph of eal pat of FRF as a function of fequency. These fequency values can be easily ead fom the gaph and using them, damping atio ζ can be expessed (see Fig. 3.9). ( η) η + iζη H H ( η ) η ζη + ( ) ( ) ( ) ( ) i η + ζη η + ζη ReH ( η ) η ( η ) + ( ζη)

47 d ReH dη ( η) Re (H(ω))... η ζ η + ζ ω Ω ζ (3.37) ω Ω + ζ (3.38) ω ω Ω ω ω ω ω + ζ ζ + ζ ζ Fig. 3.9 Detemination of Damping fom Re (H(ω)) ( ζ) + ζ ω ω ω ω ω ω ω ζ ζ ω ω ζ (3.39) ω + ω b) Detemination of damping fom half-powe points Half-powe points ae points on the plot of magnitude of H(ω), in which the magnitude deceases to the value H es, which means to the of the peak value. In the powe spectum, it would be one half of the peak value - hence the name half-powe points. If H(ω) is plotted in logaithmic scale, in these points the peak magnitude deceases by 3 db (see Fig. 3.). H H es halfpowe halfpowe log H es log H log 3 H es H halfpowe 3dB If these points ae denoted P and P and the coesponding fequencies ω and ω, then the diffeence between those fequencies ω -ω is called 3dB band of the system. Fo light damping: ω 3dB ω ω ζ Ω

48 whee ω 3dB is 3dB band. Thus: ω ω Ω ζ (3.4) log (H(ω)) P P 3 db ω δ Ω ω ω Fig. 3. Detemination of damping fom 3dB band Now, it will be poved that fequencies ω and ω ae the same fequencies (fo slight damping) as those that wee obtained in the pevious paagaph fom the extemes of eal pat FRF. Supposing it is tue, put the equations (3.37) and (3.38) into equation (3.4): ω ω Ω ζ Ω + ζ Ω ζ Ω ζ ( + ζ) ( ζ) + ζ 4 + ζ ζ 4ζ 4ζ 4ζ ζ Fo light damping (ζ <,5) applies : & To detemine damping fo lightly damped systems, the simplified equation (3.4) can be used even when eading fom eal pat FRF plot. If this equation is to be valid, the following should apply: ω ω δ (3.4) As the plot of FRF magnitude is symmetical in the vicinity of esonance, also applies: ω Ω δ (3.4) ω Ω + δ (3.43)

49 3..3 Single Degee-of-feedom System with Hysteetic (Stuctual) Damping Foced Vibation Close inspection of the behaviou of eal stuctues suggests that the viscous damping model used above is not vey epesentative when applied to MDOF systems. Real stuctues exhibit a fequency dependence that is not descibed by the standad viscous dashpot. A lage vaiety of mateials, when subjected to cyclic stess (fo stains below the elastic limit), exhibits a stess-stain elationship which is chaacteised by a hysteesis loop. The enegy dissipated pe cycle due to intenal fiction in the mateial is popotional to the aea within the hysteesis loop, and hence the name hysteetic damping. Intenal fiction is independent of the ate of stain (independent of fequency) and ove a significant fequency ange is popotional to the displacement. Thus the damping foce is popotional to the elastic foce, since enegy is dissipated, it must be in phase with the velocity. Thus fo simple hamonic motion, the damping foce is given by kγ i γ kx x& (3.44) ω whee γ is called stuctual damping loss facto. Note: In liteatue, stuctual damping loss facto is often denoted as η, but hee γ will be used in ode not to confuse it with tuning coefficient. Hysteetic model povides a much simple analysis fo MDOF systems but it pesents difficulties to a igoous fee vibation analysis. Theefoe only foced vibation analysis will be pefomed. Equation of motion fo a SDOF system with stuctual damping can be witten: kγ m & x ( t) + x& ( t) + kx( t) f ( t) (3.45) ω iωt whee ( ) f x x& & x t Fe hamonic excitation foce iωt ( t) Xe expected solution and its deivatives: iωt ( t) iωxe t ( t) Xe i ω ω Equation of motion can be also ewite as: ( t) + k( + iγ) x( t) f ( t) m & x whee k ( + iγ) is called the complex stiffness. (3.46) By setting the expected solution into the equation of motion is obtained: ( mω + k( + iγ) ) X F Afte dividing by stiffness k and using equaiton (3.5):

50 ω Ω + iγ X F k Then, amplitude of complex displacement is: F X η + iγ k (3.47) X η + iγ X st X X amplitude of displacement (3.48) st ( η ) + γ Following pocedue is analogous to the pocedue fom chapte 3... fo a system with viscous damping. x x x F time histoy of displacement (3.49) η + iγ k iωt iωt ( t) Xe e ( t) ( t) η ( η ) + γ ( η ) ( η ) F k γ + γ F i e k iωt iωt e (3.5) + γ γ θ actg phase lag between displacement and excitation foce (3.5) η ( ω θ) ( ) F i t x t e (3.5) ( ) η + γ k The expession in the squae backets is again the absolute value of fequency esponse, o magnification facto, and again it has the meaning of a dimensionless atio between displacement amplitude X and static displacement F/k. H η (3.53) ( ) ( η ) + γ In Fig. 3., vaious types of plots of FRF with the same amount of damping as it was in Fig When a system is stuctually damped, esonant peak in the plot of FRF amplitude does not shift to the left with inceasing amount of damping, but it still emains on tuning value η. But, as damping inceases, the plot does not stat with amplitude equal to coesponding to static displacement, but with a value less then. Simila diffeences ae evident in all types of plots

51 3 3 Re (H(η)) 6 6 (H(η)) ReH( η, γ) ReH( η, γ) ReH( η, γ3) ReH( η, γ4) ReH( η, γ5). eal pat magh( η, γ) magh( η, γ) η magh( η, γ3) magh( η, γ4) magh( η, γ5) amplitude Im (H(η)) η η (H(η)) H(η) η η 3 η ImH( η, γ) ImH( η, γ) ImH( η, γ3) ImH( η, γ4) ImH( η, γ5) η imaginay pat Re (H(η)) Bodeho plot gaf phase ImH( η, γ3) ImH( η, γ4) ImH( η, γ5) 3 ImH( η, γ) 4 Nyquist plot Im (H(η)) 3 ReH( η, γ3), ReH( η, γ4), ReH( η, γ5), ReH( η, γ) 3 Fig. 3. Vaious Foms of Displaying FRF - Stuctually (Hysteetically) Damped System - 5 -

52 3..3. Detemination of Damping fom FRF Plots Using the same pocedue as in chapte 3...4, damping fom the plot of eal pat of H(η) as a function of tuning coefficient η will be detemined. ( η) η + iγ H Re (H(ω)) H ( η) η ( η ) + γ ( η ) γ + γ i ω ω ω ReH ( η ) η ( η ) + γ Ω Fig. 3. Detemination of Damping fom Re (H(ω)) Detemining ω and ω : d Re H dη ( η) η γ η + γ ω Ω γ (3.54) ω Ω + γ (3.55) Using the same pocedue as fo viscously damped system leads to: ω ω γ (3.56) ω ω + Fo light damping appoximately applies: ζ γ (3.57) 3..4 Vaious Foms of FRF fo Single Degee-of-feedom System Fo all thee types of systems - undamped, viscously and hysteetically damped - fequency esponse function in the fom of eceptance, i.e. with displacement as esponse paamete, has been deived. Duing modal tests, the esponse is mostly measued using acceleometes, so FRF in the fom of inetance, with acceleation as esponse paamete, is moe common. In chapte.3, elations between the individual foms of FRF wee discussed togethe with the fact that in fequency domain, foms of FRF deived fom eceptance ae obtained by simple multiplication by iω (see eq..6 and.7). Fom plots of FRF amplitude as a function of - 5 -

53 fequency, fom of FRF can be detected only in logaithmic scale (see Fig. 3.3). In othe plots, shifting by 9 is evident fo each deivative. Thus, e.g. Nyquist plot is in the ight halfplane fo mobility FRF and in the uppe half-plane fo inetance FRF etc eceptance 4 α(f) [kg - s - ] magα( f).. α(f) [db]. log kg. s. magα( f) f f [Hz].. f f [Hz] mobility Y(f) [kg - s - ] magy( f).5 Y(f) [db]. log kg. s. magy( f) f f [Hz] 8 inetance f f [Hz] A(f) [kg - ] maga( f) 75 5 A(f) [db]. log( kg. maga( f) ) f Fig. 3.3 f [Hz] f Vaious Foms of FRF Accoding to Response Paamete f [Hz] 3. Multi Degee-of-feedom System (MDOF) Real stuctues possess a lot of degees of feedom and a lot of equations ae necessay fo thei desciption. Theefoe, matix fom is pefeably used fo MDOF systems as it enables to wite a single matix equation instead a numbe of equations

54 3.. Undamped MDOF System Fo undamped MDOF system of N degees of feedom, the equation of motion in matix fom is: [ M ]{ x(t) & } + [ K]{ x(t) } { f (t)} & (3.58) whee [ M ] and [ K ] ae the mass and stiffness matices of ode NxN and { x (t)} and { f (t)} ae vectos of time-vaying displacements and foces of ode N Fee vibation In ode to detemine modal popeties of the system, fee vibation solution will be consideed by putting { f (t)} { } In that case, solution can be expected in the fom { } { } i ω x(t) X e t { } { } e i ω & x ω X t whee { X } is a vecto of N time-independent amplitudes. This assumes that the system is able to vibate on a single fequency ω. Substituting the homogeneous solution into equation of motion gives: ([ K] ω [ M] ){ X} { } (3.59) The only non-tivial solution is: det [ K] ω [ M] When substituing (3.6) ω λ (3.6) then: [ K] λ[ M] det chaacteistic equation of the system The chaacteistic equation can be tanscibed to the fom: N d N λ N + d N λ d (3.6) By solving this chaacteistic equation, N values of λ i, which ae called eigenvalues, can be detemined. Undamped natual fequencies can be obtained fom eigenvalues as: i λ i Ω (3.63) Substituting any of these back into the equation (3.59) yields a coesponding set of elative Ψ, the so-called mode shape (o eigenvecto) coesponding to that values fo { X }, i.e. { } natual fequency. The complete solution can be expessed in two NxN matices: [ ] Ω... spectal matix (eigenvalue matix) - diagonal

55 Ψ... modal matix (matix of modal shapes); its fom is [{ Ψ } { Ψ}... { Ψ} { Ψ} ]... N whee Ω is the th eigenvalue, o natual fequency squaed, and { Ψ } is the th eigenvecto that descibes the coesponding mode shape. Thee ae vaious numeical pocedues, that convet the spatial model epesented by matices [ M ] and [ ] K to the modal model epesented by matices [ ] Ω and [ ] A spectal matix is a unique one, but a modal matix is not. Wheeas the natual fequencies ae fixed quantities, the mode shapes ae subject to an indeteminate scaling facto which does not affect the shape of the vibation mode, only its amplitude. Thus, a mode shape vecto of descibes exactly the same vibation mode as What detemines how the eigenvectos ae scaled, o nomalized, is lagely govened by the numeical pocedues followed by the eigensolution. The actual amplitudes of vibation depend on the initial conditions and positions and magnitudes of exciting foces. The pocedue of obtaining eigenvalues and eigenvectos will be illustated on the example of two degee-of-feedom undamped system (see Fig.3.4) etc. Ψ.. k f (t) k f (t) k 3 m m x (t) x (t) Fig. 3.4 Two Degee-of-feedom System Equations of motion of this system ae: ( k + k ) x k x m & (3.64) x+ f ( k + k 3 ) x m & (3.65) x k x + f o in matix fom: m && x k + k + m && x k k k + k 3 x x f f (3.66) Conside numeical values: m 5 kg m kg k k N/m k 3 4 N/m

56 Substituting in eq. (3.66), fo fee vibation (i.e. f and f ) yields to: + x x 6 4 x x 5 && && λ X X X X λ λ (3.66a) λ λ ( )( ) ( )( ) λ λ 7 5 λ + λ λ,4 s - Ω 4, s - λ s - Ω s - Substituting back λ and λ into eq. (3.66a) will give the two natual mode shapes: Fo Ω : X X,4 6,4 5 4 X X X X +... One of these equations is enough. X X Thus the mode shape fo Ω is { } Ψ Fo Ω : X X X X X X Thus the mode shape fo Ω is { } Ψ / Thus the entie solution is given by matices: [ ] Ω,4 [ ] Ψ / Mode shapes ae shown in Fig It can be seen that the masses move eithe in phase o 8º out of phase. Since the masses each thei maximum displacements simultaneously, the nodal points (points, which do not move) ae clealy defined.

57 x Ω,4 s x x Ω s nodal point x Fig. 3.5 Mode Shapes fo the two Degee-of-feedom System 3... Othogonal Popeties of Eigenvectos Solution of the equation ([ K ] λ[ M] ){ X} { } eigenvectos. A paticula th mode will satisfy: [ K]{ } λ [ M]{ Ψ} yields N eigenvalues and N coesponding Ψ (3.67) Pemultiply eq. (3.67) by the tanspose of anothe, s th mode shape: T T { } s [ K]{ Ψ} λ { Ψ} s [ M]{ Ψ} Ψ (3.68) Similaly, s th mode will satisfy (afte pemultiplying by the tanspose of th eigenvecto) : T T { } [ K]{ Ψ} s λs{ Ψ} [ M]{ Ψ} s Ψ (3.69) As [M] and [K] ae symmetic matices, it applies: T T { } [ K]{ Ψ} s { Ψ} s [ K]{ Ψ} Ψ and T T { Ψ } [ M]{ Ψ} s { Ψ} s [ M]{ Ψ} Theefoe subtacting eq. (3.69) fom eq. (3.68) yields to: T ( λ λs ){ Ψ} s [ M]{ } Ψ (3.7) It is obvious that fo λ λ s (two diffeent natual fequencies) applies: T { } [ M]{ Ψ} Ψ and (3.7) s T { } [ K]{ Ψ} Ψ (3.7) s Equations (3.7) and (3.7) define the othogonal popeties of the mode shapes with espect to the system mass and stiffness matices espectively

58 In the case of λ λ s it applies: T T { Ψ } [ K]{ Ψ} s λ { Ψ} [ M]{ Ψ} s Theefoe: T { } [ K ]{ Ψ} s K Ψ... genealized (modal) stiffness of the th mode (3.73) T { } [ M ]{ Ψ} s M Ψ... genealized (modal) mass of the th mode (3.74) K λ Ω (3.75) M In matix notation: T { } [ K ]{ Ψ} [ K ] Ψ (3.76) T { } [ M ]{ Ψ} [ M ] Ψ (3.77) [ Ω ] [ M ] [ ] K (3.78) The numeical values of the mode shapes calculated above will be used to detemine the genealized mass and genealized stiffness of both modes: 5 / M / M M M 5kg 7,5kg Then K K Ω Ω M M,4 5 7,5 6 N / m 7,5 N / m Nomalization of Mode Shapes As the mode shapes ae abitay scaled, the values M and K ae not unique (in contay to the atio K / M ), it is not advisable to efe to a paticula genealized mass o stiffness. This poblem is eliminated by so-called nomalization of mode shapes. If one of the elements of the eigenvecto is assigned a cetain value, the est of elements ae also fixed because the ation between any two elements is constant. This pocess of adjusting the elements of the natual modes to make thei amplitude unique is called nomalization. Thee ae seveal ways how to do it, e.g.: - mass nomalization (to unity modal mass) - the lagest element of the mode shape is set to unity - the length of the mode vecto is set to unity Mass nomalization This type of nomalization is pobably the most common and has most elevance to modal testing. The mass-nomalized eigenvectos ae witten as Φ and have the paticula popety that

59 T [ ] [ M] [ Φ] [ I] Φ (3.79) T [ ] [ K] [ Φ] [ Ω ] Φ (3.8) The elation between the mass-nomalized mode shape fo mode, { Φ }, and its moe geneal fom, { Ψ }, is simply: o Φ (3.8) M { } { Ψ} [ ] [ Ψ] [ M ] Φ (3.8) Mass-nomalized shapes can be deived by denoting them u i and substituting to the equation (3.77): T { Ψ } [ m ] { Ψ} { Ψ} M X X T { Φ } [ ] { Φ} m 5 u u { u u } u u { 5u u } 5u + u u 5 u /5 { Φ} /5 /5 The same can be obtained by substituting the modal mass into equation (3.8): { Φ} { Ψ} M 5 /5 /5 { Φ} { Ψ} M 5 / /5 / Foced Response Analysis of Multi Degee-of-feedom System Suppose that the stuctue is excited hamonically by a set of sinusoidal foces all at the same fequency ω, but with individual amplitudes and phases. Then:

60 { } { } i ω f (t) F e t and solution is assumed to exist of the fom: { } { } i ω x(t) X e t whee {F} and {X} ae N vectos of time-independent complex amplitudes. The equation of motion then becomes: ([ ] [ ]) { } i t { } e i ω K ω M X e ω F t (3.83) o, eaanging to solve fo the unknown esponses ( ) { F} { X} [ K] ω [ M] which may be witten as { X} [ α( ω) ] { F} (3.84) (3.85) whee α( ω) is the N N eceptance FRF matix fo the system and constitutes its esponse model. The geneal element in the eceptance FRF matix, αjk(ω), is defined as follows: X j α jk ( ω) ; F m ; m... N ; m k F k and as such epesents an individual eceptance FRF expession vey simila to that defined ealie fo the SDOF system. It is possible to detemine values fo the elements of α( ω) at any fequency of inteest simply by substituting the appopiate values into: ( ) [ ( ω) ] [ K] ω [ M] α (3.86) Howeve, this involves invesion of the system matix at each fequency and this has seveal disadvantages, namely: - it becomes costly fo lage-ode systems (lot of DOFs) - it is inefficient if only a few of the individual FRF expessions ae equied - it povides no insight into the fom of the vaious FRF popeties Fo these easons, an altenative way of deiving the vaious FRF paametes is used which makes use of the modal popeties of the system. Come out of inveted eq. (3.86): ([ ] [ ]) [ ) K ω M α( ω ] Pemultiply both sides by Φ T and postmultiply both sides by Φ to obtain T ( ) [ Φ] [ Φ] [ α( ω) ] [ Φ] T [( Ω ω )] [ Φ] [ α( ω ] [ Φ] T [ Φ] [ K ] ω [ M] ) - 6 -

61 which leads to [ α( ω) ] [ Φ] [( Ω ω )] [ Φ] T (3.87) It is clea fom this equation that the eceptance matix α( ω) is symmetic and this will be ecognized as the pinciple of ecipocity which applies to many stuctual chaacteistics. Its implications ae that: X X j k α jk α kj (3.88) Fk Fj Equation (3.87) enables to compute any individual FRF paamete, α following fomula: o α jk N ( Φ ) ( Φ ) N j k ( Ψj ) ( Ψk ) Ω ω m ( Ω ω jk( ω), using the ( ω) (3.89) ) α jk ( ω) A N jk Ω ω A... modal constant, esiduum (3.9) jk In the following example it will be poved that the same function α could be obtained by both ways - by diect invesion (eq. 3.86) and fom patial faction fom (eq. 3.89). Equations of motion of foced undamped vibation of the system fom Fig. 3.4 ae: ( k + k ω m ) X + ( k ) X F ( k ) X + ( k + k 3 ω m ) X F which yields: X F F α ( ω) ω 4 m m ω ( m k + m k + m k + m k ) + ( k k + k k + k k ) k 3 + k 3 ω m 3 3 numeically (fo m 5 kg, m kg, k k N/m, k 3 4 N/m): X F F α 6 ω 7ω + 5ω ( ω) 4 Now, the modal summation fomula (3.89) will be used togethe with the esults obtained ealie: α ( ω) Ω ( Φ ) ( Φ ) ω + Ω numeically (fo Ω.4 s -, Ω s -, Φ / 5, Φ / 5 ): ω - 6 -

62 α /5.4 ω which is the same expession as above. /5 ω 6 ω 7ω + 5ω ( ω) Chaacteistics and Pesentation of Multi Degee-of-Feedom FRF Data As fo an SDOF system, fo MDOF system thee ae also thee altenatives of FRF with using eithe displacement o velocity o acceleation as a esponse paamete, thus obtaining eceptance, mobility o inetance espectively. These thee foms ae exactly in the same elation as descibed ealie, thus: [ ( ω) ] iω[ α( ω) ] Y (3.9) [ ( ω) ] iω[ Y( ω) ] ω [ α( ω) ] A (3.9) Accoding to the place and diection of excitation and esponse (place + diection DOF), fou types of FRF can be defined: - point - coodinates of excitation and esponse ae equal (e.g. point No ) diect - diections of excitation and esponse ae equal (e.g. DOF X) coss - diections of excitation and esponse ae diffeent (e.g. excitation at DOF Z, esponse at DOF X). - tansfe - coodinates of excitation and esponse ae diffeent diect (e,g, excitation at DOF X, esponse at DOF 4X) coss (e,g, excitation at DOF X, esponse at DOF 4Z) It is helpful to examine the fom which FRF data takes when pesented in vaious gaphical fomats. This knowledge is necessay in assessing the validity and intepeting measued data. Fo the simplest case of an undamped system, fo which the eceptance expession is given N ( Φ j) ( Φ k ) by equation (3.89): α jk ( ω) Ω ω Using the type of log-log plot, individual tems in the FRF seies can be plotted as sepaate cuves. The total FRF cuve is a summation of all the individual cuves. Howeve, the exact shape of the cuve is not so simple to deduce as fist appeas because a pat of the infomation (phase) is not shown. In fact, in some sections of each cuve, the eceptance is actually positive in sign and in othes is negative but thee is no indication of this on the logaithmic plot which only shows the modulus. Howeve, when addition of the individual components is made to detemine the complete eceptance expession, the signs of vaious tems ae of consideable impotance. Examine some of the impotant featues using a simple example with just two modes: [ Φ]. Two FRF plots, point eceptance α and tansfe eceptance α will be / ceated

63 Expessions fo the eceptances ae: α ( ω) + α ( ) ω Ω ω Ω ω ω. 5 Ω ω fom which it can be seen that the main diffeence between the point and tansfe eceptances is in the sign of the modal constant (the numeato) of the second mode. As the plots only show the modulus, they ae appaently insensitive to this diffeence. Howeve, when the two tems ae added to poduce the actual FRF fo the MDOF system, the following chaacteistics will aise, which is illustated in Fig In this figue, eceptance is plotted, but the following emaks apply fo all types of FRF (eceptance, mobility and inetance). When consideing a point eceptance, the numeato in the eq. (3.89) fo all modes will always be positive, as it is a squae of modal vecto element. In tansfe eceptance, the numeato can be eithe positive o negative accoding to the signs of elements of eigenvectos. Ω log α.4998 b( ω ).. b_( ω ) 3 b_( ω ) e b( ω ) log α b_( ω ) b_( ω ).. 3 ω log ω 4.38e-5 5 Fig.3.6 ω log ω Point (uppe) and Tansfe Receptance (below) FRF Plot fo Undamped DOF System Point Receptance At fequencies below the fist natual fequency, both tems in the sum have the same sign and thus ae additive, making the total FRF cuve highe than each component, but as the plot uses a logaithmic scale, the contibution of the second mode at these low fequencies is elatively insgnificant. Hence, the total FRF cuve is only slightly above that fo the fist tem. A simila agument and esult apply at the high fequency end, above the second natual fequency, whee the total plot is just above that fo the second tem alone. Howeve, in the egion between the two esonances, the two components have opposite signs to each othe so that they ae subtactive, and at the point whee they coss, thei sum is zeo since they ae of equal magnitude but of opposite sign thee. On a logaithmic plot of this type, this poduces the antiesonance chaacteistic which eflects that of esonance. In the immediate vicinity of

64 any esonance, the contibution of the tem whose natual fequency is neaby is so much geate than the othe one that the total is, in effect, the same as that one tem. Physically, the esponse of the MDOF system just at one of its natual fequencies is dominated by that mode and othe modes have vey little influence (it applies fo undamped o vey slightly damped systems). Tansfe Receptance Simila easoning as fo point eceptance can be applied when pogessing along the fequency ange with the only diffeence that the signs of the two tems in the sum ae opposite in this case. Thus, at vey low and vey high fequencies, the total FRF cuve lies just below that of the neaest individual component while in the egion between the esonances, the two components now have the same sign and so the cancelling-out featue is not encounteed and only a minimum, athe than antiesonance, occus. The pinciples illustated hee may be extended to any numbe of degees of feedom. Thee is a fundamental ule that if two consecutive modes have the same sign fo the modal constants, then thee will be an antiesonance at some fequency between the natual fequencies of those two modes. If they have opposite signs, thee will not be an antiesonance but just a minimum. The most impotant featue of antiesonance is pehaps the fact that thee is a phase change associated with it, as well as a vey low magnitude. It is also inteesting to detemine what contols whethe a paticula FRF will have positive o negative modal constants, and thus whethe it will exhibit antiesonances o not. A consideable insight may be gained by consideing the oigin of the modal constant: it is the poduct of two eigenvecto elements, one at the esponse point and the othe at the excitation point. Fo point mobility, the total modal constant fo evey mode must be positive, it being the squae of a numbe. This means that fo point FRF, thee must be an antiesonance following evey esonance, without exception. The situation fo tansfe FRFs is less categoical because the modal constant will sometimes be positive and sometimes negative, depending upon whethe the excitation and esponse move in phase o not. Thus, we expect tansfe FRF measuements to show a mixtue of antiesonances and minima (valleys). Howeve, the mixtue can be anticipated to some extent because it can be shown that the futhe apat ae the two points in question, the moe likely ae the two eigenvecto elements to altenate in sign as pogessed though the modes. Thus, it might be expected that the tansfe FRF between two positions widely sepaated on the stuctue will exhibit fewe antiesonances than the one fo two points elatively close togethe. A clea example of this is given in Fig. 3.7 fo a 4DOF system, showing a complete set of FRFs fo excitation at one exteme point in the fom of mobility. Finally, it should be emaked that if eithe the excitation o the esponse coodinates happen to coincide with a node fo one of the modes (i.e. Φ j Φ k ), then this mode will not appea as a esonance on the FRF plot. In that case, A jk and so the only esponse which will be encounteed at o nea ω Ω will be due to the off-esonant contibution of all the othe modes

65 The fom of the FRF plot of a damped system is quite simila to those fo an undamped system descibed above. The esonances and antiesonances ae blunted by inclusion of damping, and the phase angles (not shown) ae no longe exactly o 8, but the geneal appeaance of the plot is an extension of that fo the system without damping. This applies as long as the modes ae elatively well sepaated. This condition is satisfied unless the sepaation between adjacent natual fequencies (expessed as a pecentage of thei mean) is of the same ode as, o less than, the modal damping factos, in which case it becomes difficult to distinguish the individual modes. Fig. 3.8 shows a eceptance plot of a DOF system - the same system as in Fig. 3.6, with damping added. f(t) m m m 3 m 4 x (t) x (t) x 3 (t) x 4 (t) a( ω ) a_( ω ) a_( ω ) a_3( ω ) a_4( ω ) e e a3( ω ) a3_( ω ) a3_( ω ) a3_3( ω ) a3_4( ω ) log Y log Y Y. ω Fig a( ω ) a_( ω ) a_( ω ) a_3( ω ) a_4( ω ) Y 3 a4( ω ) Y 4 a4_( ω ). a4_( ω ). a4_3( ω ) 3 a4_4( ω ) 4 Mobility Plots of a 4DOF System. ω e ω log ω log ω log Y log Y4 Y. ω log ω log ω

66 log α.477 b( ω ).. b_( ω ) b_( ω ) e b( ω ) log α b_( ω ) b_( ω ) ω log ω 5.86e-6 6 ω log ω Fig. 3.8 Point (uppe) and Tansfe Receptance (below) FRF Plot fo Damped DOF system As fo the SDOF case, it is inteesting to examine the fom of Nyquist cicle fo an MDOF system as well. Nyquist plot of DOF system is shown in Fig point eceptance on the left and tansfe eceptance on the ight. Receptance of a popotionally damped system is plotted in a solid line, eceptance of non-popotionally damped system in a dashed line (in that case, modal cicles ae otated). Non-popotional damping will be discussed in detail in chapte Im α Im α Re α Re α Fig.3.9 Nyquist Plots of Point and Tansfe Receptances 3D plot of the point eceptance of a 3DOF popotionally damped system and its pojections to the individual planes is shown in Fig

67 Fig.3. 3D Receptance Plot of a 3DOF System 3..3 Damped Multi Degee-of-Feedom System Popotional Viscous Damping A special type of damping that is quite easy to include in analysis is so-called popotional damping. The advantage of using a popotional damping model in the analysis of stuctues is that the modes of such a stuctue ae almost identical to those of the undamped vesion of the model. Specifically, the mode shapes ae identical and the natual fequencies ae vey simila to those of the simple undamped system. It is possible to deive modal popeties of a popotionally damped system by fully analysing the undamped vesion and then making a coection fo the pesence of damping. While this pocedue is often used in the theoetical analysis of stuctues, it should be mentioned that it is only valid in the case of this special type o distibution of damping, which may not apply to eal stuctues studied in modal tests. Adding a viscous damping matix [B]to the geneal equation of motion fo an MDOF system, following equation applies : [ M ]{ x& } + [ B]{ x& } + [ K]{ x} { f} & (3.93) Fist, the case whee the damping matix is diectly popotional to the stiffness matix, will be discussed: [ B] β[ K] (3.94) If the damping matix is pe- and post-multiplied by the eigenvecto matix Ψ in just the same way as it was done peviously fo the mass and stiffness matices, it becomes:

68 T [ ] [ B ][ Ψ] β[ k ] [ b ] Ψ (3.95) whee the diagonal elements b epesent the modal damping of the vaious modes of the system. The fact that this matix is also diagonal means that the undamped system mode shapes ae also those of the damped system, and this is a paticula featue of this type of damping. This can easily be demonstated. Substituting modal coodinates {p} fo {x} { x} [ Ψ] { p} (3.96) into the equation of motion and pe-multiplying it by [ Ψ ] T leads to : [ m ] { p& } + [ b ] { p& } + [ k ] { p} { } & (3.97) fom whee the th equation is : m & p + b p& + k p (3.98) which is a equation of a single degee-of-feedom system, o of a single mode of the system. This mode has a complex eigenvalue p δ ± iω with an oscillatoy pat: Ω Ω Ω ζ k b ζ (3.99) βω m k m and a decay pat: β δ ζ Ω Ω (3.) A simple extension of the steps pefomed in foced esponse analysis of undamped systems (equations 3.83 to 3.89) lead to the definition fo the geneal eceptance FRF as: o [ ( )] [ ] ω K + iωb ω M α (3.) α jk Ψ Ψ Φ Φ ( ω) (3.) N N j k Ω ω + iωω ζ j k ( k ω m ) + i( ωb ) which has a vey simila fom to that fo the undamped system except that now it becomes complex in the denominato as a esult of the inclusion of damping. Geneal Foms of Popotional Damping Othe distibutions of damping bing about the same esult and they ae included in the classification popotional damping. The usual definition of popotional damping is that the damping matix [B] should be of the fom: B β K + γ M (3.3) [ ] [ ] [ ] In this case, the damped system will have eigenvalues and eigenvectos as follows: Ω ; Ω ζ βω γ ζ + (3.4) Ω damped undamped and [ Ψ ] [ Ψ ]

69 Distibution of damping of this type is often plausible fom the pactical standpoint - the actual damping mechanisms ae usually analogous to stiffness elements (fo intenal mateial o hysteetic damping) o to mass elements (fo fiction damping). Thee is a moe geneal definition of the condition equied fo the damped system to possess the same mode shapes as its undamped countepat, and that is: ([ M] [ K] ) [ M] [ B] ( ) ([ M] [ B] ) ([ M] [ K] ) (3.5) although it is moe difficult to make a diect physical intepetation of its fom Popotional Hysteetic Damping An identical pocedue can be used fo an MDOF system with popotional hysteetic damping, poducing the same essential esults. If the geneal system equations of motion ae expessed as [ M ]{ x& } + [ K + ih]{ x} { f} & (3.6) and the hysteetic damping matix [H] is popotional, typically: [ H] β [ K] + γ [ M] (3.7) then the mode shapes fo the damped system ae again identical to those of the undamped system and the eigenvalues take the complex fom: ( + iη ) λ Ω Ω k + m Ω γ η β (3.8) Note: η is the hysteetic damping loss facto. In the chapte 3..3., this loss facto was denoted as γ (and in the entie chapte 3., η was used fo tuning coefficient). In the following text, the loss facto will be denoted as η in accodance with common notation in liteatue. The geneal FRF expession is: α jk ( ω) Ψ Ψ N N j k k ω m + iηk Ω ω + iηω j k ( ) Φ Φ (3.9) Hysteetic Damping - Geneal Case As stated above, the case of popotional damping is a paticula one which does not always apply. It is justified in a theoetical analysis because it is ealistic and also because of a lack of any moe accuate model. Howeve, it is impotant to conside the most geneal case in ode to be able to intepet and analyse coectly the data obseved on eal stuctues. The geneal equation of motion fo an MDOF system with hysteetic damping and hamonic excitation is: [ ]{ } [ ]{ } [ ]{ } { } i ω M x& + K x + i H x F e t & (3.) Now, fist conside the case whee thee is no excitation and assume a solution of the fom: { } { } i λ x X e t (3.)

70 Substituted to the equation of motion, this tial solution leads to a complex eigenpoblem whose solution is in the fom of two matices (as fo the ealie undamped case), containing the eigenvalues and eigenvectos. In this case, both matices ae complex, meaning that each natual fequency and each mode shape is descibed in tems of complex quantities. The th eigenvalue is witten as ( + iη ) λ Ω (3.) whee Ω is the natual fequency and η is the damping loss facto fo that mode. The natual fequency Ω is not necessaily equal to the natual fequency of the undamped system, Ω, as was the case fo popotional hysteetic damping, although the two values will geneally be vey close in pactice. The complex mode shapes means that the amplitude of each DOF has both magnitude and phase angle. This is only vey slightly diffeent fom the undamped case whee thee is also both magnitude and phase, but the phase angle is eithe o 8, which can be completely descibed using eal numbes ( - positive magnitude, 8 - negative magnitude). Eigensolution of the damped system possess the same type of othogonality popeties as those demonstated in chapte 3... fo the undamped system and may be defined by equations: T [ ] [ M ][ Ψ] [ m ] Ψ (3.3) T [ ] [ K + ih][ Ψ] [ k ] Ψ (3.4) Again, the modal mass and stiffness paametes (now complex) depend upon nomalisation of the mode shape vectos fo thei magnitudes but always obey the elationship: k m λ (3.5) and a set of mass-nomalized eigenvectos can be defined as: { Φ} m { Ψ} (3.6) MDOF System with Geneal Hysteetic Damping - Foced Response Solution The equation of motion fo a foced esponse analysis of an MDOF system with hamonic excitation and esponse is: [ ]{ } i t { } e i ω K ih ω M X e ω F t + (3.7) Again, a diect solution to this poblem may be obtained by using the equations of motion to give: [ ]{ F} { X} [ K + ih ω M] { F} α( ω) (3.8) - 7 -

71 but this is again vey inefficient fo numeical application and the pocedue by multiplying both sides of the equation by the eigenvectos will be used. Following the same pocedues as between equations (3.86) and (3.87), it can be witten: [ ] [ Φ] T [ α( ω) ] [ Φ] ( λ ω ) (3.9) Fom this matix equation, any FRF element α jk (ω) can be extacted and expessed explicitly in a seies fom: α jk ( ω) Φ Φ N j k Ω ω + iηω (3.) which can also be ewitten in vaious altenative ways, such as: α jk ( ω) Ψ Ψ N j k m ( Ω ω + iηω ) o α jk ( ω) A N jk Ω ω + iηω (3.) In these expessions, both the numeato and denominato ae complex as a esult of the complexity of the eigenvectos. It is in this espect that the geneal damping case diffes fom that fo popotional damping MDOF Systems - Summay fo Vaious Types of Damping An analysis of an MDOF system with a geneal fom of viscous damping is omitted in this text, because it constitutes a much moe difficult poblem than that of an analysis of an MDOF system with a geneal fom of hysteetic damping. Only esults ae stated hee - in table 3.., definitions of FRFs and "natual fequencies" ae summaized fo all types of damping. The basic definition of "natual fequency" deives fom the undamped system's eigenvalues which yield the fequencies at which fee vibation of the system can take place. These undamped system natual fequencies ae given by the squae oots of the eigenvalues and identified by the symbol Ω and they occu in expessions fo both fee vibation esponse: x( t) N X e Ω i t and fo foced vibation, the FRF: N α( ω) A ω Ω (3.) (3.3) Fo damped systems, the situation is moe complicated and leads to two altenative chaacteistic fequency paametes being defined - both called "natual fequencies" - one fo fee vibation (Ω ) and the second fo foced vibation (Ω ')

72 NATURAL FREQ. SYSTEM EQUATION FOR FRF C D fee foced Ω Ω undamped popotional hysteetic popotional viscous geneal hysteetic geneal α α jk jk α jk ( ω) ( ω) ( ω) α jk Φ Φ N j k Ω ω Φ Φ N j k Ω ω + iηω Φ Φ N j k Ω ω + iωω ζ ( ω) viscous α jk ( ω) Φ N j k Ω ω + iηω R + i Φ ω Ω N jk jk Ω ω + iωω ζ S eal constant eal constant eal constant complex constant complex (ω) Ω Ω eal constant eal (ω) eal constant eal (ω) Ω ζ Ω Ω Ω Ω Ω Ω ζ Ω Tab 3. FRF Fomulae and Natual Fequencies fo All Types of Damping The natual fequency Ω constitutes the oscillatoy pat of the fee vibation chaacteistic which, being complex, contains an exponential decay tem as well. Thus: N x (t) δ Ω X e t e i t (3.4) whee Ω may o may not be equal to Ω, depending on the type and distibution of the damping. The "natual fequency" Ω ' comes fom the geneal fom of the FRF expession which, combining all the pevious cases, may be witten in the fom: C α( ω) (3.5) id N Ω ω + Hee, C may be eal o complex and D will be eal; both may be constant o fequency dependent and Ω ' will, in geneal, be diffeent to both Ω and Ω. Table 3. summaizes systems with all the above mentioned types od damping Excitation by a Geneal Foce Vecto Now, a hysteetically damped MDOF system will be consideed again. Its equation of motion in the case of hamonic excitation has the fom (3.7): [ ]{ } i t { } e i ω K + ih ω M X e ω F t If the system is excited simultaneously at seveal points (athe than at just one, as in the case of the individual FRF expession), the solution is given by eq. (3.8): - 7 -

73 [ ]{ F} { X} [ K + ih ω M] { F} α( ω) A moe explicit solution can be deived in the fom: { } T { Φ} { F}{ Φ} N Ω ω + iηω X (3.6) This equation pemits the calculation of one o moe individual esponses to excitation of seveal simultaneous hamonic foces, all of which must have the same fequency but may vay in amplitude and phase. The esulting vecto of esponses is sometimes efeed to as foced vibation mode o, moe commonly, as opeating deflection shape (ODS). When the excitation fequency is close to one of the system's natual fequencies, the ODS will eflect the shape of the neaby mode because one tem in the seies of (3.6) will dominate, but it will not be identical to it because of the contibutions of all the othe modes Excitation by a Vecto of Mono-Phased Foces Conside a special case whee the stuctue will be excited by a vecto of mono-phased foces. All the foces will have the same fequency and phase, only thei amplitudes may diffe. In this case, it would be of inteest to know whethe thee exist some conditions unde which it is possible to obtain a similaly mono-phased esponse (the whole system esponding with a single phase angle). So, let the foce and esponse vectos be epesented by: { } { } i ω f F e t (3.7) i { } { } ( ω t X e ϕ) x (3.8) whee {F} and {X} ae vectos of eal quantities. Substituting them into equation of motion, (3.7) yields to : [ ]{ } i ω K ih M X e t e i ϕ { F} e i ω ω t iϕ [ K + ih ω M]{ X} e { F} [ K ih ω M]{ X} ( cos ϕ i sin ϕ) { F} + (3.9) + (3.3) Afte splitting (3.3) into eal and imaginay pat: [ K ω M] cosϕ + [ H] [ K ω M] sin ϕ + [ H] ( sin ϕ){ X} { F} ( cosϕ){ X} { } (3.3) (3.3) Equation (3.3) can be consideed as an eigenvalue poblem which has "oots" ϕ s and coesponding "vectos" { κ } s. These may be inseted back into (3.3) in ode to establish the fom of the mono-phased foce vecto necessay to bing out the mono-phased esponse vecto descibed by { κ } s. Thus, a set of N mono-phased foce vectos is obtained, each of

74 which esults in a mono-phased esponse chaacteistics, when applied as excitation to the system. The equations used to obtain the above mentioned solution ae functions of fequency and thus each solution applies only at one specific fequency, ω s. A situation of paticula inteest occus when a phase lag ϕ between all the foces and all the esponses is exactly 9. In this case, the eq. (3.3) educes to: [ K ω M]{ X} { } (3.33) Solving this equation, natual fequencies and mode shapes of an undamped system ae detemined. This is a vey impotant esult evealing that it is always possible to find a set of mono-phased foces which will cause a mono-phased set of esponses and, moeove, if these two sets of mono-phased paametes ae sepaated by exactly 9, then the fequency at which the system is vibating is identical to one of its undamped natual fequencies and the displacement shape is the coesponding undamped mode shape. This vey impotant esult is the basis fo many of the multi-shake test pocedues used to isolate the undamped modes of stuctues fo compaison with the theoetical pediction. It is commonly used in the aicaft industy. It should be emphasized that, by this method, undamped modes ae diectly obtained, wheeas almost all othe methods extact the actual damped modes of the system unde test. The physical pinciple of this method is that the foce vecto is chosen so that it exactly balances all the damping foces whateve these may be and so the pinciple applies equally to all types of damping

75 4. Modal Test In this chapte, the pocedue how to pefom modal test will be discussed. This pocedue involves thee o fou main stages: ) pepaation of the measued stuctue and ceating a geometical model fo measuement ) measuement itself 3) identification of modal paametes fom the measued data 4) (veification of the obtained modal model, its compaison with a computational model etc.) All these stages will be discussed step by step. It is clea that using vaious pocedues in the individual stages will highly depend on the pupose of modal test (see chapte.). 4. Pepaation 4.. Pepaation of the Measued Stuctue Vaious types of how to suppot the stuctue duing measuement wee biefly mentioned in chapte.4. Hee, the individual possibilities will be discussed in moe detail Fee Suppot Fee suppot (fee-fee conditions) is theoetically such a type of suppot whee the tested object is not attached to gound at any of its coodinates and is feely suspended in space. In theoetical analysis, a feely suppoted body exhibits 6 igid body modes, i.e. 3 displacements in the diection of 3 coodinates and 3 otations aound the coodinate axes. All those six modes have the natual fequency equal to zeo. In pactice, fee suppot is ealized eithe by putting the body on a vey soft pad (e.g. foam) o suspending it on soft spings. It is obvious that fequencies of igid body modes will not be equal to zeo in that case, but the values will be vey low. Such suppot is consideed to be fee if the highest natual fequency of igid body modes is less than % of the value of the lowest defomation natual fequency. So, if the fist bending mode of a beam is e.g. 5 Hz, all of the igid body modes should be less than 5 Hz. When this condition is satisfied, the influence of suppot to defomation natual fequencies is negligible. Damping of the individual modes, athe than natual fequencies, could be influenced by the suppot. Thus, if the measuement is pefomed with the aim to detemine pecise values of damping, the influence of the suppot could be minimized by placing soft suspension spings into nodal points. But, nodal points ae diffeent fo each of the modes, so fo obtaining as pecise values of damping as possible it would be necessay to measue each mode with diffeent placing of suspension spings. Fee suppot is both the simplest and the most suitable if the modal model obtained fom measuements is to be compaed with the computational modal model. Thus, it is woth to use it wheneve it is possible

76 4... Gounded (Fixed) Suppot Gounded (fixed, clamped) suppot is theoetically such a type of suppot whee some points on the body (some DOFs) ae completely fixed by connecting to the gound. This could not be eached in pactice, so the suppot is consideed to be fixed if the esponse of the fixed DOFs is less than % of the esponse of the othe DOFs. This type of suppot causes difficulties when compaing expeimental modal model with the computational modal model, because the diffeences in both models could be caused namely by diffeent bounday conditions. But, sometimes it is necessay to use this type of suppot, if modal popeties have no impotance when the stuctue is feely suppoted (e.g. fo tubine blades). Anothe difficulty with such a type of suppot is with epeatability of the measuements. Despite any effot (tighten the scews connecting the stuctue with the measuement base using a toque wench, etc.), % epeatability is failing if disassembly and eassembly of the measuement base is pefomed. Accoding to expeience, natual fequencies of the individual modes could diffe afte such mount - demount in the ange up to ±5% Suppot in-situ This type of suppot is the simplest as fo pepaation - thee is no pepaation, measuement is pefomed in the actual opeational conditions. This type of suppot is used when thee is no othe possibility (when measuing a vey heavy stuctue, lage machine, etc.) o, when modal popeties in opeational conditions ae of inteest. It is clea that it is even moe difficult to compae an expeimental model with its computational model with this type of suppot than it is with fixed suppot. 4.. Pepaation of Expeimental Model In this chapte, a model will denote a geometical model of the measued stuctue with defined points and degees of feedom in which measuements ae to be pefomed. This means no kind of mathematical model (spatial, modal o esponse). A mesh of points epesenting the measued stuctue is to be selected. In each of these points, it should be decided in which diections the measuements ae to be pefomed, i.e. degees of feedom ae to be defined. Most often, only the tansvesal degees of feedom ae measued (i.e. diections X, Y and Z). In special cases, otational degees of feedom could be measued as well, but these equie special tansduce and pocedues that ae not common and ae not suppoted by most of modal softwae packages. The density of the mesh of the measuement points highly depends on the fequency ange of the measuements o, moe pecisely defined, on the numbe of modes that would be identified - it is impotant to conside that the highe the mode, the moe nodal lines possess its mode shape and the moe points ae necessay to display it ealistically. So, the ule fo defining the density of the mesh (i.e. numbe of points) is: define just enough points to be able to eliably identify all of the modes in the measued fequency ange, but not too many points (o DOFs), because edundant points mean a moe time-consuming modal test. It should be mentioned, that the density of mesh does not affect the pecision of the obtained modal paametes! It only affects the quality of displaying mode shapes and thus the

77 possibility to identify them coectly. It is a completely diffeent situation fom finite element calculations whee the density of the mesh highly influences the pecision of the solution. efeence DOF 97Z Fig. 4. Example of the Geometical Model fo Modal Test A mesh of points should be dawn on the measued stuctue (see Fig. 4.) and the same model should be ceated in the modal test softwae. Also, so-called efeence DOF is to be defined - this is the DOF, in which point diect FRF is measued, which means that the DOF of excitation and of esponse is the same. When a modal test is pefomed using impact excitation, an acceleomete is usually fixed in the efeence point. When a dynamic excite is used fo excitation, it is connected to the stuctue in the efeence point. Requiements fo the location of the efeence point ae somewhat contadictoy:. It should be placed in a way that thee is an adequate esponse by all of the modes so that the signal to noise atio is as good as possible.. It should be placed in a way that the influence of attaching acceleomete o dynamic excite is as low as possible. It is obvious that these two equiements ae in contadiction, because the highest influence on the stuctue caused by attaching acceleomete o excite would be at position, whee the stuctue exhibits the highest esponse. Thus, in pactice it is necessay to choose a easonable compomise between those two equiements. Moeove, when the mass of acceleomete in compaison to the mass of the measued stuctue is negligible, its influence is also negligible. The fist equiement is elated to the fact that one must be caeful that the efeence point would not also be a nodal point of one mode fom those which ae of inteest. In this case, the esponse of this mode is zeo and it would not be possible to identify it. One of the ways to avoid this situation is to appoximately know the mode shapes in advance, e.g. fom a computational model. Anothe option, if mode shapes ae not known and can not be estimated, is to ty diffeent locations of the efeence point befoe the entie modal test and watch if the numbe of esonances in the measued FRF is stable. If some of the esonances disappeas at any point, it means that at this point is a nodal point of the paticula mode shape and that this point can not seve as a efeence

78 4. Measuement Techniques This chapte deals with measuement techniques which ae used fo modal testing. Thee ae two types of vibation measuement: - Those in which just one type of paamete is measued (usually the esponse levels) - Those in which both input and esponse output paametes ae measued. Accoding to the basic elationship: RESPONSE PROPERTIES INPUT it is clea that only when two of the thee tems in this equation have been measued, it can be defined completely what is going on in the vibation of the test object. If we measue only the esponse, we ae unable to say whethe a paticulaly high esponse level is due to a stong excitation o to a esonance of the stuctue. Nevetheless, both types of measuements have thei applications and much of the equipment and instuments used is the same in both cases. This text is focused on the second type of measuement, whee both excitation and esponse ae measued simultaneously so that the basic equation can be used to deduce the system popeties diectly fom the measued data. Within this categoy thee ae a numbe of diffeent appoaches which can be used, but it should be stated with a method efeed as the single-point excitation (although this point may change its location duing the modal test). Using this method, eithe one ow o one column of the fequency esponse function matix is measued. Thee ae two pincipially identical modifications of the single-point excitation method: - SISO (Single Input Single Output) - SIMO (Single Input Multiple Output) - the numbe of outputs (esponses) depends on the numbe of channels that ae at disposal on the analyze, i.e. how many esponses can be captued simultaneously. The pinciple of signal post-pocessing is the same as fo SISO - the classical FRF is used accoding to eq. (.). Anothe type of measuement is so called MIMO (Multiple Input Multiple Output), duing which excitation at multiple points is applied simultaneously. This type of measuements is essential fo modal testing in these cases: - Lage stuctues that ae impossible to be excited using only one excite. - Complex stuctues that exhibit local modes. Local modes ae modes when only a pat of stuctue vibates and it is impossible to excite all these modes simultaneously with a single excite. - Symmetical stuctues that exhibit multiple modes (two o moe modes at the same fequency). In ode to isolate these modes, it is necessay to have as many efeence points as is the numbe of modes at the same fequency. MIMO method is a common standad in aicaft and automotive industies, but it has a slightly diffeent theoetical backgound than single-point excitation and thus it will not be discussed in detail in this text

79 4.. Basic Measuement Setup The expeimental setup used fo FRF measuements basically consists of thee o fou majo items: - an excitation mechanism - a tansduction system, to measue the vaious paametes of inteest - an analyze, to extact the desied infomation fom the measued signals - (computing system, to post-pocess the measued data, extact modal paametes, animate mode shapes etc.) acceleomete testpiece foce tansduce impact hamme analyze Fig. 4. Expeimental Setup Whee Impact Hamme is Used fo Excitation A typical expeimental setup fo excitation using impact an hamme is shown in Fig. 4.. In this case, the acceleomete is fixed in the efeence point and the stuctue is subsequently excited at all the points. It povides one ow of the FRF matix. A typical expeimental setup fo excitation using dynamic excite is shown in Fig In this case, the excite is fixed in the efeence point and the esponse is subsequently measued at all the points (o at all the points simultaneously, depending on how many analyze's channels ae at disposal). It povides one column of the FRF matix. acceleomete testpiece analyze foce tansduce dynamic excite Fig. 4.3 powe amplifie Expeimental Setup Whee Dynamic Excite is Used fo Excitation

80 4... Excitation Techniques Thee ae seveal ways how to excite vibation of a stuctue. They can be divided into two majo goups:. impact excitation - using impact (modal) hamme - the most common method - eleasing fom the defomed position - e.g. by cutting acoss the suspension cable - hitting by a falling mass - using a pendulum impacto. excitation using an attached excite - electomagnetic excite - the most common method - electo-hydaulic excite - mechanical excite - eccentic otating masses Thee ae othe, non-standad excitation methods that ae used fo lage stuctues (bidges, off-shoe platfoms etc.): - using jet engines - natual excitation (by wind, sea waves, taffic) They ae used in so called opeational modal analysis (see chapte 6) and lead to an unscaled modal model Impact Excitation Using Impact Hamme Using an impact hamme is the simplest and fastest way of exciting a stuctue into vibation. It equies no pepaation wok and thus is vey suitable to use in opeational conditions. Moeove, it does not influence the stuctue by attaching the excite, which is an advantage itself. The hamme consists of a head, foce tansduce, tip and handle. An impacto can also be used - it is basically a hamme without a handle (see Fig. 4.4). The equipment of a hamme is usually completed with a set of tips of diffeent stiffness and with a set of heads of diffeent masses. A foce tansduce detects the magnitude of the foce felt by the impacto which is assumed to be equal and opposite to that expeienced by the stuctue. The magnitude of the impact is basically detemined by the mass of the hamme head and the velocity with which it is moving when it hits the stuctue. The opeato contols the velocity athe than the foce level itself, and so an appopiate way of adjusting the ode of the foce level is by vaying the mass of the hamme head. The fequency ange which is effectively excited by this type of device is contolled by the stiffness of the contacted sufaces and the mass of the hamme head: thee is a system esonance at a fequency given by contact stiffness impacto mass above which it is difficult to delive enegy into the test stuctue. When the hamme tip impacts the test stuctue, this will expeience a foce pulse which is substantially that of a half-sine shape, as shown in Fig. 4.5 (left). A pulse of this type has a fequency content of the fom illustated in Fig. 4.5 (ight)

81 which is essentially flat up to a cetain fequency (f c ) and then it is of uncetain stength above this fequency. This means that it is elatively ineffective at exciting vibations in the fequency ange above f c and so it is necessay to have some contol ove this paamete. It can be shown that thee is a diect elationship between the fist cut-off fequency, f c, and the duation of the pulse, T c, and that in ode to aise the fequency ange it is necessay to induce a shote pulse length. This, in tun, is elated to the stiffness (not the hadness) of the contacting sufaces and the mass of the impacto head. The stiffe the mateials, the shote will be the duation of the pulse and the highe will be the fequency ange coveed by the impact. Similaly, the lighte the impacto mass, the highe the effective fequency ange. It is fo this pupose that a set of diffeent hamme tips and heads ae used to enable the egulation of the fequency ange. Geneally, it can be said that a tip as soft as possible has to be used in ode to supply input enegy only to the fequency ange of inteest. Using a stiffe tip than necessay leads to the fact that the supplied enegy causes vibations outside the fequency band of inteest at the expense of those within this zone. impacto impact hamme head Fig. 4.4 foce tansduce tip Impacto and Impact Hamme Details a(t) G AA (f) T c t f c f Fig. 4.5 Impact Foce Pulse and its Spectum Thee ae some disadvantages associated with using an impact hamme: - Contol of the fequency ange of excitation is limited and, moeove, it is aely possible to use fequency zoom. - Cest facto is vey high and due to a high peak level of the acting foce thee is a dange of causing a local damage to the stuctue and exciting its nonlinea behaviou

82 - Window (weighting) functions have to be used both fo input and output signal. When using impact excitation, diffeent weighting functions ae used fo input and output signals. The input signal is weighted with tansient window that seves fo suppessing noise to zeo in the time peiod when the impact foce is not acting but the measuement of esponse is in pogess. In ode to be sue that the shift and length of the tansient window ae set coectly, it is woth to expand the vetical axis so that level of noise is visible. Fo the output signal (esponse), exponential window is often used to impove the analysis by minimizing leakage eo that is caused by tuncating the time signal. The length of the exponential window (its time constant τ) should be set with egad to the equiement that the signal at its end should be attenuated to the level of noise o at least by 4 db. The windows fo both signals should stated at the same time if thee is no system delay. Weighting windows ae shown in Fig Tansient weighting of input signal Exponential weighting of output signal Fig. 4.6 Weighting Windows fo Impact Excitation By applying exponential weighting, an electonic damping is added to the system. Unde this condition, the damping value detemined by measuement will be ove-estimated and fo obtaining the tue value, compensation fo the exponential window should be pefomed (see Fig. 4.7). Coection of the decay constant: δ δ m δ w δ... tue value δ m... measued value δ w... influence of exp. w. τ w Coection of damping atio: δ δ m δ w ζ ζ m ζ Ω Ω w b(t) shift window function oiginal signal weighted signal length τ w sample time T t č Fig. 4.7 Compensation fo Exponential Weighting - 8 -

83 Some of the disadvantages of application of impact hamme excitation can be avoided by applying so-called andom impact excitation. This means seveal consecutive hits duing the ecod (see Fig. 4.8). In this case, Hanning window is used both fo input and output signals in ode to push the signals to zeo in thei edges and minimize the eo caused by leakage. As the windows applied to the input and output signals ae the same, thei influence on FRF will be cancelled out and no compensation fo weighting window is necessay when damping is estimated. Random impact excitation intoduces moe enegy than a single impact duing one ecod and the cest facto is lowe. All advantages of impact excitation emain unchanged, namely its easy use in opeational conditions. Moeove, this type of excitation may be used fo measuements in a naow fequency band o with fequency zoom. In these cases, ecod time is quite long and if only a single impact was used, the stuctue would vibate fo much shote time peiod than the measuement peiod was. By applying moe hits, this poblem is avoided, but still thee is a isk that most of the enegy supplied to the stuctue will be out of the measuement ange, because contol ove the fequency ange of excitation is always limited when an impact hamme is used. ecoded analyzed excitation esponse t t T m t t Fig. 4.8 Signals Involved in Random Impact Excitation 4... Excitation with the Help of Dynamic Excite Pehaps the most common type of excite is the electomagnetic (o electodynamic) shake in which the supplied input signal is conveted to an altenating magnetic field whee a coil is placed which is attached to the dive pat of the device and to the stuctue. In this case, the fequency and amplitude of excitation ae contolled independently of each othe, giving moe opeational flexibility. It is useful because it is often bette to vay the level of the excitation as esonances ae passed though. Howeve, it must be mentioned that electical impedance of these devices vaies with the amplitude of motion of the moving coil and so it is not possible to deduce the excitation foce eithe fom a measuement of the voltage applied to the shake o by measuing the cuent passing though the shake because this measues the foce applied not to the stuctue itself, but to the assembly of stuctue and shake dive. The diffeence between this foce and that applied to the stuctue is quite small, but nea the esonance a vey little foce is equied to poduce a lage esponse and without alteing the

84 settings on the powe amplifie o signal geneato, thee is a maked eduction in the foce level at fequencies adjacent to the stuctue's natual fequencies. As a esult, measuement of foce at fequencies adjacent to natual fequencies inclines to be contaminated by noise. So, the foce acting on the stuctue should be measued as close as possible to the stuctue's suface, the same as with impact excitation. Geneally, the lage the shake, the geate the foce which may be geneated fo exciting the stuctue, but the woking fequency ange is limited at the same time. An effective excitation is possible only as long as the moving pats of the excite emain a igid mass. Once the fequency of vibation appoaches and passes the fist natual fequency of the coil and dive platfom, thee is a sevee attenuation of the foce which is available fo diving the test object and although some excitation is possible above this citical fequency, it does impose a natual limit on the useful woking ange of the device. This fequency is of couse lowe fo lage shakes. In special cases, it is appopiate to use an electohydaulic excite. These ae cases of testing stuctues o mateials, the nomal vibation ambience of which is associated with highe static load that may quite often alte thei dynamic chaacteistics o even thei geomety. An advantage of electohydaulic excites is thei ability to apply simultaneously a static load as well as dynamic vibatoy load that is necessay in such cases. Anothe advantage that may hydaulic excites offe is the possibility of poviding a elatively long stoke, theeby pemitting the excitation of stuctues at lage amplitudes. On the othe hand, hydaulic excites tend to be limited in opeational fequency ange above khz, wheeas electomagnetic excites can opeate well into the 3-5 khz egion, depending on thei size. Hydaulic excites ae moe complex and expensive, although geneally compact and lightweight compaed with electomagnetic excites. Anothe type of excite that is woth to be mentioned is a mechanical excite. It is ealized by means of eccentic otating masses (unbalances) and it is able to geneate the pescibed foce at vaious fequencies, although its contol is not much flexible. The amplitude of the exciting foce is given by the unbalanced mass and it can be changed only by changing eithe the mass o its adius which both is impossible to pefom duing opeation. This type of excite is also elatively ineffective at lowe fequencies, because the exciting foce depends on the squae of otational speed. Howeve, if vibation amplitudes caused by this type of excite is not too lage in efeence to the obit of otating mass, amplitude and phase of the excitation foce ae exactly known and equie no additional measuements unlike othe type of excites. A mechanical excite is used to measue lage stuctues such as bidges o bedplates of tubogeneatos. Attachment of an Excite to a Stuctue When using an electomagnetic o electohydaulic excite, it is necessay to connect the diving platfom of the shake to the stuctue, usually incopoating a foce tansduce. Some pecautions must be taken in ode to avoid the intoduction of unwanted excitations o inadvetent modification of the stuctue. Fom the definition of a single FRF as a atio between the hamonic esponse at DOF j caused by a single hamonic foce applied in DOF k

85 it is evident that this single foce must be the only excitation of the stuctue and this equiement should by met in the modal test. Although it may seem that the excite is capable of applying the foce in one diection only, as it is essentially a unidiectional device, thee exists a poblem on most pactical stuctues whose motion is geneally complex and multidiectional. The poblem is that when pushed in one diection, the stuctue esponds not only in the same diection but also in othes. Such motion is expected but it is possible that it can give ise to a seconday fom of excitation if the shake is incoectly attached to the stuctue. The moving pat of the shake is usually vey mobile along the axis of its dive but vey stiff in the othe diections. Thus, if the stuctue wishes to espond both in line of action of the excite and in lateal diection, then the stiffness of the excite will cause esisting foces o moments to be geneated which ae, in effect, exeted on the stuctue in the fom of seconday excitation. The esponse tansduces know nothing of this and they pick up the total esponse, which is caused not only by the diving foce which is known, but also by the seconday and unknown foces. foce tansduce excite dive od plug excite testpiece Fig. 4.9 Excite Attachment and Dive Rod Assembly The solution is to attach the shake to the stuctue though a dive od o simila connecto which has the chaacteistic of being stiff in one diection (that of the intended excitation) while at the same time being elatively flexible in the othe five diections. A suitable dive od, o stinge, is shown in Fig It is made of mm diamete wie of the length of about to 5 mm. Cae must be taken not to ove-compensate: if the dive od is too long o too flexible, it begins to intoduce the effects of its own esonances into the measuements and contaminates the genuine data. Anothe equiement that has to be met in ode to have the pecise measuements of the excitation foce is to place the foce tansduce as close to the measued stuctue as possible. Coect aangement of the connection that is necessay fo eliable FRF measuements is shown in Fig. 4. and also in a pactical example in Fig

86 testpiece acceleomete at efeence DOF foce tansduce excite dive od Fig. 4. Placing of Foce Tansduce and Attachment of the Excite to the Testpiece Anothe consideation which concens the shake is the question of how it should be suppoted, o mounted, in elation to the test stuctue. Seveal possibilities ae shown in Fig thee of them ae acceptable and one is unsatisfactoy. Geneally, it can be said that eithe the shake o the stuctue should be feely suppoted. In Fig, 4. (left), and also in Fig. 4., the shake is fixed to the gound while the test stuctue is suppoted by a soft suspension. It is the most satisfactoy aangement and it is often used fo light stuctues that can be suppoted feely. In the middle of Fig. 4., two aangements in which the shake is esiliently suppoted ae shown. In this case, the measued stuctue can be eithe gounded o feely suppoted. The poblem with this aangement is that the eaction foce causes a movement of the shake body which, at low fequencies, can be of lage displacement. This in tun causes a eduction in the foce acting on the stuctue and theeby adding an additional inetia mass to the shake may be necessay. The aangement in Fig. 4. (ight) is unsatisfactoy fo the modal test, because eaction foces in shake's suppot intoduce an additional excitation to the stuctue that is not measued by the foce tansduce. ideal configuation suitable configuation unsatisfactoy configuation fee stuctue fee shake fee suppoted stuctue gounded shake gounded stuctue gounded shake gounded stuctue fee shake Fig. 4. Vaious Mounting Aangements fo Excite with Respect to Reaction Foces

87 Fig. 4. Pactical Realization - Fee Stuctue, Fixed Excite Types of Excitation Signals When an attached shake is used fo excitation, thee ae vaious possibilities egading types of the excitation signals. The types of signals can be divided into (see Fig.4.3): - hamonic (sinusoidal) - spectum contains only a single fequency of the sinewave - boadband - spectum contains a band of fequencies. It futhe splits into: o impulse (tansient) single pulse (o impact) peiodic pulse andom impact o andom o pseudoandom o swept sine (chip) Impulse excitation is mostly applied in the fom of an impact o andom impact using an impact hamme (see chapte 4...), but it can also be applied with the help of an attached shake. In this case, it would be most likely a single o peiodic pulse. All of the above mentioned signals ae usually povided by a signal geneato which is usually pat of the analyze. The signal fom the geneato goes though the powe amplifie, then to the dynamic excite and then to the stuctue. Note.: Swept sine signal can also be poduced by a mechanical vibation excite, while natue excitation by sea waves o oad taffic poduces a andom signal. Sine signal The single sine signal of constant fequency can hadly be applied in modal tests. This type of signal can be used in the fom of stepped sine in FRA analyzes (FRA Fequency Response Analyze) that, in contast to FFT analyzes, do not pefom the Fouie tansfom of signals, but diectly measue the steady-state esponse of the system to the steady-state hamonic excitation. The fequency of excitation is changed step by step and the atio of the esponse to the excitation is ecoded. In this way, subsequently, the entie FRF in the

88 fequency ange of inteest is obtained. This pocedue is vey time consuming, but it is basically the only option if the nonlineaities in stuctues ae to be examined in detail. The fequency ange of the measuement can be set almost abitaily, thus this pocedue is applied with the use of the fequency zoom to examine fequency egions nea esonances. a(t) sine a(t) pulse t t t a(t) swept sine a(t) peiodic pulse t T t t a(t) andom a(t) impact a(t) pseudoandom t a(t) andom impact t T T T T t t Fig. 4.3 Basic Types of Excitation Signals Random Signal Random signal is chaacteized by its powe spectal density G AA (f) and the amplitude pobability density p(a) with Gaussian nomal distibution (see Fig. 4.4). Cest facto and signal to noise atio ae quite good with this type of signal. a(t) G AA (f) p(a) baseband G AA (f) t zoom fequency ange f fequency ange f Fig. 4.4 Random Signal - Time Histoy and Spectum

89 Random signal can be geneated in a limited fequency band accoding to the fequency ange of inteest. It can be used fo baseband measuements as well as fo zoom measuements. The signal is not peiodic within the ecod time, so it is necessay to apply Hanning window to both input and output signals to minimize the leakage eo. Changes in amplitudes and phases of the excitation signal ae andom, thus aveaging is necessay. The influence of the eventual nonlineaities is aveaged out and the ideal lineaized FRF estimate is obtained. Pseudoandom signal Pseudoandom signal is essentially a piece of andom signal that is epeating with the peiod T. This peiod is equal to the ecod length in the analyze. Consequently, no leakage eo occus because the signal is peiodic in the ecod time so that no weighting windows ae necessay (ectangula window no window). This implies that the signal only contains fequencies coinciding with the lines computed in the analyze. Nomally it is designed to have the same level in each line. At a given fequency, a system will always be excited at the same level since each ecoding contains the same infomation. Theefoe no aveaging of non-lineaities can be obtained and this type of signal is only suitable fo a pefectly linea system. As with the andom signal, the pseudoandom signal can be used both fo baseband and zoom measuements. A special type of pseudoandom signal is a chip signal. It is a swept sine signal with a high sweep ate when sweep fom the minimal to the maximal fequency is epeated each peiod T and this peiod is equal to the ecod length Tansduces Used fo Excitation Foce and Response Measuements A detailed desciption of the individual types of tansduces is not the subject of this text. It should only be emphasized hee that each senso measues exactly what is happening with it itself. It should be ensued by pope attachment of the senso to the stuctue so that this would be the same as what is happening to the stuctue and what should be measued. Futhe, it is impotant to ealize that each tansduce has its own esonant fequency which mostly depends on the mass of the tansduce and which is futhe moe o less influenced by the attachment of the tansduce to the stuctue. The effective fequency ange in which a tansduce may be used is to appoximately /3 of the value of this fequency. Vaious ways of tansduce attachment that ae soted accoding to how much they educe the tansduce's natual fequency (fom the best to the wost) ae: - scew (stud) - special cement - thin double adhesive stud - beeswax (up to 4 C only) - magnet Beeswax is vey often used in modal tests fo attachment of acceleometes because it is quite undemanding and quick and does not significantly educe the effective fequency ange of the tansduce. On the contay, attachment with the use of magnet is not appopiate and it

90 is mostly used fo opeational measuement but only up to about 5 Hz. If a stuctue with a cuved suface is to be measued, mounting with swivel base can be used. When an impact hamme is used fo excitation, a foce tansduce is an integal pat of the hamme and it is not attached to the stuctue. When a dynamic excite is used fo excitation, a foce tansduce should be attached to the stuctue, as descibed in chapte 4... (Fig. 4.). It is mostly attached with a scew. If it is not possible to dill a thead fo a scew into the measued stuctue, it is possible to use so-called "plug". It is a small cylinde, one base of which is smooth and cemented to the suface of the measued stuctue and a scew thead is dilled into its second base. The foce tansduce is scewed into this thead as shown in Fig Anothe aspect that should be consideed in the context of selecting tansduces fo modal tests is the mass of the tansduce in elation to the mass of the measued stuctue to avoid dynamic popeties of the measued stuctue to be alteed significantly by the tansduce. It was aleady mentioned in chapte 4... egading the selection of efeence point, whee the influence of tansduce placement to the dynamic popeties of the stuctue was discussed. Mass of the tansduce plays a ole especially in light stuctues. Geneally, it should be less than % of the mass of the stuctue. Fom the pinciple of design of acceleometes, it is obvious that the smalle the tansduce, the lowe its sensitivity but highe the fequency ange of the measuement. Fo special applications, it could happen that a tansduce that would be light enough not to alte the stuctue would have at the same time inadequate sensitivity. In these cases, a non-contact tansduce can be used, e.g. lase dopple vibomete Analyze Two types of analyzes - FRA o FFT - can be used to pefom modal tests. Fequency Response Analyze (FRA) This type of analyze has been aleady mentioned in chapte in elation to the stepped-sine excitation. It does not pefom Fouie tansfom of time signals. The pinciple of its opeation is as follows: The souce o command signal is a sine-wave at the desied fequency. Measued signals fom the foce and esponse tansduces undego a digital filteing pocess duing which all components with fequency diffeent to that of the command signal ae eliminated. Non-synchonous component ejection is impoved by filteing ove a longe peiod of time. This is quantified by the numbe of cycles of the command signal duing which the computations ae pefomed. By this pocess, a vey accuate measuement of the signal component at the desied fequency is obtained. FRF at the desied fequency is diectly given by the atio of amplitudes of the esponse and foce signals. Then, the fequency of the command signal is changed by a step coesponding to fequency esolution and the pocess is epeated. Step by step, the entie fequency band of inteest is measued. This is a vey accuate and vey time consuming measuement. That is why this type of analyze is not commonly used fo modal tests, but only fo special applications, namely fo studying nonlineaities

91 Fast Fouie Tansfom (FFT) Analyze (fequency, spectal analyze) The pinciple of opeation of this type of analyze was descibed in detail in the chapte - Dual-channel Analysis. Basic pinciples can be summaized as: - All the fequency components pesent in the complex time vaying signal ae measued simultaneously. - The output is a spectum containing a finite numbe of components, descibing the elative amplitudes of the entie ange of fequencies pesent in the signal. - Calculates additional functions and all calculations ae based on the discete Fouie tansfom. - Signal should always be subjected to the anti-aliasing filte pio to enteing the A/D convete. 4.. Pepaation of Measuement Afte the tested stuctue is pepaed (suppoted) and the geometical model is ceated, the measuement can stat. Remembe that when a stuctue is excited with an attached shake, the shake and the foce tansduce ae placed in the efeence point (DOF) and the esponse tansduce (usually acceleomete) is placed successively to all the points (see Fig. 4.5). If the numbe of measued DOFs is not geate than analyze's channels that ae at disposal, esponses at all the DOFs can be captued simultaneously. In any case, one column of the FRF matix is obtained. [ H] H... H Hn... Fig. 4.5 Pepaation of Measuements - Shake Excitation When hamme excitation is used, the acceleomete is usually placed in the efeence point and the stuctue is excited successively in all the points, thus poducing one ow of the FRF matix (see Fig. 4.6). Howeve, this is not a stict ule. If the tested stuctue is quite complex and some of the DOFs that have to be measued ae hadly accessible, it might be easie to place an acceleomete to them athe than to induce an impact in them. In that case, the stuctue might be excited with an impact hamme in a fixed (efeence) point and the esponse acceleomete might be oving. Likewise, if a tiaxial acceleomete is used to captue esponses, it can not be used as efeence, because only one degee-of-feedom is the efeence one (supposing SISO o SIMO technique). Thus, when a tiaxial acceleomete is - 9 -

92 used, it is always a oving one and the DOF of excitation is fixed, egadless whethe hamme o shake excitation is applied. Pio to stating with the entie modal test, it is appopiate to pefom some checks in ode to be sue that the measued data will be coect. With hamme excitation, it is simple to pefom the ecipocity check. Theoetically, if a stuctue is excited in DOF i and the esponse is captued in DOF j, it should be the same as the ecipocal situation (excitation in DOF j and esponse in DOF i). In both case, an identical FRF should be obtained - H ij (f)h ji (f). The most impotant check that should neve be omitted is that of checking the accuacy of efeence point measuement. [ H] H H...H n Fig. 4.6 Pepaation of Measuements - Hamme Excitation 4... Refeence Point Measuement In chapte 4.., the equiements egading the efeence point wee discussed. It is a good idea to stat a modal test with efeence point measuement and to pefom all the possible checks to ensue that this measuement is coect pio to continuing with measuements in all othe points. One should always keep in mind that fom low-quality data, it is not possible to obtain a eliable modal model whateve sophisticated post-pocessing is used. Fom this pespective, the cae given to the initial checks of the accuacy of data appeas to be excellent investment. The coect efeence point should exhibit the following featues (see Fig. 4.7): - Thee is an antiesonance behind each esonance in an FRF plot when displayed in db. - In a phase FRF plot, the phase only vaies in the ange of 8. - In the plot of imaginay pat of inetance o eceptance and in the plot of eal pat of mobility, all the peaks ae of the same signs. Thee ae some othe equiements that apply to all measuements, not only to the efeence point measuement. Howeve, if they ae satisfied fo the efeence point measuement, they ae usually satisfied fo all the othe measuement as well. These ae the following: - Coheence function (eq..) should have the value as close to as possible. Geneally, this cannot be achieved in the whole fequency ange. Usually, even fo - 9 -

93 good quality measuements, coheence in antiesonances is fa below (see Fig. 4.7) as a esult of the fact that the level of the signal is at these fequencies compaable to the noise level. On the contay, nea esonances the coheence function use to be close to even fo mesuements of lowe quality. - Nyquist plot should daw a distinct cicle section fo each esonance. In Fig. 4.7 (below) it can be seen that one of the modes (it is the fist mode actually) didn't daw a vey distinct cicle. This is the sign of the leakage eo and elated insufficient fequency esolution of measuement. (Of couse, it has its effect to coheence function as well.) This case will be discussed in moe detail in chapte afte leaning about modal cicle popeties. [db/, (m/s²)/n]fequency Response H(Rectangle ( ).+Z, Rectangle ().+Z) (Magnitude) Modal : Measuement : Inpu t : Moda l FFT Analyze 5 4 Mag H jj (f) [db] [(m/s²)/n] Imag H jj (f) [kg - ] k 8 Fequency Response H(Rectangle ( ).+Z, Rectangle ().+Z) (Imaginay Pat) Modal : Measuement : Inpu t : Moda l FFT Analyze ,k,6k k,4k,8k 3,k [Hz] [Degee] Fequency Response H(Rectangle ( ).+Z, Rectangle ().+Z) (Phase ) Modal : Measuement : Inpu t : Moda l FFT Analyze phase H jj (f) [ ] 4 8,k,6k k,4k,8k 3,k [Hz] [ ] Coheence (Rectangle ().+Z, Rectangle ().+Z) Modal : Measuement coheence : Inpu t : Moda l FFT Analyze,8,6,4, 8 m 6 m 4 m m 4 8,k,6k k,4k,8k 3,k [Hz] 4 Modal : Measuement : Input : Modal FFT Analyze Im H ij (f) 4 8,k,6k k,4k,8k 3,k [Hz] 3 Nyquist plot Re H ij (f) Fig. 4.7 Checking of Accuacy of Refeence Point Measuement 4.3 Expeimental Modal Analysis - Modal Paametes Identification Afte acquiing all the data by pefoming measuements at all the DOFs, thus obtaining eithe one ow o one column of the FRF matix, modal analysis softwae is to be used to

94 post-pocess this data. This pat of modal test is called expeimental modal analysis as this is the stage of the expeimental appoach coesponding to the stages called modal analysis also in theoetical appoach. In both cases, modal analysis leads to identification of modal popeties of the system. Howeve, it should be noticed that these two pocesses ae somehow diffeent: expeimental appoach deals with cuve-fitting theoetical expessions to the actual measued data, while theoetical analysis deals with the eigenvalue poblem. Nowadays, a lot of softwae packages fo the expeimental modal analysis exist and the analyst is not supposed to cay out this stage without softwae suppot. In each softwae package, seveal methods fo post-pocessing of the measued data ae at disposal, and it is the esponsibility of the analyst to choose the most appopiate method fo each application. In inceasing complexity, the methods involve the analysis of a pat of a single FRF cuve encompassing single esonance, then of a complete cuve encompassing seveal esonances and, finally, of a set of many FRF plots all on the same stuctue. In evey case, howeve, the task undetaken is basically the same: to find the coefficients in a theoetical expession fo the FRF which most closely matches the measued data. This task is most eadily tackled by using the patial faction seies-fom fo the FRF, as developed in chapte 3 fo diffeent types of system. The paticula advantage of the seies-fom FRF appoach is that the coefficients thus detemined ae diectly elated to the modal popeties of the system unde test, and these ae geneally the vey paametes that ae sought. In this text, only thee of all the existing methods will be descibed Single Degee-of-Feedom Appoach Thee ae seveal modal analysis methods that exploit the same basic assumption: that in the vicinity of esonance, the entie esponse of the system is dominated by the neaest mode. These methods can be futhe divided into: - those based on an assumption that all the esponse is given by this single mode (e.g. the simplest method call peak-amplitude o peak-picking) - those based on an assumption that contibution of othe modes ae epesented by a single appoximation (e.g. the cicle-fit method) Peak-picking method This is a method that woks adequately fo stuctues whose FRFs exhibit well sepaated modes which ae not so lightly damped that accuate measuements at esonances ae difficult to obtain but which, on the othe hand, ae not so heavily damped that the esponse at a esonance is stongly influenced by moe than one mode. The applicability of this method is limited, but it can be useful in moe difficult cases fo obtaining initial estimates to the paametes equied, theeby speeding up moe geneal cuve-fitting pocedues which equie stating estimates. This method is also used in identification of opeational deflection shapes. Its application is as follows:

95 ) On the FRF plot, the individual esonant peaks ae identified and fequencies with maximum esponse ae conside to be natual fequencies Ω. ) Local maximum value H of FRF at the natual fequency is noted. 3) 3dB bandwidth ω is detected and damping value is detemined fom half-powe points - ω a and ω b (eq. 3.4): ( ω ω ) ω ζ a b η Ω Ω 4) An estimate fo the modal constant can be obtained by assuming that the total esponse in the esonant egion is attibuted to a single tem in the geneal FRF seies (eq. N A jk 3.): α jk ( ω) fo ωω Ω ω + iη Ω η Modal constant can be found fom H A η Ω, thus A H Ω η Cicle-fit Method Fo the geneal SDOF system, a Nyquist plot of fequency esponse popeties poduces cicle-like cuves and in two special cases it poduces an exact cicle (mobility of the viscously damped system and eceptance of the hysteetically damped system). The MDOF systems also poduce Nyquist plots of FRF data which include sections of nea-cicula acs coesponding to the egions nea the natual fequencies. This chaacteistic povides the basis of one of the most impotant types of modal analysis - SDOF cicle-fit method. In this text, the descibed pocedue will be based on a system with stuctual damping and thus the eceptance fom of FRF shall be used. Howeve, if it is equied to use a model incopoating viscous damping, then mobility fom of FRF should be used. Although this gives a diffeent geneal appeaance to the plots - as they ae otated by 9º on the complex plane - most of the following analysis and comments apply equally to both cases. Some of the moe disciminating modal analysis packages offe the choice between the two types of damping and simply take eceptance o mobility data fo the cicle-fitting accoding to the selection. The cicle-fit method exploits the fact that in the vicinity of a esonance, the behaviou of most systems is dominated by a single mode. Algebaically, this means that the magnitude of the FRF is effectively contolled by one of the tems in the seies, that being the one elating to the mode whose esonance is being obseved. The assumption can be expessed as follows. Fom eq. (3.): α jk ( ω) A N s jk s Ωs ω + iηsωs This can be ewitten, without simplification, as N A jk s A jk α jk ( ω) + Ω ω + iη Ω Ω ω + iη s,s s sωs (4.a) (4.b)

96 The SDOF assumption is that fo a small ange of fequency in the vicinity of the natual fequency of the th mode, the second of the two tems in (4.b) is nealy independent of fequency ω, and the expession fo the eceptance may be witten as: α jk A ω Ω (4.) Ω ω + iη Ω jk ( ω) + B jk The total eceptance plot may be teated as a cicle with the same popeties as the modal cicle fo the specific mode in question but which is displaced fom the oigin of the complex plane by an amount detemined by the contibution of all the othe modes. It cannot be said that othe modes ae unimpotant o negligible - quite the evese, thei influence can be consideable - but athe that thei combined effect can be epesented as a constant tem aound this esonance. (by Popeties of modal cicle: Assuming a system with stuctual damping, the basic function unde consideation is: N α( ω) ω Ω + η i Ω since the only effect of including the modal constant A ) and to otate it (by A jk jk jk (4.3) A is to scale the size of the cicle ). A plot of the quantity α(ω) is given in Fig It can be seen that fo any fequency ω, following elationships may be witten: ( α) η ( α) Im tg( γ ) (4.4a) Re ω Ω tg Re ( ) ( α) 9 γ ( α) ( θ / ) η tg (4.4b) Im fom which is obtained: ( η tg( θ / ) ) ω Ω ω Ω (4.4c) Im (α) γ ω Re (α) ½θ a Im (α) ½θ b Re (α) θ ω ω a θ a θ b Ω ω b Fig. 4.8 Popeties of Modal Cicle

97 Diffeentiating the equation 4.4c with espect to θ yields to: dω dθ ω Ω η Ω + η The ecipocal of this quantity is the measue of the ate at which the locus sweeps aound the cicula ac. It can be seen to each a maximum value (maximum sweep ate) when ω Ω, the natual fequency of the mode. This is shown by futhe diffeentiation, this time with espect to the fequency: (4.5) d dω dω dθ po Ω ω (4.6) The above popety poves useful in analysing MDOF systems data since, in geneal, it is not known exactly whee is the natual fequency, but if elative spacing of the measued data points aound the cicula ac nea each esonance can be examined, it should be possible to detemine its value. Damping can be detemined with the help of two points on the cicle - ω a above esonance and ω b below esonance. Substituting to (4.4b) leads to: tg ( θ / ) tg( θ / ) b ω Ω η b a ωa Ω η and fom these two equations, an expession fo the damping of the mode can be obtained: ωa ωb η (4.7) Ω ( tg( θ / ) + tg( θ / ) ) a b This is an exact expession, and applies fo all levels of damping. If light damping is consideed (loss facto about -3%), the expession (4.7) simplifies to: ωa ωb η (4.8) Ω ( tg( θ / ) + tg( θ / ) ) a When the two point ae consideed fo which familia fomula is obtained: ω ω b θ a θb 9 (the half-powe points), the η (4.9a) Ω o, if the damping is not light: ωa ωb η (4.9b) Ω

98 The final popety elates to the diamete of modal cicle that is fo the quantity specified by the eq. (4.3) given as. When scaled by a modal constant added in the numeato, the Ω η diamete will be: D jk A jk Ω η (4.) and, as mentioned befoe, the whole cicle will be otated so that the pincipal diamete - the one that passes though the point petinent to the natual fequency - is oiented at an angle A jk to the negative imaginay axis. This means that if A is negative, the cicle will lie in the uppe half-plane and this is a situation that cannot aise fo a point FRF, only fo a tansfe FRF. The constant B jk fom the eq. (4.) is detemined as a distance of the "top" of the main diamete of the modal cicle fom the oigin (see Fig. 4.9). Im(α) Re(α) A jk B jk D jk Ω Fig. 4.9 Shift and Rotation of the Modal Cicle When modal paametes ae extacted fom the measued data, typically not the whole modal cicle is available. Fo a system with well sepaated modes it can be expected that each esonance will fom a lage pat of cicle, but with inceasing modal intefeence when close modes o high level of damping occus, it should be expected that only small cicula sections will be detectable (maybe 45 o 6 ). If the Nyquist plot will not fom a distinct cicula section in the vicinity of the esonance, the identification of modal paametes is poblematic. In Fig 4.7 (at the bottom) is the Nyquist plot fom measuement of a system with well sepaated modes. It is the efeence point measuement and all the othe plots in this figue indicate measuement of sufficient quality. Howeve, the Nyquist plot of the fist mode did not fom a distinct cicula pat. In this case, it is due to leakage eo and associated insufficient fequency esolution. In the figue, fou points closest to the esonance ae maked with the ed colou and it can be seen that the sweep ate aound the esonance is so high that it did not manage to fom a cicle. Fom this measuement, the accuate value of natual fequency could be identified without any poblem, but the damping estimate would not be eliable. To fix this situation, the fequency ange of the measuement should be deceased. This will cause polonging of the measuement time, consequently suppessing leakage eo and obtaining fine fequency esolution of the measuement

99 4.3. MDOF System Appoximation Methods Thee is a lot of situations in which the SDOF appoach to the modal analysis is eithe inadequate o inappopiate. Seveal altenative methods that can commonly be classified as multi degee-of-feedom appoximation can be used in these cases. One of the paticula cases is that fo vey lightly damped systems, fo which measuements in esonances ae inaccuate and difficult to obtain. These ae not of concen of this text. The opposite case epesents systems with vey closely coupled modes, fo which appoximation by a single mode is inappopiate. By closely coupled modes ae meant those modes whose natual fequencies ae vey close to each othe o which ae elatively heavily damped o both. With these systems, the esponse is not detemined by a single mode (o by one membe of FRF seies) even in esonance. Hee, eithe a simple extension of a SDOF method o a geneal appoximation appoach can be used. Its pinciple will be biefly descibed below. Geneal appoximation appoach The individual measued FRF data will be denoted as: m m α ( ω ) α (4.a) jk l l while the coesponding "theoetical" values will be denoted as α jk A l (4.b) M ( ω ) αl whee the coefficients detemined. The membe R ω l M jk m s jk + R s m Ω ω + η Ω ω s l i s s K jk l R jk A, A,..., Ω jk jk, Ω,..., η, η,..., R K jk a R M jk ae all to be epesents the effect of low-fequency modes (those that lie below the lowe limit of the measued fequency band) and the membe R K jk epesents the effect of high-fequency modes (those that lie above the uppe limit of the measued fequency band). An individual eo, ε l, can be defined as ε l m ( α α ) l l (4.) and this will be expessed as a scala value: E l ε (4.3) l If a weighting facto w l is added to each fequency point unde investigation, the pocess of appoximation has to detemine the values of the unknown coefficients in (4.) so that the oveall eo

100 E p l wl El (4.4) is minimized. This is achieved by deivative of the expession (4.4) with espect to each of the unknowns sepaately, thus ceating a set of as many equations as is the numbe of unknowns, each in the fom: de dq ; q A, A,..., etc. (4.5) jk jk A set of equations ceated like this is not, unfotunately, linea fo many of the coefficients (all paametes Ω s a η s ) and thus it can not be solved diectly. Theefoe, vaious algoithms use thei own pocedues and apply vaious simplifications. Most of them use some fom of iteative solution, some of them use lineaization and nealy all of them ely on good initial estimates. 4.4 Modal Model Whicheve of the appoximation method has been used, the appoximation pocess should esult in a consistent modal model. When global methods ae used, the consistent model is thei diect output. But when simple SDOF methods have been used, some additional steps should be done such as aveaging of natual fequencies and modal damping values obtained fom the individual FRF chaacteistics. Howeve, all these steps ae built-in in softwae packages fo expeimental modal analysis and the analyst need not to cae fo them. - spectal matix: eigenvalues - natual fequencies and damping - ae on the diagonal - modal matix: its columns ae eigenvectos petinent to individual natual fequencies 4.4. Pesentation of the Obtained Modal Model The actual output fom modal analysis softwae can be epesented in the fom of tables. Such tables (esults fom StaStuct softwae) ae shown in Figs. 4. and 4.. In Fig. 4. is a table with the estimated natual fequencies and damping values. Damping is stated in two foms thee: damp [Hz] detemines one half of 3dB band while damp [%] is the damping atio multiplied by. Othe softwae packages may give damping values in diffeent foms, e.g. as loss factos η. In Fig. 4. is a table of mode shapes whee the individual mode shapes ae given numeically - in the fom of elative displacements and phase angles in the individual degees of feedom. In the displayed table, values fo the fist mode fo the few fist points can be seen. Data oiginates fom the measuement when a tiaxial acceleomete was used so that each point has 3 DOF - diections X, Y and Z. Softwae packages usually enable to expot and impot data files in the standad Univesal file fomat (UFF) and many of them enable data exchange with othe softwae packages as well, not only fo expeimental modal analysis, but also fo a finite element analysis (quite common is communication with ANSYS softwae etc.). - -

101 Pesenting esults in a table fom may be useful, but in the case of mode shapes it is not vey tanspaent and it is only used fo numeical compaison of mode shapes obtained fom expeiment and fom computations (fo moe details see chapte 4.4.3). Fo mode shape inspection and assessment, its dawing, eithe static o animated, is used. All softwae packages fo modal analysis enable to display mode shapes in animation and some of them enable thei expot in AVI fomat as well. If the mode shape has to be pesented in the pinted fom, it is only possible to use static pictues, e.g. such as in Fig hee, the two utmost positions of the 3 d mode shape fom the table in Fig. 4. ae shown; the undefomed stuctue is in ed, the mode shape in black colou. Fig. 4. STAR Softwae Results - Table of Natual Fequencies and Damping Values Fig. 4. Static Display of the Mode Shape - -

102 Fig. 4. STAR Softwae Results - Table of Mode Shapes 4.4. Veification of the Obtained Modal Model Just the same as the accuacy of the efeence point measuement was checked befoe the modal test poceeded to all the othe measuements, it is now appopiate to eview, at least biefly, the accuacy of the obtained modal model, pio to compaing it with anothe model, mostly with a computational model obtained by finite element calculations. If a stuctue whose modes ae well sepaated is measued (and moeove, if it is measued feely suppoted), pobably no poblems with modal paametes identification occu and the obtained modal model will be most likely coect. When a moe complex stuctue with coupled modes and with a lage amount of (nonpopotional) damping is measued, the pocess of modal paametes extaction will be moe complicated. It is often necessay o at least useful to ty vaious appoximation methods and choose the one that gives the best esults. But which will be the best to choose? If the extacted mode shapes ae inspected visually in animation, they should exhibit systematic motion. Some of the "mode shapes" could exhibit chaotic motion, could be something like "bumpy", etc. If this happens, thee ae two possibilities: - It is so-called computational mode o false mode. If the softwae is asked to extact the mode (o two o moe modes) in the given fequency ange, it always meets this - -

103 equiement. But it might happen that if, let's say, thee modes ae equied and thee ae only two tue modes in the fequency band, the thid extacted mode is a false mode. To decide which of the two above possibilities apply fo the paticula case, the analyst should mainly ely on his expeience, but thee ae some possibilities that could help, e.g.: - If computationally deived modal paametes ae at disposal fo the given stuctue, the knowledge of how the mode shapes should look like is available and all the edundant modes may be excluded fom the measued set. It sounds logical, but it is not so clea: When the stuctue is measued suppoted othewise than in fee conditions, the actual bounday conditions could be diffeent fom that used in computation and, consequently, the computed mode shapes could diffe as well. This situation could be discoveed by obtaining nice smooth mode shapes fom the measuement that ae missing in the computational model. In this case, eo will be most likely on the computational side. In any case, the knowledge of theoetical mode shapes is helpful in ejecting the false mode shapes fom the expeimental model (o in ealizing that some of the modes should be identified moe caefully). - Complexity of the mode shapes could be examined. It means to examine to which extent the displacements in the mode shapes ae complex numbes. Geneally, it can be said that the less complex the mode shape is, the moe pobably it is coect. This assumption applies especially fo lightly damped stuctues measued in fee conditions. Fo such stuctues, complex mode shape may not occu. On the contay, when a stuctue is heavily damped and the damping is non-popotionally distibuted (e.g. a stuctue compising ubbe pats), high complexity of mode shapes is expected and it might not signify an appoximation eo. A complex mode shape is easily identified in animation by its waving movement. This visual effect oiginates fom the fact that the displacements in the individual points do not each thei utmost positions simultaneously. Consequently, nodal lines change thei positions. Thus, complex mode shapes cannot be displayed using static pictues, it is only possible to display them in animation. A poo attempt to display them statically can be done by so called nomalization - displacements ae tansfomed to the eal numbes by assigning zeo to phase angles of those displacements that have phase angles moe close to zeo (thus obtaining positive amplitude) and by assigning 8 to phase angles of those displacements that have phase angles moe close to 8 (thus obtaining negative amplitude). Doing this, a eal appoximation of the complex shapes is obtained. It can be seen in Fig 4. that the mode shape stated thee is vey slightly complex, as the phase angles ae mostly vey close to o 8. If two o moe mode shapes look simila, it again might be an appoximation eo. It might happen when the softwae identifies false modes fom the eason descibed above (moe modes ae equied in the fequency bands than eally exist). In this case, it is woth to examine the MAC matix. It will be discussed in detail in the next chapte, because its main pupose is in compaing of two models obtained by diffeent methods, but it can be used also fo a single model only. In this case, all values on the main diagonal will be equal to and all the othe values should be numbes close to zeo. If a numbe close to appeas out of the - 3 -

104 main diagonal, it most likely indicates a false mode. This applies fo modes whose fequency ae quite close. Diffeent situation occus when two mode shapes look vey simila, but thei fequencies ae quite apat fom each othe. In this case, the mode shapes diffe vey likely in something that was not measued (e.g. movement in diection Y if only measuements in diections X and Z wee pefomed). The geometical model used fo modal test is usually quite simple and it could easily happen that it would not captue all the details of the movement. Thus, two modes could appea as being the same even if they ae not. In this case, it is not a measuement o appoximation eo, but the consequence of insufficiently fine model Compaison of the Expeimentally and Computationally Deived Models In chapte. it was stated that a modal test is vey often pefomed with the aim to compae dynamic behaviou of the stuctue deived fom computations and those that is eally obseved in pactise. Sometimes, this pocess is called veification (updating) of the theoetical model and it takes seveal steps: - compaison of the dynamic popeties - expeimental vs. theoetical model - quantification of the diffeences between those two data sets - making changes in one of the sets of esults in ode to achieve bette coespondence If this is achieved, the theoetical model can be consideed as veified and thus it is eady fo use in a subsequent analysis. Compaison of dynamic popeties of the expeimental and theoetical models can be pefomed fo all the thee types of dynamic models (spatial, modal and esponse). Fom the opposite pocedues used in expeimental and theoetical analysis it is clea that what is the most convenient type of model fo a theoetical analysis would be wost accessible fo an expeimental analysis and vice vesa. A theoetical analysis stats with the spatial model, but it is quite difficult to achieve it though the measued data - it equies application of additional data pocessing pocedues that ae usually not the pat of softwae packages. On the contay, fequency esponse functions that ae diectly obtained by measuements ae elatively tedious to acquie in the theoetical model. Theefoe, the most common fom fo compaing the two sets of data is the modal model. Seveal possibilities of compaing modal models ae stated below Compaison of Natual Fequencies Quite obvious is to compae measued vs. calculated natual fequencies. This is often done by a simple tabelation of both esult sets, but a moe useful fom is to daw the expeimental value vs. the theoetical value fo all the modes included into compaison, as is shown in Fig In this way, not only the degee of consistency between the two sets of esults can be seen, but also the natue and possible causes of the discepances. The dawn points should lie on the line with the slope o nea this line. If they lie nea a line with the slope othe than, the cause of this discepancy is quite sue in incoect mateial popeties used fo computation. If the points ae widely spead along the line, thee is a seious eo in the model that epesents the stuctue and a basic eassessment should be done. Paticula - 4 -

105 attention has to be paid to the case whee the points deviate slightly fom the ideal line, but in a systematic athe than andom manne, since this situation implies that thee is a specific featue which is esponsible fo the deviation and can not simply be attibuted to expeimental eos. If the scatte is small and andomly distibuted along the line of the 45 slope, the values ae supposed to oiginate fom the nomal pocess of modelling and measuement. Geneally, the highe (in ode) is the fequency, the lage is the diffeence between theoetical (computed) and measued values. The diffeences should have the tendency that the theoetical values of fequencies ae highe than the measued ones, because usually damping is not included in the theoetical values wheeas measued fequencies ae always damped and thus of lowe values. Compaison sovnání vlastních of Natual fekvencí Fequencies 5 computed vypočtené [Hz] [Hz] measued naměřené [Hz] Fig. 4.3 Gaphical Compaison of Natual Fequencies When most of the points ae nea the ideal line (blue points in Fig. 4.3) and some points ae fa fom this line (ed point in Fig. 4.3), it could by the case that diffeent modes ae compaed. This is quite a fequent mistake that students allow to happen. It should be emphasized that if modes ae identified fom measuements and modes in the same fequency band ae identified fom theoetical analysis, it cannot be guaanteed that the st mode may be assigned to the st one fom the diffeent analysis, the th to the th etc. It could happen that: - The ode of two modes close in fequency is changed in the two compaed models. - One mode is missing in the expeimental model (maybe because the efeence point was at the nodal line of that mode), and at the same time one mode is missing in the theoetical model (because its fequency in theoetical analysis is above the fequency ange in which compaison is pefomed). Thus, the numbe of modes is the same in both models, but not all of them ceate petinent pais to be compaed. It is clea fom the above that the identified modes cannot be compaed automatically, because in all the methods discussed hee it would lead to eoneous conclusions. It is - 5 -

106 essential to check the mode shapes visually and match the petinent modes togethe pio to stating with any compaison of the two models. Doing this, the isk of compaing two modes that do not match is eliminated. This isk is quite high, as it is quite a common situation that some modes ae missing in the expeimental model. Nevetheless, if the modes ae coectly matched, the fact of a missing mode in the expeimental model cannot be consideed as a mistake that would pevent good compaison of the two models o veifying the theoetical model. It is quite sufficient to veify and update the theoetical model in accodance with those modes that ae at disposal (it is supposed that some highe modes might be missing athe than lowe ones) Gaphical Compaison of Modal Shapes One of the possibilities how to compae the mode shapes is to daw the defomed shapes fo both models - theoetical and expeimental - and ovelay the two pictues. A disadvantage of this appoach is that even though the diffeences can be seen, it is difficult to intepet them and often the obtained pictues ae vey confusing as thee is a lot of infomation in them. A moe convenient method is simila to that fo gaphical compaison of natual fequencies. Mode shape elements ae dawn to the x-y plot, again measued vs. computed, and in an ideal case, they should again be scatteed nea the line with the slope. Fo this compaison, it is necessay to choose those points fom the theoetical model (which usually has much moe degees-of-feedom than expeimental model) that coincide with the expeimental model. Gaphical Compaison of the Mode Shapes 5 Theoetical Mode Shapes mode # mode #3 mode #4-5 - Expeimental Mode Shapes Fig. 4.4 Gaphical Compaison of the Mode Shapes The natue of deviations fom the ideal state can again quite clealy indicate the cause of the discepancy: if the points lie nea a line with the slope othe than, then one of the compaed mode shapes is not nomalized to the unity modal mass o thee is some othe scale - 6 -

107 eo. If the scatte is lage, then thee is a sevee inaccuacy in one of the data sets and if the scatte is excessive, two diffeent modal vectos that do not belong to the same mode might be compaed. In Fig. 4.4, compaison of thee mode shapes, each of diffeent colou, is shown. It can be seen that the st and the 3 d shape exhibit a good accodance, while this is wose fo the 4 th shape. This is quite a typical esult, as with inceasing ode of modal shapes it is moe difficult to achieve accodance Numeical Compaison of Mode Shapes As an altenative to the above intoduced gaphical appoach, numeical compaison of the mode shapes can be used. The expessions (4.6) and (4.7) below suppose that mode shapes data may be complex. The expeimental (measued) mode shape is denoted as {φ X } and the theoetical (computed, analytical) mode shape is denoted as {φ A }. These concepts ae in fact useful fo all types of compaisons, not only expeiment vs. theoy, but they can be used to compae any pai of mode shape estimates. The fist citeion deals with the quantity called Modal Scale Facto (MSF) and it epesents the slope of the best line fitted to the points in Fig This quantity is defined as (two foms ae possible accoding to which of the mode shapes is taken as efeence): n * ( φχ ) ( ) j φα j j MSF ( Χ, Α) ( A,X) n * ( φα ) ( φα ) j j j n j n ( φ ) ( φ ) ( φx ) ( φx ) j j A j * X j MSF (4.6) It should be noted that this citeion povides no indication concening the quality of fitting the line to the points, but only its slope. The second citeion is called Mode Shape Coelation Coefficient (MSCC) o Modal Assuance Citeion (MAC) and povides the measue of distances of point fom the ideal line in the least squae sense. It is defined as: ( A,X) n j ( φ ) ( φ ) Χ j Α * j MAC (4.7) n n * * ( ) ( ) ( ) ( ) φx φ φ φ j X j A j A j j j and it is a scala value, although the data of mode shapes ae complex. As the modal scale facto does not indicate the degee of coespondence, the modal assuance citeion does not distinguish between a andom scatte esponsible fo the discepancies and systematic discepancies eithe. So even though these paametes ae useful means to quantify the compaison between the two data sets of mode shapes, they do not give a complete insight and should be consideed pimaily in connection with plots such as the one shown in Fig 4.. It is woth to conside two special cases: () that when two mode shapes ae identical and () that when two mode shapes diffe by a simple scala multiple. In the case () applies: * j - 7 -

108 { φ } { } X φ A fom which it can be seen that ( X,A) MSF( A,X) MSF and also that MAC ( X,A) In the case (), {Φ X } γ{φ A } and it is found out that MSF ( X,A) γ, whilst MSF ( A,X) γ but since the two modes ae still quite pefectly coelated, it still applies: MAC ( X,A) In pactice, a typical data would be less ideal than these and what is expected is that if the involved expeimental and theoetical mode shapes would eally belong to the same mode, then the MAC value would be close to, whilst if they would belong to diffeent modes, the MAC value would be close to zeo. If a set of m x expeimental modes and a set of m A theoetical modes ae taken into account, a MAC matix m X m A can be computed and displayed in a matix that would clealy indicate which expeimental mode matches to which theoetical mode. It is difficult to give exact values that should the MAC values be to ensue good esults. Geneally, the values geate than.9 should be obtained fo the same modes and values less than.5 fo the diffeent modes. Fig. 4.5 MAC Matix - Gaphical Fom It is woth to mention some possible causes of impefect esults of these calculations. Apat fom the obvious eason that the model is defective, MAC values less than can be caused by: - non-lineaities in the tested stuctues - noise in measued data that was not aveaged out - weak modal analysis of the measued data - incoect selection of DOFs included into coelation - 8 -

109 In Fig. 4.5 is an example of MAC matix that comes fom compaison of two sets of expeimental data, both fom the same stuctue, one fom a modal test when hamme excitation was used, the second when shake excitation was used. The stuctue had vey simila mode shapes of the 3 d and the 4 th mode, and these modes wee spaced only.4 Hz in fequency. Thei identification was vey difficult and it can be seen fom the MAC matix whee the element (3,3) is less than.8 and the element (3,4) is geate than.4. Note.: MAC citeion has aleady been mentioned in the chapte If MAC citeion is applied on one set of esults, the values on the main diagonal ae all exactly equal to and significant ae the out-of-diagonal values only - they should be close to zeo. If they ae not, thee is a isk of false mode existence

110 5. Opeational Deflection Shapes (ODS) The issue of visualization the opeational deflection shapes belongs to the aea of vibation diagnostics, whee it is used to solve specific poblems elated to opeation of machines. If this issue is sometimes assigned to a modal analysis, it is pobably because it uses the same tools, both technical and softwae. Basic equipment fo opeational deflection shapes identification compises a dual o multi-channel analyze, set of two o moe acceleometes and fo one of the ODS types, so called spectal ODS, softwae fo expeimental modal analysis can also be used. Opeational deflection shapes ae used as a diagnostic tool fo visualization of the actual dynamic behaviou of machines. This visualization seves fo bette undestanding of what is happening with the given equipment and thus povides a basis fo decisions on poblem solving. If the vibation levels ae unsatisfactoy, the ODS is to find the "weak point" of the stuctue at given opeating conditions. Application of ODS is indicated especially when the stuctue vibates on a single pedominant fequency - in this case it is possible that the system etuned itself close to the esonance. This can happen by educing the stiffness of the suppot due to mechanical looseness - it may be a loose baseplate, ancho bolt cacked, etc. If defects of this type ae suspected, they can be easily detected using ODS visualization, if well-defined model is used. Convesely, this method is not suitable if vibations ae excessive in a wide ange of opeating conditions. Anothe possible application of ODS is veification of foced esponse simulation which was pefomed on a modal model, obtained expeimentally o by calculation. Opeational deflection shapes visualize vibation behaviou of stuctues unde the actual opeational conditions. Thus, opeational deflection shapes ae about identification of the foced vibations. The system and the input excitation foces ae not obseved, no assumptions about the linea behaviou of the system ae made, only the vibation esponse of the system is measued (see Fig. 5.). The advantage of the ODS is that the dynamic behaviou of the stuctue unde the actual opeational conditions and unde actual bounday conditions is identified. The disadvantage is (in compaison with a modal test) that no model of the system is obtained and thus no estimates about its esponse unde diffeent conditions could be made. F(ω) X H(ω) i (ω)... measued vibation esponse signals Fig. 5. Signals Measued fo ODS - -

111 Opeational conditions efes to otational speed, load, pessue, tempeatue, flow, etc. These conditions could be stationay, quasi-stationay (slightly vaying otational speed, unup, un-down) o tansient (e.g. a piece of a ock falls on the measued stuctue). Accoding to the opeational conditions, the type of ODS should be chosen. Basically, thee ae two types of ODS:. Spectal ODS - deflection shapes on the individual fequencies o odes ae obtained. When the opeational conditions ae stationay, fequency specta ae used; when they ae quasi-stationay, ode specta ae used. As a esult of measuements, elative amplitudes and phase angles of the individual degees-of-feedom on the individual fequencies (o ode components) ae obtained. The individual DOFs could be measued simultaneously (if enough of acceleometes and analyze channels ae at disposal) o successively. In pinciple, two acceleometes ae sufficient fo spectal ODS measuements - one of them is the efeence and is placed in a fixed position and the second one captues the esponse in each of the DOFs.. Time ODS - defomation pocess as a function of time is obtained. This type of ODS is used fo tansient signals. It is clea that when tansient pocess is obseved, it is not possible to measue the individual DOFs successively, but they should be measued all at once. This leads to highe demands on technical equipment - multiple sensos, multichannel analyze, softwae and equipment must be special, softwae fo modal analysis cannot be used. To view the time ODS, signals fom acceleometes ae integated to povide speed o displacement. The pocedue fo application of ODS is vey simila to that fo a modal test. It includes following steps: - pepaation - measuements - post-pocessing of the measued data As fo a modal test, a geometical model should be pepaed fo measuing opeational deflection shapes. It is usually simple than that fo modal test, because ODS measuements ae focused on identifying how the machine behaves as a igid body with espect to the base plate o to othe pats of the machine-set, athe than on defomation of the machine itself. A smalle numbe of measued DOFs is sufficient fo this task. Howeve, it is necessay to include points on the suppoting stuctue of the measued stuctue, on the base plate, etc. because the most elevant infomation about the natue of the defect often oiginates fom the infomation about the movement of the stuctue with espect to the base. On the geometical model, degees-of-feedom should be defined. If tiaxial acceleomete is used, all point will be measued in thee DOFs (X, Y and Z), with mono-axial acceleomete it should be decided, which diections ae to be measued. Then, efeence DOF (position + diection) should be defined. In ode to obtained highquality measuements, it is equied fo efeence DOF to exhibit esponse of sufficiently high levels at all fequencies that ae of inteest. A efeence acceleomete (athe than a foce tansduce in modal test) is placed to the efeence DOF (see Fig. 5.). - -

112 Using a efeence acceleomete, so-called tansmissibility, T(f), is measued. It is a function vey simila to fequency esponse function, but it has the signal fom a efeence acceleomete athe than the signal fom a foce tansduce in the denominato: X i (f ) G xix G i xix ef T(f ) X (5.) ef (f ) G xix G ef x ef x ef The second possibility how to povide the efeence signal is to use a phase efeence instead of a efeence acceleomete. It is the simplest configuation fo ODS measuements as possible - single-channel analyze with the possibility of phase measuements is sufficient. In this case, autospectum with the assigned phase is measued instead of tansmissibility. Pio to stating complete measuements, it is useful to check if all the components of the measuement chain ae set coectly, fo example, by checking the tansmissibility function in the efeence degee-of-feedom. As it is a ation of two identical signals, it should be close to unity in the whole measued fequency ange. In Fig. 5.3, tansmissibilities measued at the efeence point ae shown - dak line is fo tansmissibility in efeence DOF, light lines ae fo tansmissibilities in efeence point, but in othe two diections. A tiaxial acceleomete was used fo that measuements. efeence acceleomete acceleomete fo esponse measuements Fig. 5. Example of Tansmissibility Measuements - -

113 [(m/s²)/n] tansmissibility in the efeence DOF (point and diection) Fequency Response H(Response, Foce) (Magnitude) Modal : Measuement : Input : Modal FFT Analyze tansmissibility in the same point, othe two diections [Hz] Fig. 5.3 Tansmissibility in the Refeence Point Afte all measuements ae done, post-pocessing of the measued data is pefomed eithe in special softwae fo ODS o in modal analysis softwae. If the latte is used, the peak-peak appoximation method is used to obtain opeational shapes (see chapte 4.3..). Stictly speaking, no appoximation is associated with this method - amplitudes at elevant fequencies ae ead only. Fo otating machiney, the elevant fequency used to be the otational speed and its hamonics. Results may be obtained in table fom (as in Fig. 4.), but much moe illustative is to use an animated display of the obtained shape. Opeational deflection shapes (similaly to complex mode shapes) cannot be accuately displayed in static position, because diffeent points do not each the utmost positions at the same time - it can be said that the movement of the stuctue follows the excitation foce

114 6. Opeational Modal Analysis (OMA) Opeational modal analysis is a method that enables to obtain a modal model based only on esponse measuements. The measuement pocedue is the same as fo tansmissibility measuements, but the mathematical backgound behind the opeational modal analysis is much moe complex. It can be said that only a massive incease in computational capacity of computes in ecent yeas has facilitated the emegence and expansion of this method. Opeational modal analysis, as the name suggests, is pefomed unde actual opeating conditions of the measued machine o device, and only the vibation esponses ae measued, not the input excitation foces. Nevetheless, a valid modal model of the measued system can be obtained by this method, even if unnomalized. This method is paticulaly suitable fo modal tests of lage stuctues, atifical excitation of which is diffficult o even impossible. It is successfully used fo modal tests of bidges, buldings, off-shoe platfoms, etc. In these cases, natual excitation fom taffic, wind o sea waves is boadband excitation, and this is the main pe-equisite fo a successful modal test. Nowadays, OMA is beginning to apply also in engineeing applications such as in modal tests of otating machiney, on-oad and in-fly tests, etc. In these cases, the kind of natual excitation constitutes a limitation fo the use of this method. Fo instance, the main excitation foce in otating machiney oiginates fom the otational speed and excitation on othe fequencies is vey small. Unde these cicumstances, eliability of the obtained modal model would be quite unsatisfactoy. The situation can be impoved by applying an additional boadband excitation that is not measued (see Fig. 6.). acceleomete fo esponse measuements efeence acceleomete additional boadband excitation with hamme that is not measued Fig. 6. Measuement Setup fo Opeational Modal Analysis with Additional Excitation The advantages of the opeational modal analysis in compaison to the classical modal test can be summaized as follows: - 4 -

115 No elaboate fixtuing of stuctues, shakes and foce tansduces: - no test igs needed - shot setup time - no dynamic loading fom shakes and stinges - no cest facto poblems as when using hammes - no potential destuction of stuctue Modal model can epesent eal opeating conditions: - tue bounday conditions - actual foce and vibation levels Only natual andom o unmeasued atificial excitation equied. No intefeence o inteuption of daily use. Modal testing can be applied in paallel with othe applications. OMA is inheently a poly-efeence method, as the excitation acts in multiple positions simultaneously. Identification of multiple modes is possible. Disadvantages of the OMA method should also be mentioned, namely: The obtained modal model is unscaled, thus no foce esponse and modification simulations ae possible. The method equies moe opeato's skills, pe-analysis is often needed. Lage time histoies might be equied: - moe data handling capacity needed - highe computational powe needed In opeational modal analysis, some assumptions should be accepted. They may be divided into theoetical (mathematical) and pactical. Theoetical assumptions: Stationay input foce signals can be appoximated by filteed zeo mean Gaussian white noise. - Signals ae completely descibed by thei coelation functions. - Synthesized spectal densities and coelation functions ae simila to those obtained fom expeimental data. Pactical assumptions: Boadband excitation All modes must be excited (as in classical modal test). And it is the the fact that tue boadband excitation is not always achieved that causes poblems in data analysis. If the stuctue is excited by white noise only, then all the peaks in esponse spectum indicate modes and the spectum contains only infomation about the stuctue itself (see Fig. 6. at the top). Howeve, this is geneally not the case in pactice. If the spectum of the excitation foce was measued (what is not done in opeational modal analysis), it would become evident that it is not flat but it has its spectal distibution. This - 5 -

116 fact demonstates itself in the spectum as additional modes (see Fig. 6. in the middle). But the esponse spectum is futhe contaminated by othe factos, such as influence of noise, and also fequencies of otational speed and its hamonics occu in the spectum. All these esults into the esponse spectum shown in Fig. 6.. at the bottom. It is obvious that the mathematical appaatus, howeve pefect it could be, is not able to distinguish the tue stuctue modes fom the false modes oiginating fom the uneven distibution of the exciting foces. To distinguish them, opeato's expeience o peliminay knowledge about the eal modes of the stuctue obtained fom theoetical model ae necessay. When otating machiney is examined, it is advisable to know at least the opeational deflection shapes pio to stating with the opeational modal analysis. OMA would most likely esult in a mixtue of tue stuctual modes and opeational deflection shapes, whee the latte would dominate and so it is woth to know about them. If they wee not measued in advance, they would be at least expected at the fequencies of otational speed and its hamonics. spectum of white noise esponse spectum input foce spectum esponse spectum otating pats esponse spectum input foce spectum noise Fig. 6. Response Spectum in Opeational Conditions - 6 -

117 6. Identification Methods The mathematical appaatus of an opeational modal analysis is quite complex and it is beyond the possibilities of the autho of this text to descibe the theoetical backgound in detail. Theefoe, the methods ae intoduced hee only vey biefly. Softwae package "PULSE Opeational Modal Analysis" that is used at the Depatment of Mechanics has implemented two techniques of modal paametes identification: - non-paametic - implemented as fequency domain decomposition (FDD). Modal paametes ae estimated diectly fom cuves, functional elationships o tables. - paametic - implemented as stochastic subspace identification (SSI). Modal paametes ae estimated fom a paametic model fitted to the signal pocessed data. Pocedue of the fequency domain decomposition method is as follows: - powe spectal density (PSD) estimation (see Fig. 6.3) - singula value decomposition (SVD) of PSD - identification of single degee-of-feedom (SDOF) models fom SVD - modal paamete identification fom SDOF models Singula value decomposition of PSD is descibed by the expession: G yy (iω ) j k d k iω λ j k φ k φ T k * d k + iω λ j * k φ * k φ *T k k s k φ k φ T k + s φ k * k φ *T k (6.) whee s k is constant and eal fo the given fequency. SVD is pefomed fo each fequency. [db ( m/s ) / Hz] Magnitude of Spectal Density betw een Response (7Z-) and Response (7Z-) of Data Set Measuement Fequency [Hz] Fig. 6.3 Powe Spectal Density Estimate - 7 -

118 [db ( m/s ) / Hz] 4 Singula Values of Spectal Density Matix of Data Set Measuement Fequency [Hz] Fig. 6.4 Singula Values One of the paametes that the opeato sets in the OMA softwae is the numbe of the equied singula values (thee ae thee singula values in Fig. 6.4). The theoy says that if thee is a peak in the uppe line (geen), at least one mode exists at that fequency, if thee is a peak in the second line (ed), at least two mode exist at that fequency, etc. (Autho's shot expeience with OMA has not poved this theoy yet.) 6. Pesentation of Results As the classical modal analysis, the opeational modal analysis leads to a modal model, i.e. modal and spectal matices ae obtained. A modal model can be consideed valid only if all false modes and opeational deflection shapes ae emoved fom it, as discussed above. Fig. 6.5 Table of Natual Fequencies and Damping Ratios Resulted fom OMA - 8 -