Ch10 experimental modal analysis

Applied Structural and Mechanical Vibrations Theory, Methods and Measuring Instrumentation 10 Experimental modal analysis 10 1 Introduction In almost every branch of engineering, vibration phenomena h.

10 Experimental modal analysis 10.1 Introduction In almost every branch of engineering, vibration phenomena have always been measured with two main objectives in mind: the first is to determine the vibration levels of a structure or a machine under ‘operating’ conditions, while the second is to validate theoretical models or predictions Thanks to the developments and advances in electronic instrumentation and computer resources of recent decades, both types of measurements can now be performed effectively; one should also consider that the increasing need for accurate and sophisticated measurements has been brought about by the design of lighter, more flexible and less damped structures, which are increasingly susceptible to the action of dynamic forces Experimental modal analysis (EMA) is now a major tool in the field of vibration testing As such, it was first applied in the 1940s in order to gain more insight in the dynamic behaviour of aircraft structures and, since then, it has evolved through various stages where the terms of ‘resonance testing’ or ‘mechanical impedance’ were used to define this general area of activity Modal testing is defined as the process of characterizing the dynamic behaviour of a structure in terms of its modes of vibration More specifically, EMA aims at the development of a mathematical model which describes the vibration properties of a structure from experimental data rather than from theoretical analysis; in this light, it is important to understand that a correct approach to the experimental procedures can only be decided after the objectives of the investigation have been specified in detail In other words, the right questions to ask are ‘What we need to know? What is the desired outcome of the experimental analysis?’ and ‘What are the steps that follow the experimental test and for what reason are they undertaken?’ As often happens in science and technology—and this easier said than done—posing the problem correctly generally results in considerable savings in terms of time and money The necessity of stating the problem correctly is due to the fact that modal testing can be used to investigate a large class of problems— from finite-element model verification to troubleshooting, from component substructuring to integrity assessment, from evaluation of structural Copyright © 2003 Taylor & Francis Group LLC modifications to damage detection and so forth—and therefore the final goal has a significant influence on the practical aspects of what to and how to it Obviously, the type and size of structure under test also play a major role in this regard Last but not least, it is worth noting that, on the experimenter’s part, a correct approach to EMA requires a broad knowledge of many branches of engineering which, traditionally, have often been considered as separate areas of activity If we now refer back to the introduction of Chapter 7, we can once again adopt Ewins’ definitions and note that in this chapter we will proceed along the ‘experimental route’ to vibration analysis which, schematically, goes through the following three stages: the measurement of the response properties of a given system; the extraction of its modal properties (eigenfrequencies, eigenvectors and modal damping ratios); the definition of an appropriate mathematical model which, hopefully, describes within a certain degree of accuracy some essential characteristics of the original system and can be used for further analysis 10.2 Experimental modal analysis—overview of the fundamentals In essence, EMA is the process by which an appropriate set of measurements is performed on a given structure in order to extract information on its modal characteristics, i.e natural frequencies of vibration, mode shapes and damping factors Broadly speaking, the whole process can be divided into the three main phases as defined in the preceding section, which can be synthetically restated as: data acquisition modal parameters estimation interpretation and presentation of results It is the author’s opinion that the most delicate phase is the first one In fact, no analysis can fix a set of poor experimental measurements, and it seldom happens that the experimenter is given a second chance By contrast, a good set of experimental data can always be used more than once to go through phases and A modal analysis test is performed under a controlled forced vibration condition, meaning that the structure is subject to a measurable force input and its vibratory response output is measured at a number of locations which identify the degrees of freedom of the structure Three basic assumptions are made on the structure to be tested: Copyright © 2003 Taylor & Francis Group LLC The structure is linear This assumption means that the principle of superposition holds; it implies that the structure’s response to a force input is a linear combination of its modes and also that the structure’s response to multiple input forces is the sum of the responses to the same forces applied separately In general, a wide class of structures behave linearly if the input excitation is maintained within a limited amplitude range; hence, during the test, it is important to excite the structure within this range For completeness of information, It must be pointed out that there exists an area of activity called ‘nonlinear modal analysis’ whose main objective is the same as for the linear case, i.e to establish a mathematical model of the structure under test from a set of experimental measurements In this case, however, the principle of superposition cannot be invoked and the mathematical model becomes nonunique, being dependent on vibration amplitude The structure is time invariant This assumption means that the parameters to be determined are constants and not change with time The simplest example is a mass-spring SDOF system whose mass m and spring stiffness k are assumed to be constant The structure is observable This assumption means that the input-output measurements to be made contain enough information to adequately determine the system’s dynamics Examples of systems that are not observable would include structures or machines with loose components (that may rattle) or a tank partially filled with a fluid that would slosh during measurements: if possible, these complicated behaviours should be eliminated in order to obtain a reliable modal model In addition to the assumptions above, most structures encountered in vibration testing obey Maxwell’s reciprocity relations provided that the inputs and outputs are not mixed In other words, for linear holonomic-scleronomic systems reciprocity holds if, for example, all inputs are forces and all outputs are displacements (or velocities or accelerations); by contrast, reciprocity does not apply if, say, some inputs are forces and some are displacements and if some outputs are velocities and some are displacements Unless otherwise stated, we will assume in the following that reciprocity holds; for our purposes, the main consequence of this assumption is that receptance, mobility, and accelerance and impulse response functions matrices are all symmetrical Given the assumptions above, a modal test can be performed by proceeding through phases 1–3 Since there is no such thing as ‘the right way’ valid for all circumstances, each phase poses a number of specific problems whose solutions depend, for the most part, on the final objectives of the investigation and on the desired results In phase the problem to be tackled has to with the experimental setup and the questions to be answered are, for example: how many points (degrees of freedom) are needed to achieve the desired result? how we excite the structure and how we measure its response? Copyright © 2003 Taylor & Francis Group LLC In phase 2, on the other hand, the focus is on the specific technique to be used in order to extract the modal parameters from the experimental measurements This task is now accomplished by means of commercial software packages but the user, at a minimum, should at least have an idea of how the various methods work in order to decide which technique may be adopted for his/her specific application Finally, phase has to with the physical interpretation of results and with their presentation in form of numbers, graphs, animations of the modal shapes or whatever else is required for further theoretical analysis, if any is needed 10.2.1 FRFs of SDOF systems With the exception of the available electronic instrumentation and the basic concepts of digital signal analysis—which will be considered separately in the final chapters of this book—most of the theoretical concepts needed in EMA have been introduced and discussed in previous chapters (Chapters 4, and 7) whose content is a prerequisite for the present developments Nevertheless, in the light of the fact that the first step in a large number of experimental methods in modal analysis consists of acquiring an appropriate set of frequency response functions (FRFs) of the system under investigation, this section considers briefly some characteristics of these functions Consider, for example, the receptance function of an SDOF system whose physical parameters are mass m stiffness k and damping coefficient c From eq (4.42) the magnitude of this FRF is given by (10.1a) or, alternatively (eq (4.44)) (10.1b) where, as usual, we and When or have, respectively (10.2a) and (10.3a) Copyright © 2003 Taylor & Francis Group LLC Owing to the wide dynamic range of FRFs, it is often customary to plot the magnitude of FRF functions on log-log graphs or, more precisely, in dB (where the reference value, unless otherwise stated, is unity); this circumstance has also the additional advantage that data that plot as curves on linear scales become asymptotic to straight lines on log scales and provide a simple means for identifying the stiffness and mass of simple systems In fact, eqs (10.2a) and (10.3a) become, respectively (10.2b) and (10.3b) so that in the low-frequency part of the graph we have a horizontal spring line and in the high-frequency part of the graph we have a mass line whose slope is –40 dB/decade (–12.04 dB/octave, or a downward slope of –2 on a log scale) and whose position is controlled by the value of m The stiffness and mass lines intersect at a point whose abscissa is the resonant frequency of the system, i.e when the spring and the inertia force cancel and only the damping force is left to counteract the external applied force As an example, a graph of this kind is plotted in Fig 10.1 for a system with and c=1200 N s/m (implying and note that, as expected, the stiffness line is at –120 dB, meaning that Fig 10.1 Copyright © 2003 Taylor & Francis Group LLC A similar line of reasoning applies to mobility and accelerance FRFs; mobility graphs, for example, are symmetrical about the vertical axis at ωn; in the low frequency range we note a stiffness line with an upward slope of +20 dB/decade (+6.02 dB/octave, or +1 on a log scale) while in the highfrequency range there is a mass line with a downward slope of –20 dB/ decade (–1 on a log scale) Moreover, at resonance we get (10.4) implying that there is a horizontal line of viscous damping in the logarithmic representation (in this regard, the reader can verify that a horizontal line of hysteretic damping is obtained in receptance graphs) By contrast, accelerance graphs display a stiffness line with an upward slope of +40 dB/decade in the low-frequency range and a horizontal mass line in the high frequency range The graphs of mobility and accelerance for the SDOF system considered above are shown in Figs 10.2 and 10.3 Equation (10.1a) (or (10.1b)), however, does not tell the whole story Whether we consider an SDOF or an MDOF system, we know from previous chapters that FRFs are complex functions and cannot be completely represented on a standard x–y graph The consequence is that there are three widely adopted display formats: • The Bode diagram This consists of two graphs which plot, respectively, the FRF magnitude and phase as functions of frequency The graph of Fig 10.2 Copyright © 2003 Taylor & Francis Group LLC Fig 10.3 • • magnitude versus frequency is usually displayed in log(y)–log(x) scales, dB(y)–log(x) or dB(y)–linear(x) scales (but linear-linear scales are sometimes used as well); in this regard it is worth noting that plotting the amplitude ratio in dB on a linear scale is equivalent to plotting the amplitude on a logarithmic scale The real and imaginary plots These display the FRF real and imaginary parts as functions of frequency The Nyquist diagram (or polar graph) This is a single plot which displays the FRF imaginary part as a function of the real part (this format is particularly useful in many circumstances, but has the inconvenience of not showing explicitly the frequency information (Fig 4.14); this information can be given by adding captions which indicate the values of frequency) All of the above formats are generally available in commercial software packages In the Bode diagrams, the graphs of phase angles may sometimes be a source of confusion If we adopt the phasor representation of rotating vectors (Chapter 1), we have stated on a few occasions that, provided that consistency is maintained, it is somewhat irrelevant to choose the convention (clockwise rotating vector) or (counterclockwise rotating vector) However, if the forcing function is written as it is customary (eq (4.41)) to write the displacement response as so that φ, when Copyright © 2003 Taylor & Francis Group LLC Fig 10.4 Copyright © 2003 Taylor & Francis Group LLC Fig 10.4 (continued) plotted as in Fig 4.9, is to be understood as the angle of lag of displacement behind the external force, the two extreme situations being as follows: • • When When the displacement is in phase with the force and the displacement lags behind the force of π radians and By the same token, velocity is written as where φ v is the angle of lag of velocity behind force and is given by since we know that velocity leads displacement by π/2 radians When velocity leads force by π/2 so that the velocity angle of lag behind force is on the other hand, when Similar considerations apply for the acceleration phase angle which ranges from –π ( ) to zero ( ) radians In brief, in the negative exponential convention the phase angle is positive when it is an angle of lag, negative when it is an angle of lead and for an SDOF system all phase angles plotted as functions of frequency are monotonically increasing functions The same situation arises if we adopt the positive exponential convention but we write displacement, velocity and acceleration as and respectively Copyright © 2003 Taylor & Francis Group LLC Fig 10.5 Copyright © 2003 Taylor & Francis Group LLC from eq (10.39) we get (10.40) In this regard, it is interesting to note that, since implies in this case from the definition of H2 (eq (10.29)) we get (10.41) and also, since we have also (10.42) Equations (10.41) and (10.42) show that both Gee and Gmm, although they cannot be directly measured, can be calculated (strictly speaking ‘estimated’) in terms of measured quantities Case No noise in the excitation signal, noise in the response signal Equations (10.33) now read (10.43) because m(t)=0 Evaluation of H1 gives (10.44) so that H1 is insensitive to uncorrelated noise in the response signal in this case On the other hand, by taking into account eqs (10.35) and the first of eqs (10.34) (10.45a) which, by virtue of the second of eqs (10.34) can also be written as (10.45b) Copyright © 2003 Taylor & Francis Group LLC Equations (10.45) show that H2 generally overestimates the true FRF, because the term in parentheses is usually larger than unity The phase information, however, is correct because it comes from the error-free cross-spectrum As for the previous case, the precise meaning of eqs (10.44) and (10.45) is as follows: when only the response signal is contaminated by noise, H1 is an unbiased (and preferable) estimator of H, while H2 is a biased estimator of H The coherence function is now (10.46) and (10.47) Also, we can express Grr and Gnn in terms of measured quantities because and hence, from the definition of H1, (10.48) and (10.49) Case Noise in both excitation and response signals Equations (10.33) become now (10.50) so that (10.51) and (10.52) Copyright © 2003 Taylor & Francis Group LLC It is then evident that (10.53) and that the coherence function (10.54) can be less than unity because both excitation and response noises contribute together Since H1 and H2 are, respectively, a lower and an upper bound estimator for H, the geometric mean of these two quantities provides an estimator that lies in between the two This is generally called Hv and is given by (10.55) Some conclusions can be drawn from the discussion above If we consider that at the structure’s resonances the excitation signal is particularly susceptible to noise because the structure is compliant at these values of frequency and little force is required to produce a significant displacement while, on the other hand (unless the response transducer is placed at a node for that particular mode) the response signal has generally a good signal-tonoise ratio, we can infer that better results can be expected by using the estimator H2 near the resonances By contrast, away from resonances (and, specifically, around antiresonances) the structure is stiff and the response signal, rather than the excitation signal, may be more susceptible to noise contamination Then, a better estimate of the actual FRF can be obtained by using the estimator H1 Furthermore, the coherence function is related to the variance on the estimate of H and is a statistical parameter based on averages in the quantities Gfx, Gff and Gxx The coherence for a single measurement is always unity (and is useless), even in the presence of noise In a number of subsequent measurements, this function can be assumed to be a measure of the quality of our measurements with the following considerations in mind: • With random excitation, low coherence does not necessarily imply a poor estimate of the FRF but it may just mean that more averages are needed for a reliable result (incidentally, we note that if we use a shaker with a ‘stinger’ which is too stiff in the transverse direction, the transverse shaker-structure interactions appear as noise in the excitation signal; however, this noise is not uncorrelated with the input signal and one of the assumptions of case is not valid) Copyright © 2003 Taylor & Francis Group LLC • With deterministic excitation (for example, impact or rapid ‘chirp’ sinusoidal sweeps), low coherence usually indicates bias errors such as nonlinearity, significant noise levels or, for improper windowing, leakage The considerations of this section can be extended to more general cases and, in particular to the most general case of a multiple-input multiple-output (MIMO) test configuration, i.e where there is a number n of excitation points and a number m of measured responses and all signals are measured simultaneously The main advantages of this procedure are an increase of accuracy and consistency in the estimates of the structure FRFs and a reduction in the testing time Specifically, multiple input configurations allow the separation of closely spaced modes, a circumstance in which single-input configurations incur serious difficulties MIMO techniques, however, involve the inversion of a matrix (containing the input auto- and cross-spectrum information) which, in a number of practical situations, has been found to be singular It has been shown that such an inverse exists and leads to a unique solution for FRFs when the inputs (excitations) are not correlated In this light, the concepts of partial coherence and multiple coherence are introduced and partial coherence between the inputs is used for assessing whether the inputs are correlated There are several sources to which the interested reader can refer for detailed discussion of this topic [5–10] 10.4.3 Identification of modal parameters—curve fitting Once the experimental data (typically the FRFs) have been collected, the next task of interest is to extract from this information the modal parameters of the structure: natural frequencies, damping ratios and modal amplitudes associated with each natural frequency For an SDOF system, this is an easy task; it has already been shown in Chapter and in Section 10.2.1 how the FRF can be used to obtain the values of natural frequency, mass, stiffness and damping ratio (For example, consider the simple SDOF system used to draw the FRF graphs of Section 10.2.1 Noting that for an SDOF system as the mass can be extracted from the dB(y)–log(x) graph of accelerance (Fig 10.3) by noting that the mass line in the highfrequency range is approximately at –34 dB, so that ) Similarly, suppose we have measured a column of the FRF matrix of an n-DOF system with widely spaced and lightly damped resonances, and suppose further that all n modes have been experimentally observed Among other possibilities, we can determine the natural frequencies ωj from: • • • the peaks of the magnitude graphs; the peaks of the imaginary parts of the receptance or accelerance graphs; the zero crossings of the real part of the mobility graphs Copyright © 2003 Taylor & Francis Group LLC Then we can determine the modal damping ratios ζj from any one of the magnitude graphs by calculating (10.56) where and (the so-called half-power points, or –3 dB points) are obtained from (Section 4.4.1) (10.57) Alternatively, on the real part of receptance or accelerance graphs, and are those values of frequency at which the local maximum and minimum are attained (e.g Fig 10.22a) When the ωj and ζj are known we can calculate the magnitude of the modal coefficients as outlined in Section 7.4.1 for a 3-DOF system An alternative (generally leading to more accurate results) to this method is to consider only the imaginary part of receptance or accelerance FRFs and obtain directly the jth mode shape from the ratios: (10.58) where in eq (10.58) we assumed that we have measured the mth column of the matrix H(ω) Let us clarify this point Suppose we have a 3-DOF system with natural frequencies ω1, ω2 and ω3 and that we have measured the first column of the receptance matrix R(ω), i.e the functions R11, R21 and R31 Since the real part of receptance is zero at each resonance, the response is purely imaginary and we can obtain the first mode shape (eigenvector) from (10.59a) and, respectively, the second and third eigenvectors from (10.59b) (10.59c) Copyright © 2003 Taylor & Francis Group LLC All three eigenvectors are normalized so that their first element is unity and the sign of each element is automatically taken into account It is obvious that the same results are obtained by considering any column other than the first or any row of the matrix R(ω) or, alternatively, any row or column of A(ω) Example 10.2 A simple numerical example can be given by considering once again the 2-DOF system of Section 7.9 whose mass-orthonormal eigenvectors are given in eq (7.110b) From the receptance functions R11 and R12, the calculation the eigenvectors shown above leads to (10.60) which are just the eigenvectors of eq (7.110b) with a different normalization; the minus sign in the second element of z2 comes from the fact that is negative (Fig 10.22(b)) and indicates that, in the second mode, the two masses move in opposite phase By contrast, all the peaks of the imaginary part of the point FRF (R11(ω) in this case) must have the same sign because the response and excitation are measured at the same point The procedure shown above can lead to reliable results when its basic underlying assumption is verified, namely that in the vicinity of a resonance the response is dominated by only one term of the sum (10.9) This is generally reasonable for structures with widely spaced and lightly damped resonances, a case in which the contribution of off-resonant modes can be assumed to be negligible, and we treat each resonance as if the other resonances did not exist In practice, all modal analysis software packages incorporate some ‘identification methods’, that is, numerical procedures with the specific purpose of extracting the modal parameters from a set of experimental data Some methods require the user to participate in various decisions throughout the analysis, while others—once the relevant data have been supplied—are completely automatic From a user’s point of view, however, the main concerns are not the detailed analytical aspects of the numerical procedures, but a general picture of the available possibilities in order to have an idea of which method may fit his/her particular needs Then, the choice must be based on the available hardware (to perform the test) and computing resources, the scope of the investigation, the structure under test and the format of the experimental data, not necessarily in this order In the following, we adopt this point of view, also because due consideration of the most popular methods would exceed the scope and boundaries of this chapter All the parameter identification methods are based on analytical curve fitting of the measured data Copyright © 2003 Taylor & Francis Group LLC The first and more general classification concerns the domain in which the data are treated numerically, that is, we can distinguish between frequencydomain and time-domain methods The former methods operate on the system’s response characteristics in the frequency domain, i.e on the FRFs which are generally written as (eq (7.78b)) (10.61) where, in the usual modal analysis symbolism, m is the mth eigenvalue (the term ‘pole’ is also common) and it is expressed as (eq (6.160)) (10.62) and is the mth residue On the other hand, the latter methods perform the fitting in the time domain, i.e on the impulse response functions (IRFs) and generally involve the calculation of the inverse FFT of the FRFs The basic mathematical expression is now (10.63) which is the time-domain counterpart of eq (10.61) In principle, no difference should exist between the two approaches but the numerical behaviour of the identification method and the fact that experimental measurements are always performed in a limited frequency band must be taken into account in practical applications Frequency-domain methods, in turn, can be further divided according to the number of modes that can be analysed; hence we have SDOF methods and MDOF methods SDOF methods are based on the assumption that in the frequency region around a resonance the response is dominated by the resonant term corresponding to that mode only, so that the contribution of other modes can be either completely ignored or taken into account by means of a simple approximation term In this regard, the strategy outlined for purposes of illustration at the beginning of this section is a typical SDOF method (as a matter of fact it is somewhat a mixture of the two simplest methods known as ‘peak amplitude’ and ‘quadrature response’) Since the above assumption is not always justified and SDOF methods may lead to serious inaccuracies when the structure under test is not lightly damped and has closely spaced modes, a number of MDOF methods have been devised which fit a multiple-mode form analytical expression to the experimental FRFs The distinction between SDOF and MDOF methods Copyright © 2003 Taylor & Francis Group LLC does not exist in the time-domain because all time domain methods are necessarily MDOF (from an IRF there is no way to make an a priori separation of the various modes) Then, another classification is usually made for both frequency- and timedomain methods based on the number of functions that are analysed simultaneously In fact, depending also on the set of experimental FRFs (or IRFs), we can decide to use a method which analyses one function at a time (SISO, i.e single-input single-output methods) or more functions at the same time This latter possibility comprises two cases: if the functions analysed simultaneously have been collected by exciting the structure at one fixed location and measuring its response at several different locations, the method is classified as SIMO: single-input multiple-output On the other hand, if the experimental data have been collected by exciting the structure and measuring the response at a number of different locations, we can analyse all the available functions simultaneously by using a polyreference or MIMO (multiple-input multiple-output) method Situations of multiple-input singleoutput (MISO) are also possible, but are used to a much lesser extent Finally, in both time and frequency domains, a general distinction exists between indirect and direct methods Indirect methods base the identification procedure on the modal model, while direct methods work directly on the spatial model, i.e the fundamental basic matrix equation from which all the treatment of MDOF systems is derived Without claim of completeness, some names may help the reader to find his/her way among the various possibilities In the frequency domain some popular methods are: • • • • • • • • • • • the peak amplitude method (SDOF, SISO) the quadrature response method (SDOF, SISO) the Kennedy-Pancu or circle-fitting method (SDOF, SISO) the inverse method (SDOF, SISO) Dobson’s method (SDOF, SISO) the Ewins-Gleeson method (MDOF, SISO) the complex exponential frequency-domain, or CEFD method (MDOF, SISO) the rational fraction polynomial, or RFP method (MDOF, SISO) the global rational fraction polynomial, or GRFP method (MDOF, SIMO) the global Dobson method (MDOF, SIMO) the eigensystem realization algorithm in the frequency domain, or ERAFD method (MDOF, MIMO) On the other hand, in the time domain we have: • • • the complex exponential, or CE method (SISO) the least squares complex exponential, or LSCE method (SIMO) the polyreference complex exponential, or PRCE method (MIMO) Copyright © 2003 Taylor & Francis Group LLC • • • the Ibrahim time-domain, or ITD method (SIMO) the single station time-domain, or SSTD method (SISO) the eigensystem realization algorithm, or ERA method (MIMO) All the above methods are classified as indirect; some direct methods in the frequency domain are: • • • • • the Identification of Structural System Parameters, or ISSPA method (SIMO) the spectral method (MIMO) the simultaneous frequency-domain, or SFD method (SIMO) the multimatrix method (MIMO) the frequency-domain direct parameter identification, or FDPI method (MIMO) Among the time-domain direct methods we find, for example: • • the autoregressive moving average, or ARMA method (SISO) the direct system parameter identification, or DSPI method (MIMO) As may be expected, some of the above fitting procedures are just different versions or extensions of other methods, and new methods are continuously being developed We leave the details to the specific literature cited in the references, but it is worth remembering here that the results of the fitting procedure, no matter how sophisticated, can be no better than the quality of the input data (FRFs or IRFs) In other words, the quality of the measurements is a necessary prerequisite for a reliable test, otherwise one will experience a classical case of ‘garbage in, garbage out’ In conclusion, given the appropriate computing capabilities, the choice is often a matter of personal preference and familiarity with a specific method On a very general basis, it can be said that time-domain methods tend to provide the best results when a large frequency range or a large number of modes exist in the data, whereas frequency-domain methods tend to provide the best results when the frequency range of interest is limited and/or the number of modes is relatively small Nonetheless, a major advantage of the frequency domain implementation is that we can take into account the effect of the modes outside the frequency range of interest—say, from ωa to ωb— by virtue of residual terms (not to be confused with the terms mAjk of eq (10.61)) that are incorporated in the model by writing eq (10.61) as (10.64) where n is the number of modes in the frequency range of interest, Pjk(ω) is Copyright © 2003 Taylor & Francis Group LLC the lower residual (called inertia restraint or residual inertia) and is an inverse function of the frequency squared, and Qjk is the upper residual (called residual flexibility), independent of frequency No such possibility exists in the time domain and no account is taken of the effect of modes outside the frequency range of analysis However, it can be argued that the time-domain implementation is numerically better conditioned than the frequency-domain equivalent and is generally more suited to handle noisy measurements Fig 10.28 Circle fitting of experimental FRF Fig 10.29 Experimental FRF: phase versus frequency and Nyquist plot Copyright © 2003 Taylor & Francis Group LLC We close this chapter by giving some examples of actual measurements on engineering structures Figure 10.28 shows the analytical fitting of a mode by means of the frequency-domain circle-fitting method The excellent agreement of the two curves—i.e the actual FRF and the analytical fit—is due to the fact that that mode (at about 14.4 Hz) was relatively isolated Also note how the circle is well defined in the resonant region (from 13.50 to 15.58 Hz in this case) and encompasses approximately 250° of the circle In this type of Fig 10.30 Concrete floor: (a) first experimental mode; (b) second experimental mode (Courtesy of Tecniter s.r.l., Milan, Italy.) Copyright © 2003 Taylor & Francis Group LLC Fig 10.31 Concrete floor: (a) first finite-element mode; (b) second finite-element mode (Courtesy of Tecniter s.r.l., Milan, Italy.) Copyright © 2003 Taylor & Francis Group LLC fitting, if possible, a span of at least 180° of the circle is advisable The particular FRF of this example is a point FRF and was obtained by exciting the structure and measuring its response at location no 13, i.e H13, 13 Figure 10.29, in turn, shows the unfitted curve of the same mode as seen from the FRF H13, 15, the upper curve being the phase and the lower curve being the Nyquist plot Finally, Figs 10.30(a) and (b) show the first (9.63 Hz) and second (14.72 Hz) mode of a light rectangular concrete floor (with dimensions of about 7×5 m) as obtained from a modal test Figure 10.31(a) and (b) are the same modes as obtained from an independent finite-element analysis of the same structure The experimental and theoretical (finite-element) modal frequencies were found to be in very good agreement 10.5 Summary and comments Experimental modal analysis (EMA) is a well-established field of activity in many branches of engineering Basically, it is the process of characterizing the dynamic behaviour—or some aspects thereof—of a structure from a set of experimental measurements rather than from a theoretical (typically, finiteelement) analysis A modal test can be performed for a number of reasons, from troubleshooting and avoidance of vibration problems to finite-element model verification and updating, from nondestructive testing to evaluation of design modifications, from component substructuring to evaluation of structural integrity The final scope of the investigation is the main factor that affects the quality and quantity of data to be collected In fact, in some instances, a few data of good quality can the job, resulting in substantial savings of time and money On the other hand, other situations require a sophisticated analysis, which almost necessarily implies higher demands on hardware and computing capabilities Although there exists an area of activity called ‘nonlinear modal analysis’ (which we not consider), the basic assumption of EMA is that the structure under test is linear, time invariant, observable and obeys Maxwell’s reciprocity relations We can distinguish three phases in a modal test: (1) data acquisition, (2) modal parameter estimation and (3) interpretation and presentation of results Phase is probably the most important because the quality of the final results cannot be any better that the quality of the experimental measurements On the other hand, phase 2—once the analysis has been completed—is generally a matter of good engineering judgement This chapter considers some fundamental aspects of phases and 2, starting with the properties and formats of the data that are acquired in a modal test, i.e the frequency response functions (FRFs) FRFs of SDOF and MDOF systems are considered in order to give the experimenter some guidelines for a quick check of the data during the test so that, if needed, he/ she can repeat the measurements if there seems to be something wrong Copyright © 2003 Taylor & Francis Group LLC Then, a whole section deals with the experimental procedures, describing the instrumentation and potential problems associated with: • • • • the excitation mechanism (shaker, modally tuned hammer, etc.); the type of excitation signal (sinusoidal, impact, random, periodicrandom, etc.); the measurement of the structure’s response; the use of a multichannel FFT analyser Finally, the last section analyses three selected topics of EMA: the characteristic phase lag theory, the presence of noise in the input and/or the output signals in a single-input single-output test configuration and the curvefitting methods, which are generally available on commercial software to extract and identify—starting from a set of measured FRFs or IRFs—the modal parameters of the structure under test Each part of this subject is interesting in its own right The characteristic phase lag theory shows how it can be possible to obtain the undamped frequencies and mode shapes of a structure by an appropriate selection of the input force levels in a sinusoidal excitation type of test This type of test is often performed on large structures, requires considerable hardware and experimental skill and one of its practical implementations is the so-called Asher method, whose line of reasoning is also shown The issue of (uncorrelated) noise in the input and/or at the output, in turn, is important to shed some light on the definition and properties of the most commonly adopted FRF estimators, i.e H1(ω), H2(ω) and Hv(ω), and on the meaning of the ordinary coherence function γ (ω), which are all displayed by commercial FFT analysers Finally, a subsection on the curve-fitting procedures focuses the attention on the various possibilities in which the modal parameters (natural frequencies, damping factors and mode shapes) can be extracted from a set of measurements In consideration of the fact that the user’s main concern is not, in general, the specific numerical procedure, a general classification based on a set of basic criteria—frequency or time domain, number of functions processed simultaneously, etc.—is given in order to provide some information on the most popular methods References Newland, D.E., Mechanical Vibration Analysis and Computation, Longman Scientific and Technical, 1989 Døssing, O., The enigma of dynamic mass, Sound and Vibration, Nov., 16–21, 1990 Proceedings of the International Modal Analysis Conference, 1982–1997 Available on CD-ROM; contact Mr Chris Tomko at SAVIAC (Shock and Vibration Information Analysis Center), Booz-Allen and Hamilton Inc., 3190 Fairview Park Drive (8th floor), Falls Church, VA 22042, USA Copyright © 2003 Taylor & Francis Group LLC McConnell, K.G., Vibration Testing Theory and Practice, Wiley Interscience, 1995 Allemang, R., Rost, B and Brown, D., Multiple input estimation of frequency response functions: excitation consideration, ASME Paper no 83-DET-73 Bendat, J., Solution of the multiple input-output problem, Journal of Sound and Vibration, 44(3), 311–325 Leuridan, J., Multiple input estimation of frequency response functions for experimental modal analysis: currently used methods and some new developments, Proceedings of the 9th International Seminar on Modal Analysis, Vol 2, 32 pp., Leuven, Belgium Leuridan, J., The use of principal inputs in multiple-input multi-output data analysis, International Journal of Analytical and Experimental Modal Analysis, July 1986, 1–8 Bendat, J and Piersol, S., Random Data: Analysis and Measurement Procedures, 2nd edn, John Wiley, New York, 1986 10 Mendes Maia, N.M and Montalvão e Silva, J.M (eds), Theoretical and Experimental Modal Analysis, Research Studies Press, 1997 Copyright © 2003 Taylor & Francis Group LLC ... channel analyser and a single-input single-output configuration The modal parameters are then extracted from the measured data by mathematical curve fitting (all modal analysis software packages... function in this case is given by (10.39) and, as expected, depends on the signal-to-noise ratio the lower the signal-to-noise ratio, the closer the value of coherence to unity Also, Copyright © 2003... in experimental modal analysis The intention of the preceding section was to give the reader a general overview of the many problems that should be addressed during the measurement phase of a modal