EA, length L and mass per unit length µ. In addition, k is the stiffness of the spring attached to the right end, and the idea is to estimate the first two eigenvalues of this system. An easy and reasonable choice of two Ritz functions is represented by the polynomials (9.54) which satisfy all the boundary conditions—and hence are two comparison functions—for the fixed-free rod (eqs (8.48)). However, they are only admissible functions for the present case, whose boundary conditions read (9.55) and it is evident that both functions eq (9.54) do not satisfy the natural boundary condition of axial force balance at x=L. A point worthy of notice is that, in this case, the coefficients k ij are given by (9.56) In fact, if we consider two comparison functions f and g (i.e. two functions Fig. 9.2 Example 9.3: longitudinal vibration of a rod. Copyright © 2003 Taylor & Francis Group LLC that satisfy eqs (9.55)), we can write where we have integrated by parts and taken into account the boundary conditions (9.55). The last expression is defined for admissible functions and is precisely the counterpart of the first of eq (9.49) for the case at hand. This result should not be surprising because the localized spring must contribute to the total potential energy of the system. A final comment to note is that in the case of an elastic element—say, for example a beam in transverse vibration—with s localized springs and m localized masses, the coefficients k ij and m ij are obtained as (9.58) where are the stiffness coefficients of the springs acting at the locations x=x l and are the localized masses at the locations x=x r . 9.4 Summary and comments On one hand, by means of increasingly sophisticated computational techniques, the power of modern computers allows the analysis and the solution of complex structural dynamics problems. On the other hand, this possibility may give the analyst a feeling of exactness and objectivity which is, to say the least, potentially dangerous. As a matter of fact, the user has limited control on the various steps of the computational procedures and sometimes—in the author’s opinion—he does not even receive great help from the manuals that accompany the software packages. The numerical procedures themselves, in turn, are never ‘fully tested’ for two main reasons: (9.57) Copyright © 2003 Taylor & Francis Group LLC first, because this is often an impossible task (furthermore, the software designer cannot be aware of the ways in which his software will be used) and, second, because of cost and time problems. So, it is always wise to look at the results of a complex numerical analysis with a critical eye. In this light, the importance of approximate methods cannot be overstated. This is why, even in the era of computers, a chapter on ‘classical’ approximate methods is never out of place. Here, the term ‘classical’ refers to methods that have been developed many years before the advent of digital computers (e.g. the fundamental text of Lord Rayleigh[8]) and whose ‘only’ requirements are a little patience, a good insight into the physics of the problem and, when necessary, a limited use of computer resources. Hence, discussion of the ubiquitous finite-element method—which is also an approximation method in its own right—is not included in this chapter. Our attention is mainly focused on the Rayleigh and Rayleigh-Ritz methods, which are both based on the mathematical properties of the Rayleigh quotient (Sections 9.2 and 9.2.1)—a concept that pervades all branches of structural dynamics. For a given system, the Rayleigh method is used to obtain an approximate value for the first eigenvalue, while the Rayleigh- Ritz method is used to estimate the lowest eigenvalues and eigenvectors. Both methods start with an initial assumption on the vibration shape(s) of the system under study and their effectiveness is due to the stationarity property of the Raleigh quotient which guarantees that a reasonable guess of these trial shape(s) leads to acceptable results. Moreover, when the initial assumption seems too crude, both methods can be used iteratively in order to obtain better approximations of the ‘true’ values. In the light of the fact that—unless the assumed shape coincides with the true eigenshape—the Rayleigh method always leads to an overestimate of the first eigenvalue, Section 9.2.2 considers Dunkerley’s formula which, in turn, always leads to an underestimate of the first eigenvalue. Although its use is generally limited to positive definite systems with lumped masses, Dunkerley’s formula can also be useful when we need to verify that the fundamental frequency of a given system is higher than a given prescribed value. The Rayleigh and Rayleigh-Ritz methods apply equally well to both discrete and continuous systems, and so does the assumed modes method, which is closely related to the Rayleigh-Ritz method but uses a set of time dependent generalized coordinates in conjunction with Lagrange equations. However, for continuous systems the problem of boundary conditions must be considered when we choose the set of Ritz trial functions. Boundary conditions, in turn, can be classified as geometric (or essential) or as natural (or force). Geometric boundary conditions arise from constraints on the displacements and/or slopes at the boundary of a physical body, while natural boundary conditions arise from force balance at the boundary. Since the accuracy of the result depends on how well the chosen shapes approximate the real eigenfunctions, it may seem appropriate to choose a set of trial functions which satisfy all the boundary conditions of the problem at hand, Copyright © 2003 Taylor & Francis Group LLC i.e. a set of ‘comparison functions’. However, natural boundary conditions are much more difficult to satisfy than geometric ones and the common practice is to choose a set of Ritz functions which satisfy only the geometric boundary conditions, meaning that the choice is made from the much broader class of ‘admissible functions’. Again, this possibility ultimately relies on the stationarity property of the Rayleigh quotient and allows more freedom of choice to the analyst, often at the price of a negligible loss of accuracy for most practical purposes. Furthermore, when we adopt a modal approach to solve a forced vibration problem, a judicious choice of admissible Ritz functions may lead to an approximation of the true response which is just as good (or even better) as the approximation that we can obtain by choosing a set of comparison functions. This is because the response of the system depends both on the eigenfunctions of the system and on the spatial distribution of the forcing function(s). References 1. Bathe, K.J., Finite Element Procedures, Prentice Hall, Englewood Cliffs, NJ, 1996. 2. Spyrakos, C., Finite Element Modeling in Engineering Practice, Algor Publishing Division, Pittsburgh, PA, 1996. 3. Weaver, W. and Johnston, P.R., Structural Dynamics by Finite Elements, Prentice Hall, Englewood Cliffs, NJ, 1987. 4. Wilkinson, J.H., The Algebraic Eigenvalue Problem, Oxford University Press, 1965. 5. Meirovich, L., Principles and Techniques of Vibration, Prentice Hall, Englewood Cliffs, NJ, 1997. 6. Humar, J.L., Dynamics of Structures, Prentice Hall, Englewood Cliffs, NJ, 1990. 7. Meirovitch, L. and Kwak, M.K., On the convergence of the classical Rayleigh- Ritz method and finite element method, AIAA Journal, 28(8), 1509–1516, 1990. 8. Rayleigh, Lord J.W.S., The Theory of Sound, Vols 1 and 2, Dover, New York, 1945. Copyright © 2003 Taylor & Francis Group LLC 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 do 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 do and how to do 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: 1. the measurement of the response properties of a given system; 2. the extraction of its modal properties (eigenfrequencies, eigenvectors and modal damping ratios); 3. 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: 1. data acquisition 2. modal parameters estimation 3. 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 2 and 3. 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 1. 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. 2. The structure is time invariant. This assumption means that the parameters to be determined are constants and do 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. 3. 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 1 the problem to be tackled has to do with the experimental set- up and the questions to be answered are, for example: how many points (degrees of freedom) are needed to achieve the desired result? how do we excite the structure and how do 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 3 has to do 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, 6 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, and When or we 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 [...]... derivatives and substituting in eq (10. 11), we get (10. 13) so that, after use of well known trigonometric identities, we can separate the terms in sin ωt and cos ωt to obtain (10. 14) where the unknowns are the vectors z, f0 and the phase angle θ Specifically, from the second of eqs (10. 14) it is immediately evident that when we have (10. 15a) and, from the first of eq (10. 14), (10. 15b) Equation (10. 15a)... mobility has a peak at resonance On the other hand, the imaginary part of the receptance and accelerance has a peak at resonance, while that of the mobility has a zero crossing Referring once again to the SDOF system considered before in this section, examples of such plots are shown in Figs 10. 5(a) and (b) (receptance), 10. 6(a) and (b) (mobility) and 10. 7(a) and (b) (accelerance) Finally, the Nyquist... 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... that these graphs have been drawn by using the ‘Bode’ command of MATLAB® Magnitude graphs are the same as Figs 10. 1, 10. 2 and 10. 3 and the label ‘Gain’ on the y-axis comes from the terminology commonly adopted in the electrical engineering community The characteristic features of real and imaginary plots are that the real part of the receptance and accelerance has a zero crossing at the resonant frequency,... 1/(2kγ)) and diameter As a hint, define and note that (10. 5) As an example, Fig 10. 15 shows the Nyquist plot of the FRF receptance (10. 6) Fig 10. 12 Copyright © 2003 Taylor & Francis Group LLC and y directions (the structure, for example, could be a beam with rectangular cross section, z being the longitudinal direction of the beam) Either: 1 we can perform two separate tests, one in the x direction and. .. by using only 200 spectral lines and no information is lost on the shape of the curves Finally, from the graphs of mobility of Figs 10. 9 and 10. 13 we can easily obtain the value of viscous damping In fact, eq (4.97) shows that the diameter D of the mobility circle is 1/c, observing that in Fig 10. 9 and in Fig 10. 13 we get, as expected, c=1200 N s/m in the first case and c=4000 N s/m in the second case... direction and in both cases we would be dealing with a 2-DOF system, or 2 we can perform a single test in which the x and y directions are considered together, and in this case we would be dealing with a 4-DOF system If all FRFs are measured, each test of option 1 results in two direct point FRFs (H11 and H22) and two direct transfer FRFs (H12 and H21) Strictly speaking, no distinction between direct and. .. Copyright © 2003 Taylor & Francis Group LLC Fig 10. 7 Copyright © 2003 Taylor & Francis Group LLC mobility traces out an exact circle (see also eqs (4.95), (4.96) and (4.97)), while receptance and accelerance curves are distorted circles and tend to become more distorted as damping is increased Figures 10. 12 10. 14 illustrate this situation: stiffness and mass are as before, but now c=4000 N s/m, i.e... eigenvalues and eigenvectors are, respectively (10. 19) and (10. 20) (note that the eigenvectors have not been mass-orthonormalized; however, this is irrelevant for our purposes) Then, from eq (10. 18) the force configuration needed to isolate the first mode is given by (10. 21) The quantity of interest to us is the ratio Similarly, for the minus sign indicating that the the second mode, we get force applied. .. this 2-DOF system (Section 7.9), the graphs of mobilities M11, M12 and accelerances A11, A12 are shown in Figs 10. 16 10. 19: part (a) of each figure plots the magnitude on dB(y)–linear(x) scales, while part (b) plots the magnitude on dB(y)–log(x) scales The reader is invited to draw the Copyright © 2003 Taylor & Francis Group LLC Fig 10. 17 Although this may not be immediately evident in our example, . shown in Figs. 10. 5(a) and (b) (receptance), 10. 6(a) and (b) (mobility) and 10. 7(a) and (b) (accelerance). Finally, the Nyquist plots for the same SDOF system are shown in Figs. 10. 8 10. 10. All these. 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. 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)),