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First, I would like to acknowledge the many contributions made by my colleague at Imperial College, Mehmet Imregun, in particular in the implementation of most of the techniques described in this book in the software package MODENT and its associated routines.

These capabilities have greatly enhanced the shorl courses which I have continued to present throughout the period, many in collaboration with another colleague at X Imperial College, David Robb. A major influcnce in the subject has been provided by the Dynamic Testing Agency the DTA , an organisation founded in to help ensure the use of ‘best practice’ in all forms of dynamic testing.

Our activities in encouraging and promoting the subject of Modal Testing were, in fact, the initial stimulus for the founding of this Agency which now provides an excellent focus for ali who wish to benefit from a technology which has matured considerably in the past decade. I would like to acknowledge the major contributions of many colleagues in the DTA to the development of the subject and, especially, to the distillation and recording of what is usually referred to as the ‘expertise’ of the most experienced practitioners into a set of Handbooks which usefully supplement this textbook by enabling the reader-turncd-user io get the maximum benefit from what is a very powerful tool.

Lastly, I wish to record my real appreciation ofthe patience, forbearance and long-suffering of my wife, Brenda, who has so often seen the other sidc of these endeavours: the anti-social hours when a book like this tends to get written and revised , the endless working groups and meetings and the many, many occasions when the experience-gathering phases of teaching and practising the subject in any one of the four corners of the globe mean yet another weckend when the home jobs are neglected.

Analysers 3. Analysis 1 – Peak-Amplitude 4. Since the very early days of awareness of vibrations, experimental observations have been made for the two major objectives of a determining the nature and extent of vibration response level s and b verifying theoretical models and predictions. First, there are a numbcr of strudures, from turbine bladcs to suspcnsion bridges, for which structural integrity is of paramount concern, and for which a thorough and precise knowledge of the dynamic characteristics is essential.

Then, there is an cven wider set of components or assemblies for which vibration is directly related to performance, either by virtue of causing temporary malfunction during excessive motion or by creating disturbance or discomfort, including that of noise. For all these examples, it is import. The two vibration measurement objectives indicated above represent two corresponding types of test. The first is one where vibration forces or, more usually, responses are measured during ‘operation’ of the machine or structure under study, whilo the second is a test where the structure or component is vibrated with a known cxcitation, often out of its normal service environment.

This second type of test is generally made under much more closely-controlled conditions than the former and consequently yields more accurate and dctailed information. This type of tcst – including both the data acquisition and its subsequent analysis – is 2 nowadays called ‘Modal Testing’ and is tbe subject of thc following tcxt. While we shall be defining the specific quantities and parameters used as we proceed, it is perbaps appropriare to stare clearly at this pointjust what we mean by the name ‘Modal Testing’.

It is used here to encompass “the processes involved in testing components or structures with the objective of obtaining a mathematical description of their dynamic or vibration behaviour”. The forro of the ‘mathematical description’ or model varies considerably from one application to the next: it can be an estimate of natural frequency and damping factor in one case and a full mass-spring- dashpot modcl for.

Although the name is relatively new, the principies of modal testing were laid down many years ago. These have evolved through various phases when descriptions such as ‘Resonance Testing’ and ‘Mechanical Irnpedance Methods’ were used to describe the general area of activity. One of the more important mlestones in the development of the subject was provided by the paper in 7 by Kennedy and Pancu [1]’”. The methods described there found applicaton in the accurate determination of natural frequcncies and damping leveis in aircraft structures and were not out- dated for many ycars, untl the rapid advance of measurement, and analyss tcchniques in t,he s.

This activity paved the way for more precise measurements and thus more powerful applications. A paper by Bisbop and Gladwell in 2 described the state of the theory of resonance testing which, at that time, was considerably in advance of its practical implementation. Another work of the sarne period but from a totally different viewpoint was the book by Salter 3 in which a relatively non-analytical approach to the interpretation of measured data was proposed.

Whil st more demanding of the user than today’s computer- assisted automation of the sarne tasks, Salter’s approach rewarded with a considerable physica1 insight into the vibration of the strncture thus studied. However, by thcre had been major advances in transducers, electronics and digital analysers and the current techniques of modal testing were establishcd.

There are a great many papers which relate to ihis period, as workcrs made further advances and applications, and a bibliography of several hundrcd such references now exists 4,5. The following pagcs set out to bring together the major featurcs of all aspects of the subject to providc a comprchensive guide to both the theory and the practice ofmodal testing. We have adopted a particular notation and terminology throughout this work but in order to facilitate ‘translation’ for compatibility with other references, and in particular the manuais of various analysis equipment and software in widespread use, the alternative names will be indicated as the various parameters are introduced.

Thcre are many applications to which the results from a modal test may be put and several ofthese are, in fact, quite powerful. However, it is important to remember that no single test or analysis procedure is ‘best’ for all cases and so it is very important that a clear objective is defined bcfore any test is undertaken so that the optimum methods or techniques may be uscd.

This process is best dealt with by considcring in some detail the following questions: what is the desired outcome from the study of which the modal test is a part? First then, it is appropriate to review the major applicaton areas in current use. A Perhaps thc single most commonly used application is the measurement of vibration modes in order to compare these with corresponding data produced by a finite element or other theoretical model. This application is often born out of a nced or desire to validate the theoretical model prior to its use for predicting response leveis to complcx excitations, sucb as shock, or other further stages of analysis.

It is generally felt that corroboration of the major modes of vbration by tests can provide reassurance of the basic validity of the model which may then be put to further use.

For this specifi. At this stage, accurare mode shape data are not essential. It is gcncrally not po11hiblc lo ‘prcdict’ thc dumping in cach modo of v1brnlton frorn a thcorclicul modcl and so thcrc is nothing with which to compare mcasurcmcnts of modal damping from thc tosts. However, such information is usoCul as it can bo incorporated into the theoretical modol, albeit as an approximation, prior to that being called upon to predict specific response levels which are often significantly influenced by the damping.

B Many cases of experiment-theory comparison stop at the stage of obtaining a set of results by each route and simply comparing thom. Sometimes, an attempt will be made to adjust or correct lhe theoretical model in order to bring its modal properties closer into line with the measured results, but this is usually done using a trial-and-error approach. A logical evolution of the procedure outlined above is the correlation, rather than the comparison, of experimental and theoretical results.

By this is meant a process whoreby tho two sots of data are combincd, quantitatively, in order to identify specifically the causes of the discrepancies between predicted and measured properties. Such an application is clearly more powerful than its less ambitious forerunner but, equally, will be more demancling in terms of the accuracy required in the data taken from the modal test.

Specifically, a much more precise description of the mode shape data the ‘eigenvectors’ is required than is generally necossary to depict or describe the general shape in pictorial form. C The next application area to be reviewed is that of using a modal test in order to produce a mathematical model of a component which may then be used to incorpora te that component into a structural assembly.

This is oflen referred to as a ‘substructuring process’ and is widely used in theoretical analysis of complox structures. Here again, it is a fully quantitativo model that is sought – with accurate data required for natural frequencies, modal dampings and mode shapes – and now has the added constraint that all modes must be included simultaneously. It is not sufficient to confine the model to certain individual modes – as may be done for the provious comparisons or correlations – since out-of-range modes will inluence the structure’s dynamic behaviour in a given frequency range of interest for the complete assembly.

Also, it is not possible to ignore certain modes which exist in the range of interest but which may present some difficulties for measurement or analysis. This 5 application is altogether more demancling than the previous ones and is oflen undorestimated, and so inappropriately tackled, with the result that the results do not match up to expectations.

D There is a variant of lhe previous application which is becoming of considcrablo interest and potential and that is to the generation of a model which may be used for predicting the effects of modifications to the original structure, as tested.

Theoretically, this falls into the sarne class of process as substructuring and has the sarne requiremonts of data accuracy and quantity.

However, sometimes the modification proceduro involves relatively minar changes to the original, in order to fine tune a structure’s dynamics, and this situation can relax the data roquirements somewhat. One particular consideration which applies to both this and the previous case concerna the need for information about rotational motion, i. Thesc are automatically included in theoretical analyses but are generally ignored in experimentally-based studies for the simple roason that they are so much more difficult to measure.

Nevertheless, they are generally an essential feature in coupling or moclification applications. E A different application for the model produced by a modal test is that of force dotermination.

Thero are a number of situations where knowledge of the dynamic forces causing vibration is required but where direct measuroment of these forces is not practical. For these cases, one solution is offered by a process whereby measurements of the response caused by the forces are combined with a mathematical description of the transfer functions of lhe structure in order to deduce the forces.

This process can be very sensitivo to the accuracy of thc model uscd for the structure and so it is often essential that the modol itself be derived from measurements in other words, via a modal test. The last step, e , is usually taken in order to reduce a vast quantity of actual measurements to a small and efficient data set usually referred to as the ‘modal model’. Examples of this problcm may bc found particularly in E , force determination, and in C , subsystem coupling.

The solution adopted is to generate a model ofthe test structure using ‘raw’ turn, may well demand additional measurements bcing made, and additional care to ensure the necessary accuracy. However, the subject we are exploring now demands a high level of understanding and competence in all three and cannot achieve its full potential without the proper and judicious rrxture of the necessary components.

For example: when taking measurements of excitation and response levels, a full knowledge of how the measured data are to be processed can bc essential if the correct decisions are to be made as to the quality and suitability of that data.

Again, a thorough undcrstanding of the various forms and trends adopted by plots of frequency. Throughout this work, we shall repeat and re-emphasise the need for this integration of lheoretical and experimental skills. Indeed, the route chosen to devclop and to explain the details of the subject takes us first through an extensivo review of the necessary theoretical foundation of structural vibration. This theory is regarded as a necessary prerequisite to t he subsequent studies of measurement techniques, signal processing and data analysis.

Already, here, there is a bewildering choice of test methods – harmonic, random, transient excitations, for example – vying for choice as the ‘best’ method in each application. If the experimenter is not to be Ieft at the mercy of the sophisticated digital analysis equipment now widely available, he must fully acquaint himself witb the methods, limitations and implications of the various techniques now widely used in this measurement phase.

Next, we consider the ‘Analysis’ stage whcre the measured data invariably, frequency response functions or mobilities are subjected to a range of curve-fitting procedures in an attempt to find the mathematical model which provides the closest description of the actually-observed behaviour. There are many approaches, or algoritbms, for this phase and, as is usually the case, no single one is ideal for all problems.

Thus, an appreciation of the alternatives is a necessary requirement for the experimenter wishing to make optimum use ofhis time and resources. Generally, though not always, an analysis i s conducted on each measured curve individually.

If this is the case, thcn there is a furlher step in the process which we refer to as ‘Modelling’. This is the final stage where all the measured and processed data are combined to yield the most compact and efficient description of the end result – a mathcmatical model of the test structure – that is practicable. This last phase, like the carlicr ones, involves some type of averaging as the means by which a large quantity of measured data are reduced to a relative]y small mathemalical model.

This is an operation which must be used with some care. The averaging process is a valid and valuable technique provided that the data thus treated contain random variations: data with systematic trends, such as are caused by non-linearities, should not be averaged in the same way.

We shall discuss this problem later on. Thus we have attempted to describe the underlying philosophy of our approach to modal testing and are now in a position to revicw the highlights of lhe three maio phascs: theory, measurement and analysis in order to provide a brief overviow of the entire subject.

This will highlight the key features of each arcu of activity und is included for a number of reasons. First, it providos Lho scrious studcnt with a non-detailed review of the different subjects to bc dcalt with, cnabling him to see the context of each without being distracted by thc rninutiae, and thus acts as a useful introduction to the full study.

Secondly, it provides him with a breakdown of the subject int-0 identifiable topics which are then useful as milestones for the process of acquiring a comprchensive ability and understanding of the techniques.

Lastly, it also serves to provide the non-specialist or manager with an explanation of the subject, trying to remove some of the mystery and folklore which may have developed. We begin with the theoretical basis of the subject since, as has already been emphasised, a good grasp of this aspect is an essential prerequisitc for successful modal testing.

It is very important that a clcar distinction is made between the free vibration and the forced vibration analyses, these usually being two successive stages in a full vibration analysis.

As usual with vibration studies, we start with the single degree-of-freedom SDOF system and use this familiar model to introduce the general noLation and analysis procedures which are later extended to the more general multi degree-of- freedom MDOF systems.

For the SDOF system, a frcc vibration analysis yields its natural frequency and damping factor while a particular of forced response analysis, assuming a harmonic excitation, leads to Lhe definition of the frequency response function – such as mobility, the ratio of velocity response to force input.

These two types of rcsult are referred to as “modal properLics” and “frequency response respectively and they constitutc the basis of all our studies. Before leaving thc SDOF model, it is appropriate to consider Lhe forro which a ploL ofthe mobiliLy of other type of frequency response function takes. Three alternative ways of plotting this information are shown in Fig.

Ncxt we consider Lhe more general class of systems with more than one degree-of-freedom. For these, it is customary that the spatial properties – the values of the mass, stiffuess and damper elements which make up the model – be expressed as matrices. The first phase of three in the vibration ana1ysis of such systems is that of setting up the governing equations of motion which, in effect, means determining the elements of the above matrices. This is a process which does exist for the SDOF system but is often trivial.

The second phase is that of performing a free vibration analysis using the equations of motion. This analysis produces first a set of N natural frequencies and damping factors N being the number of degrees of freedom, or equations of motion and secondly a matching set of ‘shape’ vectors, each one of these bcing associated with a specific natural frequency and damping factor.

The complete free vibration solution is conveniently contained in two matrices, r. One element from the diagonal matrix. There are many detailed variations to this general picture, depending upon the type and distribution of damping, but all cases can be described in the sarne general way.

By solving the equations of motion when harmonic forcing is app1ied, we are able to describe the complete solution by a single matrix, known as the “frequency response matrix” H co ], al though unlike the previous two matrix descriptions, the elements of this matrix are not constants but are frequency-dependent, each element being itself a frequency response or mobility function.

Thus, element H jk co represents the harmonic response in one of the coordinates, x j caused by a single harmonic force applied ata different coordinate, fk. Both these harmonic quantities are described using complex algebra to accommodate the magnitude and phase information, as also is H jk e. Each such quantity is referred to as a frequency response function, or FRF for short.

The particular relevance of these specific response characteristics is the fact that they are the quantities which we are most likely to be able to measure in practice. It is, of course, possible to describe each individual frequency response function in terms of the various mass, stiffness and damping elements of the system but the rel evant expressioos are extremely complex. However, it transpires that the sarne exprcssions can be drastically simplified if we use the modal properties instead of the spatial properties and it is possible to write a general expression for any FRF, H jk ro as: 1.

Here again, it is appropriate to consider the form which a plot of such an expression as 1. Thus we find that by making a thorough study of the theory of structural vibration, we are able to ‘predict’ what we might expect to find if we make mobility-type FRF measurements on actual hardware. Indeed, we shall see later how these predictions can be quite detailed, to the point where it is possible to comment on the likely quality of measured data.

We have outlined above the major aspects of the ‘theoretical route’ of vibration analysis. There are also a number of topics which need to be covered dealing with aspects of signal processing, non harmonic response characteristics and non linear behaviour, but these may be regarded as additional details which may be required in particular cases while the above-mentioned items are fundamental and central to any application of modal testing.

Thus ‘the main measurement techniques which must be devised and developed are those which will permit us to make direct measurements of the various mobility properties of the test structure. Essentially, there are three aspects of the measurement process which demand particular attention in order to ensure the acquisition of the high- quality data which are required for the next stage – data analysis. These are: i the mechanical aspects of supporting and correctly exciting the structure; ii the correct transduction of the quantities to be measured – force input and motion response; and iii the signal processing which is appropriate to the type of test used.

Gtntralor 8. Usually, one of three options is chosen for the support: free, or unrestrained, which usually means suspended on very soft springs ; grounded, which requires its rigid clamping at certain points; or in situ, where the testpiece is connected to some other structure or component which presents a non-rigid attachment.

The choice itself will often be decided by various factors. Amongst these may be a desire to correlate the test results with theory and in this case it should be remembered that free boundaries are much easier to simulate in the test condition than are clamped, or grounded ones. Also, if tests are being made on one component which forms part of an assembly, these may well be required for the free-free condition.

The mechanics of the excitation are achieved either by connecting a vibration generator, or shaker, or by using some form of transient input, such as a hammer blow or sudden release from a deformed position. Both approaches have advantages and disadvantages and it can be very important to choose the best one in each case.

Transducers are very important elements in the system as it is essential that accurate measurements be made of both the input to the structure and ofits response. Nowadays, piezoelectric transducers are widely used to detect both force and acceleration and the a j o r problems associated with them are to ensure that they interfer e with the test structure as little as possible and that their performance is adequate for the ranges of frequency und ompliludc of lhe lcsl.

Tbe mobility parameters to be measured can be obtained directly by applying a harmonic excitation and then measuring the resulting harmonic response. This type of test is often referred to as ‘sinewave testing’ and it requires the attachment to the structure of a shaker. The frequency range is covered either by stepping from one frequency to the next, or by slowly sweeping the frequency continuously, in both cases allowing quasi-steady conditions to be attained.

Alternative excitation procedures are now widely used. Periodic, pseudo-random or random excitation signals often replace the sine-wave approach and are made practical by the existence of cornplex signal processing analysers which are capable of resolving the frequency content of both input and response signals, using Fourier analysis, and thereby deducing the mobi1ity pararneters required. A further extension of this development is possible using impulsive or transient excitations which may be applied without connecting a shaker to the structure.

AH of these latter possibilities offer shorter testing times but great care must be exercised in their use as there are many steps at which errors may be incurred by incorrect application.

Once again, a sound understanding of the theoretical basis – this time of signal processing – is necessary to ensure successful use of these advanced techniques. As was the case with the theoretical review, the measurement process also contains many dctailed features which will be described below. Here, we have just outlined the central and most important topics to 0 be considered.

One final observation which must be made is that in modal testing applications of vibration measurements, perhaps more than many others, accuracy of the measured data is of paramount importance.

This is so because these data are generally to be submitted to a range of analysis procedures, outlined in the next section, in order to extract the resulta eventually sought.

Some of these analysis processes are themselves quite complex and can seldom be regarded as insensitive to the accuracy of the input data. By way of a note of caution, Fig.

This is quite separate from the signal It is a procedure whereby the measured mob1hties are analysed in such a way as to find a theoretical which closely resembles the behaviour of the actual testpicce. Th1s process itself falls into two stages: first, to identify the appropriate type of model and second, to determine the appropriate parameters of the chosen model. Most of the effort goes into this second stage, whi ch is widely referred to as ‘experimental modal analysis’. We have seen from our review of the theoretical aspects that we ‘predict’ or better ‘anticipate’ the form of the mobility plots for a multi- degree-of-ft:eedom and we have also seen that these directly related to the modal properties of that system.

The great maJor1ty of modal analysis effort involves the matching or curve-fi. A completely general curve-fi. These coefficients are, of course, closely related to the modal properties of the system.

However, although such approaches are made, they are inefficient and neither exploit the particular properties of resonant systems nor take due account of the unequal quality of the various measured points in the data set Hm ro 1 , Hm ro2 , Thus there is no single modal analysis method but rather a selection, each being the most appropriate in differing conditions. The most widespread and one of the most useful approaches is that known as the ‘Single-Degree-of-Freedom Curve-Fit’ or, often, the ‘Circle Fit’ procedure.

This metbod uses the fact that at frequencies dose to a natural frequency, the mobility can often be approximated to that of a single degree-of-freedom system plus a constant offset term which approximately accounts for the other modes. This process can be repeated for each resonance individually until the whole curve has been analysed.

At this stage, a theoretical regeneration of tbe mobility function is possible using the set of coefficients extracted, as illustrated in Fig. The method can be used for many of the cases encountered in practice but it becomes inadequate and inaccurate when the structure has modes which are ‘close’, a condition which is identified by the lack of an obviously- circular section on the Nyquist plot.

Under these conditions it becomes necessary to use a more complex process which accepts the simultaneous influence of more than one mode. These latter methods are referred to as ‘MDOF curve-fits’ and are naturally more complicated and require more computation time but, provided the data are accurate, they have the capability of producing more accurate estimates for the modal properties or at least for the coefficients in equation 1. Some of thc more detuilcd considerations include: compensating for slightly non-linear behaviour; simultaneously analysing more then one mobility function and curve- fitting to actual time histories rather than the processed frequency response functions.

The overall objective of the test is to determine a set of modal properties for a structure. These consist of natural frequencies, damping factors and mode shapes. The procedure consists of three steps: i measure an appropriate set of mobilities; ii analyse these using appropriate curve-fitting procedures; and iii combine the results ofthe curve-fits to construct the required model. Using our knowledge of the theoretical relationship between mobility functions and modal properties, it is possible to show that an ‘appropriate’ set of mobilities to measure consists of just one row or one column in the FRF matrix, H co ].

Tbis last option is most conveniently achieved using a hammer or other non-contacting excitation device. In practice, this relatively simple procedure’ will be embellished by various detailed additions, but the general method is always as described here. Thus it is appropriate that this first cbapter deals with the various aspects of theory which are used at the different stages of modal analysis and testing.

The majority of this chapter Sections 2. La ter sections extend the theory somewhat to take account of the different ways in which such properties can be measured 2. There are some topics of which knowledge is assumed in the main text but for which a review is providcd in the Appendices in case the current usage is unfamiliar. Before embarking on the detailed analysis, it is appropriate to put the diferent stages into context and this can be done by showing what will be called the ‘theorP.

This illustrates the three phases through which a typical vibration analysis progresses. Generally, we start with a description of the structure’s physical characte1;stics, usually in terms of its mass, stiffness and damping properties, and this is referred to as the Spatial Model.

Then it is customary to perform an analytical modal analysis of the spatial model which leads to a description of the structure’s behaviour as a set of vibration modes; the Modal Model.

This model is defined as a set of natural frequencies with corresponding vibration mode shapes and modal damping factors. Clearly, this wall dcpcnd not only upon the structure’s inherent properties but also on lho nature and magnitude of the imposed excitation and so there will be i11numcrable solutions of this type.

However, it is convenient to present an alys1s of the structure’s response to a ‘standard’ excitation from which solution for any particular case can be constructed and to describe this “” tho ltesponse Model. Thus our rcsponse model will consist of a set of frequency roHponsc functions FRFs which must be defined over the applicable ruc of frcqucncy. As ilil1 1 cl i11 l 1 ’11r 2 1, it i11 possiblc to procccd from the spatial model tht1h tlys1s.

This is the ‘1x1wrinwnt11l roull” lo v1brolion analysis which is shown in Fig. Throughout this chapter we sball describe three classes of system model: a undamped b viscously damped c hysteretically or structurally damped 24 Fig 2. Appcndix 1 1 v c s sonw l1t llw u11c of complcx algebra for ham1onic quantitics.

Now the equation ofmotion is 2. Note that this function, along with other versions of the FRF, is independent of the excitation.

There appears to be a frequency-dependence exhibited by real structures which is not described by the standard viscous dashpot and what is required, apparently, is a damper whose rate varies inversely with frequency: i. An alternative damping model is provided by the hysteretic or structural damper which has not only the advantage mentioned above, but also provides a much simpler analysis for MDOF systems, as shown below in Fig.

However, it presents difficulties to a rigorous free vibration analysis and attention is gcnerally focused on the forced response analysis.

The similarities between the FRF expressions for the different cases are evident from equations 2. We shall first discuss variations in the basic forro of the FRF and then go on to explore different ways of presenting the properties graphically.

Finally we shall examine some useful geometric properties of the resulting plots. This ratio is complexas there is both an amplitude ratio a ro anda phase angle between the two sinusoids 0a. X t iroxeirot So, V.

Table 2. Because of this, any such simple plot can only show two of the three quantities and so there are different possibilities available, ali ofwhich are used from time to time. The three most common forms of presentation are: ‘” i Modulus of FRF vs Frequency and Phase vs Frequency the Bode type ofplot, consisting oftwo graphs ;. Corresponding plots for the mobility and inertance of the sarne system are shown in Figs.

One of the problems with these properties, as with much vibration data, is the relatively wide range of values which must be encompassed no matter which type of FRF is used. The result is something of a transformation in that in each plot can now be divided into three regimes: a low-frequency straight-line characteristic; a high-frequency straight-line characteristic, and the resonant region with its abrupt magnitude and phase variations.

These have in fact been included in Fig. By referring to and interpolating between the mass- and stiffness-lines drawn on the plot, we can deduce that system a consists of a mass of 1 kg with a spring stifTness of 2.

This basic style of displaying FRF data applies to ali types of system, whether damped or not, while thc other forros are only applicable t-0 damped systems and then tend to be sensitive to the type of damping. All three forros of the FRF are shown and from these we can see how the phase change through the resonance region is characterised by a sign change in one part accompanied by a peak max or min value in the other part.

It should be noted here that the use of logarithmic scales is not feasible in this case primarily because it is necessary to accommodate 36 – Fig 2. IEN::Y Hz a b 05 -. Partly for this reason, and others which become clearer when dealing with MDOF systems, this format of display is not so widely used as the others.

As this style of presentation consists of only a single graph, the missing information in this case, frequency must be added by identifying the values of frequency corresponding to particular points on the curve. This is usually done by indicating specific points on the 1 1 curve at regular incrernents of frequency.

In the examples shown, only 11 those frequency points closest to rcsonance are clearly identifiable because those away from this area are very close together. Indeed, it is this feature – of distorting the plot so as to focus on the resonance area – that makes the Nyquist plot so attractive for modal testing applications.

It is clear from the graphs in Fig. For viscous damping, it is the mobility Y c. In the other cases, the degree of distortion from a circular 1ocus depends heavily on the amount of damping present – becoming negligible as the damping decreascs. We shall consider three specific plots: i Log mobility modulus versus frequency ii Nyquist mobility for viscous damping iii Nyquist receptance for hysteretic damping. O ffE -1 IO 20 Flg2.

As was pointed out by Salter, the basic forro of this plot can be constructed quite accurately using the reference values indicated on Fig. Take first the viscous damping case. Froro equations 2. For the hysteretic damping case we have, from equation 2. X2 k1 k2 Fig 2. NOTE that this assumes that the whole system is capable of vibrati ng ata single frequency, e.

Substitution of this condition and trial solution into the equation of motion 2. Substituting any one of these back into 2. Various numerical procedures are available which take the system 44 matrices M and [K lhe Spatial Model , and convert them to the two eigenmatrices rwn and ‘1’] which constitute the Modal Model.

It is very important to realise at this stage that one of these two matrices – the eigenvalue matrix – is unique, while the other – the eigenvector matrix – i s not. Whereas the natural frequencies are fixed quantities, the mode shapes are subject to an indeterminate scaling factor which does not afTect the sbape of the vibration mode, only its amplitude.

Thus, a mode shape vector of 1 2 1 o describcs exactJy the sarne vibration mode as and so on. This topic will be discussed in more detail below. For our 2DOF example, we find that equation 2. Substituting eit. Now, because the eigenvector matrix is subject to an arbitrary scaling factor, the values of mr and kr are not unique and so it i s inadvisable to refer to “the” generalised mass or stiffness of a particular mode.

Many eigenvalue extraction routines scale each vector so that its largest element has unit magnitude 1. Among the many scaling or normalisation processes, there i s one which has most relevance to modal testing and that is mass-normalisation.

The mass-normalised eigenvectors are writ. The equation of motion may be written 2. For our 2DOF example, the numerical results give eigenvectors which are clearly plausible. Then a.

It is clearly possible for us to determine values for the elements of [ cx ro ] at any frequency of interest simply by substituting the appropriate va1ues into 2. Here we introduce a new parameter, rAjk, which we shall refor to as a Modal Constant: in this case, that for mode r for this specific receptance linking coordinates j and k.

Note that other presentations of the theory sometimes refor to the modal constant as a ‘Residue’ together with the use of’Pole’ instead of our natural frequency. The above is a most important result and is in fact the central relationship upon which the whole subject is based. From the general equation 2. However, it is clear that an expression such as 2.

All this means that a forbidding expression such as 2. We can observe some of the above relationships through our 2DOF example. The above characteristics of both the modal and response models of an undam. The following sections will examine t he effects on these models of adding 52 various types of damping, while a discussion of the presentation MDOF frequency response data is given in Section 2. This type of damping is usually referred to as ‘proportional’ damping for reasons which will be clear later although this is a soniewhat restrictive ti tle.

The particular advantage of using a proportional damping model in t he analysis of structures is that the modes of such a structure are almost identical to those of the undamped version of t he model. Specifically, the mode shapes are identical and the nr.

If we return t o the gener al equation of motion for a MDOF system, equation 2. A general solution will be presented in the next sectiop, but here we shall examine the properties of t his equation for the case where the darnping matrix is directly proportional to the stiffness matri x; i. The fact that this matrix i s also diagonal ” ” ” thut the undamped system mode shapes ar e al so t hose of t he 1 d systcm, and this is a particular feature of tbis type of damping.

This mode bas a complex natural frequency with an t l ntory part of 11 11 dccay par t of 11g lhe not ation introduced above for the SDOF analysis. There is a more general definition of the condition required for the damped system to possess the sarne mode shapes as its undamped counterpart, and that is: 2. Finally, it can bc noted that an identical treatment can be made of a MDOF system with hysterctic damping, producing the sarne essential results. Ifthe general system equations ofmotion are expressed as: 55 IMl!

These, after all, know nothing of our for proportionality. Thus, we consider in the next two sections properties of systems with general damping elements, first of the hysteretic type, then VIRCOUS. W t rt by writing the general equation of motion for a MDOF system e s a. It is important to note that the natural frequency Wr is not necessarily equal to lhe natural frequcncy of the undamped system, Wri as was the case for proportional damping, although the two values will generally bc very close in practice.

The complex mode shapcs are at firsl more difficult to interpret but in fact what we find is that the amplitude of each coordinate has both a magnitude anda phase angle. This is only very slightly different from the undamped case as lhere we efectively have a magnitude at each point plus a phase angle which is either O or , both of which can be completely dcscribed using real numbcrs. What the inclusion of general damping efects does is to gcneralise this particular feature of the mode shape data.

This eigensolution can be seen to possess the sarne type of ortbogonality properties as those demonstrated earlier for the undamped system and may bc defi. Starting with 2. It is in this! Excitation by a general force vector Having an expression for the general term in the frequency response funct1on matrix ex jk ro , it is appropriate to consider next the analysis a situation where the system is excited simultaneously at several pomts rather than at just one, as is the case for the individual FRF expressions.

The general behaviour for this case is governed by equation 2. However, a more explicit and perhaps useful form of this solution may be derived from 2. Before leaving this section, it is worth mentioning another special case of some interest, namely that where the excitation i s a vector of mono- phased forces. Here, the complete generality admitted in the previous paragraph is restricted somewhat by insisting that ali forces have the sarne frequency and phase, although their magnitudes may vary.

What is of interest in this case is to see whether there any conditions under which it is possible to obtain a similarly mono-phased response the whole syste. This is not generally the casem the solut1on to equation 2. Thus, each solution oblained as described above applies only at one specific frequency, ro 8. However, it is particularly interesting to determine what frequencies must be considered in order that the characteristic phase lag 0 between ali the forces and all the responses is exactly 90 degrees.

Inspection of C’quation 2. Thus, we have the important result that it is always possible to find a set of mono-phased forces which will cause a mono-phased set of responses and, moreover, if these two sets of mono-phased parameters are separated by exactly 90, then the frequency at which the system is vibrating is identical to one ofits undamped natural frequencies and the displacement ‘shape’ is the corresponding undamped mode shape.

This most important result is the basis for many of the multi-shaker test procedures used particularly in tbe aircraft industry to isolate the undamped modes of structures for comparison with theoretical predictions. It is also noteworthy that this is one of the few methods for obtaining 62 directly the undamped modes as almost all other methods extract the actual damped modes of the system under test.

The physics of the technique are quite simple: the force vector is chosen so that it exactly balances all the damping forces, whatever these may be, and so the principle applies equally to other types of damping. Postscript It is often observed that the analysis for hysteretic damping is less than rigorous when applied to the free vibration situation, as we have done above.

However, it is an admissible model of damping for describing harmonic forced vibration and this is the objective of most of our studies. Moreover, it is always possible to express each of the receptance or other FRF expressions either as a ratio of two polynomials as explained in Section 2. Each of the terms in the series may be identified with one of the ‘modes’ we have defined in the earlier free vibration analysis for the system. Thus, whether or not the solution is st. As will be seen in the next section, the analysis required for the general case of viscous damping – which is more rigorous – is considerably more complicated than that used here which is, in effect, a very simple extension of the undamped case.

Exactly the sarne introductory comments apply in this case as were made at the beginning of Section 2. This is an inevitable result of the fact that all the coefficients in the matrices are real and thus any characteristic values, or roots, must either be real or occur in complex conjugate pairs. As bcfore, there is an eigenvector corresponding to each of these eigenvalues, but these also occur as complex conjugates.

Hence, we can describe the cigensolution as: o. Sometimes, the quantity C. The eigensolution possesses orthogonality properties although these, also, are different to those of the earlier cases.

However, it is interesting to examine the form they take when tpe modes r and q are a complex conjugate pair. Forced response analysis. Returning to the basic equation, 2. We shall seek a similar series expans1on. If we confine our interest to a small range of frequency in the vicinity of one of the natural frequencies i. Excitation by general force vector Although we have only fully developed the analysis for the case of a single force, the ingredients already exist for the more general case ofmulti-point excitation.

The particular case of mono-phased forces has effectively been dealt with in Section 2. The basic defmition derives from the undamped system’s eigenvalues which yield thc frequencies at which free vibration of the system can take place.

These frequencies are identified by the symbol. The former constitutes the oscillatory part of the free vibration characteristic which, being complex, contains an exponential decay term as well. REAL ro ro. We shall now consider the properties of this type of function and then examine the various means used to display the information it contains. It should be emphasised that a thorough understanding of the forrn of the different plots of FRF data is invaluable to an understanding of the modal analysis processes which are described in Chapter 4.

First, we consider the various forms of FRF. We saw in Sections 2. It is true to say that for MDOF systems we can define a complete set of dynamic stiffness or irnpedance data and indeed such data are used in some types of analysis , but it is not a simple matter to derive these inverse properties from the standard mobility typc as the following explanation demonstrates.

Stated simply: 1 Y. Such a condition is almost impossible to achieve in a practical situation. Thus, we find that the only types of FRF which we can expect to measure directly are those of the mobility or receptance type. Further, it is necessary to guard against the temptation to derive impedance-type data by measuring mobility functions and then computing their reciprocais: these are not the sarne as the elements in the matrix inverse.

We can also see from this discussion that if one changes the number of coordinates considered in a particular case in practice we will probably only measure at a small fraction of the total nwnber of degrees of freedom of a practical structure , then the mobility functions involved remain exactly the sarne but the impedances will vary. Lastly, we should just note some definitions used to distinguish the various types ofFRF.

A point mobility or receptance, etc. A transfer mobility is one where the response and excitation coordinates are different. Sometimes, these are further subdivided into direct and cross mobilities, which describe whether the types of the coordinates for response and excitation are identical – for example, whether they are both x-direction translations, or one is x-direction and the other is y-direction, etc. This knowlcdge can be invaluable in assessing the validity of and interpreting measurcd data.

We shall start with the simplest case of undamped systems, for which the receptance expression is given in equation 2. Using the type oflog- log plot described in Section 2. However, the exact shape of the curve is not quite so simple to deduce as appears at first because part of the information the phase is not shown. However, when the addition of the various components is made to determine the complete receptance expression, the signs of the various terms are obviously of considerable importance.

We shall examine some of the important features using a simple example with just two modes – in fact, based on the system used in Section 2. If we look at the expressions for the receptances we have 1, 1 74 o z ‘ E..

However, if we consider what happens when the two terms are added to produce the actual FRF for the MDOF system, we find the following characteristics. Consider first the point mobility, Fig. We see from 2. Hence, the total FRF curve is only slightly above that for the first term. A similar argument and result apply at the high frequency end, above the second natural frequency, where the total plot is just above that for the second term alone.

However, in the region between the two resonances, we have a situation where the two components have opposite sign to each other so that they are subtractive, rather than additive, and indeed at the point where they cross, their sumis zero since there they are of equal magnitude but opposite sign.

On a logarithmic plot of this type, this produces the antiresonance characteristic which is so similar to the resonance. Physically, the response of the MDOF system just at one of its natural frequencies is totally dominated by that mode and the other modes have very little influence.

Remember that at this stage we are still concerned with undamped, or effectively undamped, systems. Now consider the transfer mobility plot, Fig. We can apply similar reasoning as we progress along the frequency range with the sole difference that the signs ofthe two terms in this case are opposite. Thus, at very low frequencies and at very high frequencies, the total FRF curve lies just below that of the nearest individual component while in the region between the resonances, the two components now have the sarne sign and so we do not encounter the cancelling-out feature which gave rise to the antiresonance in the point mobility.

The principles illustrated here may be extended to any number of degrees of freedom and there is a fundamental rule which has great value and that is that if two consecutive modes have the sarne sign for the modal constants, tben there will be an antiresonance at some frequency between the natural frequencies of those two modes.

If they have opposite signs, there will not be an antiresonance, but just a minimum. The most important feature of tbe antiresonance is perhaps the fact that there is a phase change associated with it, as well as a very low magnitude. It is also very interesting to determine what controls whether a particular FRF will have positive or negative constants, and thus whether it will exhibit antiresonances or not.

Considerable insight may be gained by considering the origin of the modal constant: it is the product of two eigenvector elements, one at the response point and the other at the excitation point. Clearly, if we are considering a point mobility, then the modal constant for every mode must bc positive, it being the square of a number. This means that for a point FRF, there must be an antiresonance following every resonance, without exception. Philosophical Transactions of the Royal Society of London.

The basic theory of modal analysis is developed in order to evaluate the consequences of using different test procedures. Piezoelectric accelerometer and force transducer characteristics are reviewed … Expand.

View 2 excerpts, cites background. Kaewunruen , A. Remennikov Engineering, Materials Science. Experimental modal testing is used in relation to solving problems in railway engineering such as estimating dynamic properties of structural components. Modal testing has proven to be an effective … Expand. Experimental Modal Analysis testing-a powerful tool for identifying the in-service condition of bridge structures N.

Haritos , Victoria Engineering. The Experimental Modal Analysis EMA testing technique provides a great deal more information about the in-service condition of a bridge under test than is possible from the more traditional forms … Expand. On the modelling of dynamic structures with discontinuities V. Hiwarkar , V. Babitsky , V. Silberschmidt Engineering. A method of simulation is developed for studying the dynamics of the structures with discontinuities using Matlab—Simulink.

The concept of dynamic compliance is used for modeling the continuous … Expand. Badshah Materials Science. For the prediction of the global properties in materials, dynamic tests have shown important advantages over static tests.

Despite the superiority of dynamic tests, a high level of precision is … Expand. Identification of civil engineering structures F. This thesis presents three methods to estimate and locate damage in framed buildings, simply-supported beams and cantilever structures, based on experimental measurements of their fundamental … Expand.

View 1 excerpt, cites background. Dynamic modelling of a composite plate, a mixed numerical and experimental approach P. Swider , B. Fichoux , G. Jacquet-Richardet Engineering. Optimization of modal analysis and cross-orthogonality techniques to insure finite element model correlation to test data C. Miller Engineering.

This work summarizes the views of current authors on the multifaceted problems associated with updating a finite element model with vibration test data.

 
 

 

[(PDF) m cn -1 Modal Testing: Theory and Practice | Cheng-Yu Hsieh – replace.me

 

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Using modal testing tochniques, the modal testing theory and practice ewins download h p nnd fjjpquenc1os of practical structures can be measurect urntoly, llrHI llrn results displayed iri a variety of ways. A detailed “‘ th mat1cal model of a test slfucture can bo cTevised direc! Robcrts, University of Sussex. E11gla11d 2. Modal Testing: Theory and Practicc D. Ewins 4. Parametric Random Vibration R. Dimcntbcrg 6.

Brandon 7. Casciati mui L. Faravelli 8. Yibroacoustical Diagnostics for Machines and Structures M. Dimentberg, K. Frolov and A. England Mo dal t estin g t h eo r y a n d pract i ce All right; reo;ervcd.

No pare of 1his book may be reprod uccd by any rneansnor transmiucd. Structurn l dynamic, Mathema11car modc l’ 1. TA E97 Modal testing. Mcchanical Cn!! Press Ltd. Nc” Yort NY USA 1. Vibrntion I f’f. Singapore li.

Austral ia L11mpe. England This is nota new subject although many ofthe most powerful theorh have only recently been developed and the potential ofthe method only recently recognised. Modal testing is, in effect, the process of constructing a mathematical model to describe the vibration properties of a structure based on test data, rather than a conventional theoretical analysis.

As such, it has applications throughout the entire field of engineering. The subject matter in the book is presented in a particular order, and that for a particular reason. To this end, the readeris encouraged to make a detailed study or review of the theory in Chapter 2 before proceeding midal the more practical a spects of the measurement process in Chapter 3. Even here, however, a rclatively high level of theoretical competence is expected in order to understand properly the implications and limitations of the difTerent types of measurement method – sine, random and transient excitations and the like.

In Chapt. The analysis phase continues into Cha pte r 5 where thc overall modelling process is outlined, treating difTerent leveis of complexity of the model which are appropriate to modal testing theory and practice ewins download d1fTcrent applications.

Then, in Chapter 6, examples are given of some of 1hc many uses to which the product of a modal test may be put, a list vii which will undoubtedly grow modal testing theory and practice ewins download the next few ycars. The opening chapter is rather more general, and serves both us on introduction to the tcxt and also as an Overview of the subject. As such, it provides an introduction to the rest of the work for the serious student but also presents a condensation of the whole subject for readers who wish to be apprised of the general activities and capabilities modal testing theory and practice ewins download a modal testing facility but who lack the opportunity of the extensive ‘hands-on’ experience which is essential for mastery of the subject.

Acknowledgements It is ссылка that many people contribute to such a work such as this and I should like to acknowledge the support and assistance I have received from several friends and colleagues. I must record first my gratitude to Professor Peter Grootenhuis with whom I have had the pleasure theeory privilege of studying and working for some 20 years. Ris support and encouragement both launched and sustained me modal testing theory and practice ewins download the work which has culminated in this book.

The text has developed alongside a series of short courses on Modal Testing, which I havc given together with my colleague at Imperial College, David Robb. I am most gruteful for his helpful comments on the book and for his many practical contributions. Other university colleagues, both in London and elsewhere, have kindly provided illustrations and, on many occasions, stimulating discussion on all aspects of the subject and I must also acknowledge the very significant contributions provided by severa!

Likewise, many colleagues in industry have provided me with opportunities to work on real-life problems, adding relevance to our research, and have permitted the use of examples from some of these for illustrations here. Without the cooperation of all of these people, there would have been no book.

The actual preparation of the text has been a major undertaking and I am indebted to my secretary, Lavinia Micklem, who has cheerfully produced draft after draft of the manuscript until the publication dcadline testlng me to abandon further ‘improvements’.

Scrutiny of each draft brought to light a number of mistakes in the text: I hope that the most recent and concentrated efforts of Tssting Lacey, the publishers and myself have succeeded in reducing these to a tolerable number, although I suspect VIII IX one посетить страницу two still rcmain!

The first phase of three in the vibration ana1ysis of such systems is that of setting up the governing equations of hd songs download for pc free which, in effect, means determining the elements theorg the above matrices.

This is a process which does exist xnd the SDOF system but is often trivial. The second phase is that of performing a free vibration analysis using the equations of motion. This analysis produces first a set of N natural frequencies and damping factors N being the number practicd degrees of freedom, or equations thepry motion and secondly modal testing theory and practice ewins download matching set of ‘shape’ vectors, each one of these bcing associated with a specific natural frequency and damping factor.

There are many detailed variations to anv general picture, depending upon the type and distribution of damping, but перейти на страницу cases Читаю download deer hunter 2014 for pc full version надеюсь be described in the sarne general way. By solving the equations of motion when harmonic forcing is app1ied, we are able to describe the complete solution by a single matrix, known as the “frequency dlwnload matrix” H co ], although unlike the previous two matrix descriptions, the elements of this matrix are not constants but are frequency-dependent, each element being itself a frequency response or mobility function.

Both these harmonic quantities are described using complex algebra to accommodate the magnitude and phase information, as also is H jk e. Each such quantity is referred to as a frequency response function, or FRF for short. The particular relevance of these specific response characteristics is the fact that they are the quantities which we are most likely to be able to measure in practice. It is, of course, узнать больше to describe each individual frequency response function in terms of the various mass, stiffness and damping elements of приведу ссылку system but the relevant expressioos are extremely complex.

However, it transpires that the sarne exprcssions can be drastically simplified if we use the modal properties instead of the spatial properties and it is possible to write a general expression theoy any FRF, Pracctice jk ro as: 1.

Th1s process itself falls into two stages: first, to identify the appropriate type of model and second, to determine the appropriate parameters of the chosen model. Most of the effort goes practicw this second stage, which is widely referred to as ‘experimental modal analysis’. Modal testing theory and practice ewins download completely general curve-fi.

These coefficients are, of course, closely related to the modal properties of the system. However, although such approaches are made, they are inefficient and neither exploit the particular properties of resonant systems nor take due account of the unequal quality of the various measured points in the data set Hm ro 1Hm ro2Thus there is no single modal downlkad method but rather a selection, each being the most appropriate in differing conditions.

The most widespread and one of the most useful approaches is that known as the ‘Single-Degree-of-Freedom Curve-Fit’ or, often, the ‘Circle Fit’ procedure. This metbod uses the fact that at frequencies dose to a natural frequency, the mobility can often be approximated to that of a single degree-of-freedom system plus a constant offset term which approximately accounts for the other modes.

This process can be repeated moal each resonance individually until the whole curve has been analysed. At this stage, a theoretical regeneration of tbe mobility function is possible using the set of coefficients extracted, as illustrated in Fig. The tyeory can be used for many of the cases encountered in practice but it becomes inadequate and inaccurate when the structure has modes which are ‘close’, a condition which is identified by the lack of an obviouslycircular section on the Nyquist plot.

Under these conditions it becomes necessary to use a more complex process which accepts the simultaneous influence of more than one mode. These latter methods are referred to as ‘MDOF curve-fits’ and are naturally more complicated and require more computation time but, provided the data are accurate, they have the capability of producing more accurate estimates for the modal properties or at modal testing theory and practice ewins download for the coefficients in equation 1.

Some of thc more detuilcd considerations include: compensating for slightly non-linear behaviour; simultaneously analysing more then one mobility function and curvefitting to actual time histories rather than the processed frequency response functions. The overall objective of the test is to determine a set of modal properties for a structure.

These consist of natural frequencies, damping factors and mode shapes. The procedure consists of three steps: 1. Using our knowledge of the theoretical relationship modal testing theory and practice ewins download mobility functions and modal properties, it is possible to show that an ‘appropriate’ set of mobilities to measure consists of just one row modal testing theory and practice ewins download one column in the FRF matrix, H co ].

Tbis last option is most conveniently achieved using a hammer or other non-contacting excitation device. In practice, this relatively simple procedure’ will be embellished by various detailed нажмите для деталей, but the general method is modal testing theory and practice ewins download as described here.

Thus it is appropriate that this first cbapter deals with the various aspects of theory which are used at the different stages of modal analysis and testing. The majority of this chapter Sections 2. La ter sections extend the theory somewhat to take account of the 2025 map download pc ways in which such properties can be measured 2.

There are some topics of which knowledge is assumed in the main text but for which a review is providcd in the Appendices in case the current usage is unfamiliar. This illustrates the three phases through which a typical vibration analysis progresses.

Generally, we start with a description modal testing theory and practice ewins download the structure’s physical characte1;stics, usually in terms of its жмите, stiffness and damping properties, and this is referred to as the Spatial Model.

Then it is customary to perform an analytical modal analysis of the spatial model which leads to a description of the structure’s behaviour as a set of vibration modes; the Modal Model. This model is defined as a set of natural frequencies with corresponding vibration anr shapes and modal damping factors.

It is important to remember that this solution always по этому сообщению the various ways in which the structure is capable of 22 23 vil1rnl in” nntul’olly, i. Damplng Stlrrness Fig 2.

 
 


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