Applied Structural and Mechanical Vibrations 2009 Part 10 potx

A modal analysis test is performed under a controlled forced vibrationcondition, meaning that the structure is subject to a measurable force inputand its vibratory response output is mea

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 twoeigenvalues of this system

An easy and reasonable choice of two Ritz functions is represented by thepolynomials

(9.54)

which satisfy all the boundary conditions—and hence are two comparisonfunctions—for the fixed-free rod (eqs (8.48)) However, they are onlyadmissible 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.

that satisfy eqs (9.55)), we can write

where we have integrated by parts and taken into account the boundaryconditions (9.55) The last expression is defined for admissible functionsand 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 mustcontribute 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)

locations x=x r

9.4 Summary and comments

On one hand, by means of increasingly sophisticated computationaltechniques, the power of modern computers allows the analysis and thesolution of complex structural dynamics problems On the other hand, thispossibility 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 haslimited control on the various steps of the computational procedures andsometimes—in the author’s opinion—he does not even receive great helpfrom the manuals that accompany the software packages The numericalprocedures themselves, in turn, are never ‘fully tested’ for two main reasons:

(9.57)

first, because this is often an impossible task (furthermore, the softwaredesigner 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 thislight, the importance of approximate methods cannot be overstated This iswhy, even in the era of computers, a chapter on ‘classical’ approximatemethods is never out of place Here, the term ‘classical’ refers to methodsthat have been developed many years before the advent of digital computers(e.g the fundamental text of Lord Rayleigh[8]) and whose ‘only’ requirementsare a little patience, a good insight into the physics of the problem and,when necessary, a limited use of computer resources Hence, discussion ofthe ubiquitous finite-element method—which is also an approximationmethod in its own right—is not included in this chapter

Our attention is mainly focused on the Rayleigh and Rayleigh-Ritzmethods, which are both based on the mathematical properties of the Rayleighquotient (Sections 9.2 and 9.2.1)—a concept that pervades all branches ofstructural dynamics For a given system, the Rayleigh method is used toobtain 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) ofthe system under study and their effectiveness is due to the stationarityproperty of the Raleigh quotient which guarantees that a reasonable guess

of these trial shape(s) leads to acceptable results Moreover, when the initialassumption 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 thetrue eigenshape—the Rayleigh method always leads to an overestimate of thefirst eigenvalue, Section 9.2.2 considers Dunkerley’s formula which, in turn,always leads to an underestimate of the first eigenvalue Although its use isgenerally limited to positive definite systems with lumped masses, Dunkerley’sformula can also be useful when we need to verify that the fundamentalfrequency of a given system is higher than a given prescribed value

The Rayleigh and Rayleigh-Ritz methods apply equally well to bothdiscrete and continuous systems, and so does the assumed modes method,which is closely related to the Rayleigh-Ritz method but uses a set of timedependent 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 Boundaryconditions, in turn, can be classified as geometric (or essential) or as natural(or force) Geometric boundary conditions arise from constraints on thedisplacements and/or slopes at the boundary of a physical body, while naturalboundary conditions arise from force balance at the boundary Since theaccuracy of the result depends on how well the chosen shapes approximatethe real eigenfunctions, it may seem appropriate to choose a set of trialfunctions which satisfy all the boundary conditions of the problem at hand,

i.e a set of ‘comparison functions’ However, natural boundary conditionsare much more difficult to satisfy than geometric ones and the commonpractice is to choose a set of Ritz functions which satisfy only the geometricboundary conditions, meaning that the choice is made from the much broaderclass of ‘admissible functions’ Again, this possibility ultimately relies on thestationarity property of the Rayleigh quotient and allows more freedom ofchoice to the analyst, often at the price of a negligible loss of accuracy formost practical purposes Furthermore, when we adopt a modal approach tosolve a forced vibration problem, a judicious choice of admissible Ritzfunctions may lead to an approximation of the true response which is just asgood (or even better) as the approximation that we can obtain by choosing

a set of comparison functions This is because the response of the systemdepends both on the eigenfunctions of the system and on the spatialdistribution 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.

10 Experimental modal analysis

10.1 Introduction

In almost every branch of engineering, vibration phenomena have alwaysbeen measured with two main objectives in mind: the first is to determinethe vibration levels of a structure or a machine under ‘operating’ conditions,while the second is to validate theoretical models or predictions Thanks tothe developments and advances in electronic instrumentation and computerresources of recent decades, both types of measurements can now beperformed effectively; one should also consider that the increasing need foraccurate and sophisticated measurements has been brought about by thedesign of lighter, more flexible and less damped structures, which areincreasingly susceptible to the action of dynamic forces

Experimental modal analysis (EMA) is now a major tool in the field ofvibration testing As such, it was first applied in the 1940s in order to gainmore 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 dynamicbehaviour of a structure in terms of its modes of vibration More specifically,EMA aims at the development of a mathematical model which describes thevibration properties of a structure from experimental data rather than fromtheoretical 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 desiredoutcome of the experimental analysis?’ and ‘What are the steps that followthe experimental test and for what reason are they undertaken?’ As oftenhappens in science and technology—and this easier said than done—posingthe problem correctly generally results in considerable savings in terms oftime and money The necessity of stating the problem correctly is due to thefact that modal testing can be used to investigate a large class of problems—from finite-element model verification to troubleshooting, from componentsubstructuring to integrity assessment, from evaluation of structural

modifications to damage detection and so forth—and therefore the finalgoal has a significant influence on the practical aspects of what to do andhow 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 correctapproach to EMA requires a broad knowledge of many branches of engineeringwhich, traditionally, have often been considered as separate areas of activity

If we now refer back to the introduction of Chapter 7, we can once againadopt Ewins’ definitions and note that in this chapter we will proceed alongthe ‘experimental route’ to vibration analysis which, schematically, goesthrough the following three stages:

1 the measurement of the response properties of a given system;

2 the extraction of its modal properties (eigenfrequencies, eigenvectors andmodal 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 modalcharacteristics, i.e natural frequencies of vibration, mode shapes and dampingfactors Broadly speaking, the whole process can be divided into the threemain phases as defined in the preceding section, which can be syntheticallyrestated 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 seldomhappens that the experimenter is given a second chance By contrast, a goodset of experimental data can always be used more than once to go throughphases 2 and 3

A modal analysis test is performed under a controlled forced vibrationcondition, meaning that the structure is subject to a measurable force inputand its vibratory response output is measured at a number of locations whichidentify the degrees of freedom of the structure Three basic assumptions aremade on the structure to be tested:

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 forcesapplied separately In general, a wide class of structures behave linearly ifthe 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 thereexists an area of activity called ‘nonlinear modal analysis’ whose mainobjective is the same as for the linear case, i.e to establish a mathematicalmodel of the structure under test from a set of experimentalmeasurements In this case, however, the principle of superposition cannot

be invoked and the mathematical model becomes nonunique, beingdependent 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 adequatelydetermine the system’s dynamics Examples of systems that are notobservable would include structures or machines with loose components(that may rattle) or a tank partially filled with a fluid that would sloshduring 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 vibrationtesting obey Maxwell’s reciprocity relations provided that the inputs andoutputs are not mixed In other words, for linear holonomic-scleronomicsystems reciprocity holds if, for example, all inputs are forces and all outputsare displacements (or velocities or accelerations); by contrast, reciprocity doesnot apply if, say, some inputs are forces and some are displacements and ifsome outputs are velocities and some are displacements Unless otherwise stated,

we will assume in the following that reciprocity holds; for our purposes, themain consequence of this assumption is that receptance, mobility, andaccelerance and impulse response functions matrices are all symmetrical.Given the assumptions above, a modal test can be performed by proceedingthrough phases 1–3 Since there is no such thing as ‘the right way’ valid forall circumstances, each phase poses a number of specific problems whosesolutions depend, for the most part, on the final objectives of the investigationand 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 weexcite the structure and how do we measure its response?

In phase 2, on the other hand, the focus is on the specific technique to beused in order to extract the modal parameters from the experimentalmeasurements This task is now accomplished by means of commercialsoftware 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 andwith their presentation in form of numbers, graphs, animations of the modalshapes or whatever else is required for further theoretical analysis, if any isneeded

With the exception of the available electronic instrumentation and the basicconcepts of digital signal analysis—which will be considered separately inthe final chapters of this book—most of the theoretical concepts needed inEMA 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 ofexperimental methods in modal analysis consists of acquiring an appropriateset 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

(10.3a)

Owing to the wide dynamic range of FRFs, it is often customary to plotthe magnitude of FRF functions on log-log graphs or, more precisely, in dB(where the reference value, unless otherwise stated, is unity); this circumstancehas also the additional advantage that data that plot as curves on linearscales become asymptotic to straight lines on log scales and provide a simplemeans 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 springline and in the high-frequency part of the graph we have a mass line whoseslope 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 thedamping 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

at –120 dB, meaning that

Fig 10.1

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 high-frequency 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 logarithmicrepresentation (in this regard, the reader can verify that a horizontal line ofhysteretic damping is obtained in receptance graphs)

By contrast, accelerance graphs display a stiffness line with an upwardslope of +40 dB/decade in the low-frequency range and a horizontal massline in the high frequency range The graphs of mobility and accelerance forthe 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 previouschapters 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

Fig 10.4

plotted as in Fig 4.9, is to be understood as the angle of lag of displacementbehind the external force, the two extreme situations being as follows:

• When the displacement is in phase with the force and

• When the displacement lags behind the force of π radians and

angle of lag of velocity behind force and is given by since weknow that velocity leads displacement by π/2 radians When velocityleads force by π/2 so that the velocity angle of lag behind force is

–π ( ) to zero ( ) radians In brief, in the negative exponentialconvention the phase angle is positive when it is an angle of lag, negativewhen it is an angle of lead and for an SDOF system all phase angles plotted

as functions of frequency are monotonically increasing functions The samesituation arises if we adopt the positive exponential convention but we write

respectively

Fig 10.4 (continued)

Fig 10.5

By contrast, in the positive exponential convention, displacement, velocity

respectively, so that the phase angles must be accompanied

by a minus sign when they represent angles of lag In this light, we have that

ranges from π to zero, and all phase anglesplotted as functions of frequency are monotonically decreasing functions.Obviously, whatever convention we choose, it must be consistent withthe physical fact that—in steady-state conditions—displacement is in phasewith force when , velocity is in phase with force at resonance, acceleration

is in phase with force when and that, in all cases, velocity leadsdisplacement by π and acceleration leads velocity by π/2

In order to illustrate this situation, Figs 10.4(a), (b) and (c) show,respectively, the Bode diagrams of receptance, mobility and accelerance ofthe viscously damped SDOF system considered before Those readers whoare familiar with the MATLAB® environment have certainly noticed thatthese 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 realpart of the receptance and accelerance has a zero crossing at the resonantfrequency, while that of the mobility has a peak at resonance On the otherhand, the imaginary part of the receptance and accelerance has a peak atresonance, while that of the mobility has a zero crossing Referring onceagain to the SDOF system considered before in this section, examples ofsuch 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 plots for the same SDOF system are shown in Figs

used 800 frequency lines to cover the whole range; the ‘+’ markers on thecurves identify these sampled frequency values Since modern electronicinstrumentation converts analogue signals into digital ‘sampled’ signals at

an early stage of the measuring process (Chapters 13–15), the markers onthe curves below might represent actual data from acquired FRFs

Note that, on the curves, data points away from resonance are very closetogether (the markers overlap) while the arc spacing between markers becomeslarger and larger as we approach the resonant region This is an advantageand a disadvantage at the same time: the advantage is due to the fact that theresonant frequency can be identified on these graphs with good accuracy(i.e better than other methods) by considering the maximum rate of change

of arc length as a function of frequency, while the disadvantage is that forvery lightly damped structures the typical circular shape may be lost if thenumber of frequency lines is insufficient An example of this situation is

Fig 10.7

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 tobecome 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.

Also note that in this case the graphs have been drawn in the range0–400 rad/s by using only 200 spectral lines and no information is lost onthe shape of the curves

Finally, from the graphs of mobility of Figs 10.9 and 10.13 we can easilyobtain 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

first case and c=4000 N s/m in the second case.

It is left to the reader to show that for a hysteretically damped system it

is receptance that traces out an exact circle with centre at (0, 1/(2kγ)) and

and note that

(10.5)

As an example, Fig 10.15 shows the Nyquist plot of the FRF receptance

(10.6)

Fig 10.12

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 one in the y 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) Strictlyspeaking, no distinction between direct and cross FRFs is needed because nocross FRFs exist in this case By contrast, in option 2 we would have fourdirect point FRFs (input and output measured in the same point along thesame direction), four direct transfer FRFs (input and output at different pointsalong the same direction), four cross point FRFs (input and output at thesame point along different directions) and four cross transfer FRFs (inputand output at different points along different directions) In this case, it isconvenient to number the degrees of freedom from 1 to 4 referring, forexample, to DOFs 1 and 2 for the measurements at point 1 and 2, respectively,

along the x direction and to DOFs 3 and 4 for the measurements at points

1 and 2 along the y direction With these definitions, the dynamic behaviour

of the structure is described by the 4×4 matrix

(10.7)

where the direct point FRFs are on the main diagonal, the cross transfer FRFs

are on the secondary diagonal, the direct transfer FRFs are the elements H12,

H21, H34 and H43, and the remaining elements are the cross point FRFs

In general, the most common situation in experimental tests is the case ofMDOF systems in which input and output are measured in the same direction(i.e a test of type 1, where no cross FRFs are acquired); for this reason, inthis section we will focus our attention on such tests

In order to examine the main characteristics of FRFs of MDOF systems,

it will suffice for our purposes to consider the 2-DOF system of Section 7.9,because all the considerations that follow can be extended in a straightforwardmanner to systems with more than two degrees of freedom

As expected, all graphs of magnitude versus frequency show two peakswhich occur at the resonant frequencies of our system However, we havealready pointed out (Section 7.9) the appearance, between resonances, of an

‘inverted peak’ of antiresonance in the point FRFs R11(ω) and R22(ω) No

such antiresonance exists in the magnitude graphs of the transfer FRFs R12(ω)

and R21(ω) Moreover, it is interesting to note that a phase shift of 180° notonly occurs at each resonance, but also at each (one in our case) antiresonance

As a rule, point FRFs must have antiresonances between resonances; bycontrast, transfer FRFs may or may not have an antiresonance between twoneighbouring resonances In general, all that can be said in this latter case isthat transfer FRFs corresponding to two points which are relatively closetogether on the structure will show more antiresonances than FRFscorresponding to points that are further apart on the structure Let usinvestigate these statements in more detail

For an undamped n-DOF system, it was shown in Chapter 7 that a generalreceptance FRF is written as

negative; hence R jj becomes negative and this change of sign corresponds tothe phase shift of 180° As we move towards ω2, there will be a value of

frequency at which the sum of all (positive) terms other than the first willexactly cancel out the contribution of the first term so that the magnitude atthis point will be exactly zero This is the antiresonance As we pass thispoint and move towards values of increasing frequency, the sum (10.8)becomes positive again and this second change of sign at antiresonancecorresponds to another 180° phase shift Then—until the last resonance—the whole process repeats again and again as we keep moving in the direction

of increasing frequencies

If —depending on the type of structure and on the physical distance

necessarily have the same sign and no antiresonance may occur between any

two neighbouring resonances; when point j and point k are close together

on the structure it is more likely that the coefficients have the same sign and

there will be an antiresonance In our 2-DOF example (R12(ω) and R21(ω))the two neighbouring coefficients have different signs and there is noantiresonance between the two resonances

Referring again to 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