Fundamentals of Signal Analysis Series Introduction … · Fundamentals of Signal Analysis Series Introduction to Time, Frequency and Modal Domains Application Note 1405-1

Fundamentals of Signal Analysis Series

Introduction to Time,Frequency and Modal DomainsApplication Note 1405-1

Table of Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Section 1: The Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

Section 2: The Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

Section 3: Instrumentation for the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . .16

Section 4: The Modal Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

Section 5: Instrumentation for the Modal Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

The analysis of electrical signalsis a fundamental concern formany engineers and scientists.Even if the immediate problemis not electrical, the basicparameters of interest are oftenchanged into electrical signals bymeans of transducers. Commontransducers include accelerometersand load cells in mechanical work,EEG electrodes and blood pressureprobes in biology and medicine,and pH and conductivity probesin chemistry. The rewards fortransforming physical parametersto electrical signals are great, asmany instruments are availablefor the analysis of electricalsignals. The powerful measure-ment and analysis capabilitiesof these instruments can lead torapid understanding of thesystem under study.

You can look at electricalsignals from several differentperspectives, and each of thesedifferent ways of looking at aproblem often lends its ownunique insights.

In this application note weintroduce the concepts of thetime, frequency and modaldomains. These three waysof looking at a problem areinterchangeable; that is, noinformation is lost in changingfrom one domain to another.By changing perspective, thesolution to difficult problemscan often become quite clear.

After developing the concepts ofeach domain, we will introducethe types of instrumentationavailable. The merits of eachgeneric instrument type arediscussed to give you anappreciation of the advantagesand disadvantages of eachapproach.

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Introduction

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The traditional way of observingsignals is to view them in the timedomain. The time domain is arecord of what happens to aparameter of the system versustime. For instance, Figure 1.1shows a simple spring-masssystem where we have attached apen to the mass and pulled a pieceof paper past the pen at a constantrate. The resulting graph is arecord of the displacement of themass versus time, a time-domainview of displacement.

Such direct recording schemesare sometimes used, but usuallyit is much more practical toconvert the parameter of interestto an electrical signal using atransducer. Transducers are

commonly available to change awide variety of parameters toelectrical signals. Microphones,accelerometers, load cells,conductivity and pressureprobes are just a few examples.

This electrical signal, whichrepresents a parameter of thesystem, can be recorded on a stripchart recorder as in Figure 1.2. Wecan adjust the gain of the systemto calibrate our measurement.Then we can reproduce exactlythe results of our simple directrecording system in Figure 1.1.

Why should we use this indirectapproach? One reason is thatwe are not always measuringdisplacement. We then must

convert the desired parameter tothe displacement of the recorderpen. Usually, the easiest way to dothis is through the intermediary ofelectronics. However, even whenmeasuring displacement, wewould normally use an indirectapproach. Why? Primarily becausethe system in Figure 1.1 ishopelessly ideal. The mass mustbe large enough and the springstiff enough so that the pen’s massand drag on the paper will notaffect the results appreciably. Alsothe deflection of the mass must belarge enough to give a usableresult, otherwise a mechanicallever system to amplify the motionwould have to be added with itsattendant mass and friction.

Section 1: The Time Domain

Figure 1.1. Direct recording of displacement - a time domain view Figure 1.2. Indirect recording of displacement

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With the indirect system, you canusually select a transducer thatwill not significantly affect themeasurement. (This can go to theextreme of commercially availabledisplacement transducers that donot even contact the mass.) Youcan easily set the pen deflection toany desired value by controllingthe gain of the electronicamplifiers.

This indirect system works welluntil our measured parameterbegins to change rapidly. Becauseof the mass of the pen and recordermechanism and the powerlimitations of its drive, the pencan move only at finite velocity.If the measured parameterchanges faster than the penvelocity, the output of therecorder will be in error. Acommon way to reduce thisproblem is to eliminate the pen

and use a deflected light beam torecord on photosensitive paper.Such a device is called anoscillograph (see Figure 1.3).Since it is only necessary tomove a small, lightweight mirrorthrough a very small angle, theoscillograph can respond muchfaster than a strip chart recorder.

Another common device fordisplaying signals in the timedomain is the oscilloscope (seeFigure 1.4). Here, an electronbeam is moved using electricfields. The electron beam ismade visible by a screen ofphosphorescent material.An oscilloscope is capable ofaccurately displaying signals thatvary even more rapidly than anoscillograph can handle. This isbecause it is only necessary tomove an electron beam, not amirror.

The strip chart, oscillograph andoscilloscope all show displacementversus time. We say that changesin this displacement representthe variation of some parameterversus time. We will now look atanother way of representing thevariation of a parameter.

Figure 1.3. Simplified oscillograph operation Figure 1.4. Simplified oscilloscope operation (Horizontal deflection circuitsomitted for clarity)

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Over one hundred years ago,Baron Jean Baptiste Fouriershowed that any waveform thatexists in the real world can begenerated by adding up sinewaves. We have illustrated this inFigure 2.1 for a simple waveformcomposed of two sine waves. Bypicking the amplitudes, frequenciesand phases of these sine wavescorrectly, we can generate awaveform identical to ourdesired signal.

Conversely, we can break downour real world signal into thesesame sine waves. It can be shownthat this combination of sinewaves is unique; any real worldsignal can be represented by onlyone combination of sine waves.

Figure 2.2a is a three-dimensionalgraph of this addition of sine

waves. Two of the axes are timeand amplitude, familiar from thetime domain. The third axis,frequency, allows us to visuallyseparate the sine waves that addto give us our complex waveform.If we view this three-dimensionalgraph along the frequency axis weget the view in Figure 2.2b. This isthe time-domain view of the sinewaves. Adding them together ateach instant of time gives theoriginal waveform.

However, if we view our graphalong the time axis as in Figure2.2c, we get a totally differentpicture. Here we have axes ofamplitude versus frequency, whatis commonly called the frequencydomain. Every sine wave weseparated from the input appearsas a vertical line. Its height

represents its amplitude and itsposition represents its frequency.Since we know that each linerepresents a sine wave, we haveuniquely characterized our inputsignal in the frequency domain*.This frequency domainrepresentation of our signal iscalled the spectrum of the signal.Each sine wave line of thespectrum is called a componentof the total signal.

It is very important to understandthat we have neither gained norlost information, we are justrepresenting it differently. Weare looking at the same three-dimensional graph from differentangles. This different perspectivecan be very useful.

Section 2: The Frequency Domain

Figure 2.1. Any real waveform can be produced by adding sine waves together.

Figure 2.2. The relationship between the time and frequency domainsa) Three- dimensional coordinates showing time, frequency and amplitudeb) Time-domain viewc) Frequency-domain view

* Actually, we have lost the phase information of the sinewaves. Agilent Application Note 1405-2 explains how weget this information.

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The Need for DecibelsSince one of the major uses of the frequency domain is to resolve smallsignals in the presence of large ones, let us now address the problem ofhow we can see both large and small signals on our display simultaneously.

Suppose we wish to measure a distortion component that is 0.1% of thesignal. If we set the fundamental to full scale on a four-inch (10 cm) screen,the harmonic would be only four thousandths of an inch (0.1 mm) tall.Obviously, we could barely see such a signal, much less measure itaccurately. Yet many analyzers are available with the ability to measuresignals even smaller than this.

Since we want to be able to see all the components easily at the same time,the only answer is to change our amplitude scale. A logarithmic scale wouldcompress our large signal amplitude and expand the small ones, allowing allcomponents to be displayed at the same time.

Alexander Graham Belldiscovered that thehuman ear respondedlogarithmically to powerdifference and invented aunit, the Bel, to help himmeasure the ability ofpeople to hear. One tenthof a Bel, the deciBel (dB)is the most common unitused in the frequencydomain today. A table ofthe relationship betweenvolts, power and dB isgiven in Figure 2.3. Fromthe table we can see thatour 0.1% distortioncomponent example is 60dB below thefundamental. If we had an80 dB display as in Figure2.4, the distortioncomponent would occupy1/4 of the screen, not1/1000 as in a lineardisplay.

Figure 2.3. The relationship between decibels, power andvoltage

Figure 2.4. Small signals can be measured with a logarithmicamplitude scale

Why the Frequency Domain?Suppose we wish to measure thelevel of distortion in an audiooscillator. Or we might be tryingto detect the first sounds of abearing failing on a noisy machine.In each case, we are trying todetect a small sine wave inthe presence of large signals.Figure 2.5a shows a time domainwaveform that seems to be asingle sine wave. But Figure 2.5bshows in the frequency domainthat the same signal is composedof a large sine wave and significantother sine wave components(distortion components). Whenthese components are separatedin the frequency domain, thesmall components are easy to seebecause they are not masked bylarger ones.

The frequency domain’susefulness is not restricted toelectronics or mechanics. Allfields of science and engineeringhave measurements like thesewhere large signals mask othersin the time domain. The frequencydomain provides a useful toolfor analyzing these small, butimportant, effects.

The Frequency Domain:A Natural DomainAt first the frequency domainmay seem strange and unfamiliar,yet it is an important part ofeveryday life. Your ear-braincombination is an excellentfrequency domain analyzer.The ear-brain splits the audiospectrum into many narrow bandsand determines the power presentin each band. It can easily picksmall sounds out of loud back-ground noise thanks in part to its

frequency domain capability. Adoctor listens to your heart andbreathing for any unusual sounds.He is listening for frequencies thatwill tell him something is wrong.An experienced mechanic can dothe same thing with a machine.Using a screwdriver as astethoscope, he can hear whena bearing is failing because ofthe frequencies it produces.

So we see that the frequencydomain is not at all uncommon.We are just not used to seeing it ingraphical form. But this graphicalpresentation is really not anystranger than saying that thetemperature changed with time,like the displacement of a lineon a graph.

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Figure 2.5.a Time Domain — small signal not visible

Figure 2.5.b Frequency Domain — small signal easily resolved

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Spectrum ExamplesLet us now look at a few commonsignals in both the time andfrequency domains. In Figure 2.6a,we see that the spectrum of a sinewave is just a single line. We expectthis from the way we constructedthe frequency domain. The squarewave in Figure 2.6b is made up ofan infinite number of sine waves,all harmonically related. Thelowest frequency present is thereciprocal of the square waveperiod. These two examplesillustrate a property of thefrequency transform: a signalthat is periodic and exists for alltime has a discrete frequencyspectrum. This is in contrast tothe transient signal in Figure 2.6cwhich has a continuous spectrum.This means that the sine wavesthat make up this signal arespaced infinitesimally closetogether.

Another signal of interest is theimpulse shown in Figure 2.6d.The frequency spectrum of animpulse is flat, i.e., there is energyat all frequencies. It would,therefore, require infinite energyto generate a true impulse.Nevertheless, it is possible togenerate an approximation toan impulse that has a fairlyflat spectrum over the desiredfrequency range of interest.We will find signals with a flatspectrum useful in our nextsubject, network analysis.

Figure 2.6. Frequency spectrum examples

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Network AnalysisIf the frequency domain wererestricted to the analysis of signalspectrums, it would certainly notbe such a common engineeringtool. However, the frequencydomain is also widely used inanalyzing the behavior ofnetworks (network analysis)and in design work.

Network analysis is the generalengineering problem ofdetermining how a networkwill respond to an input.* Forinstance, we might wish todetermine how a structure willbehave in high winds. Or we mightwant to know how effective asound-absorbing wall we areplanning to purchase would bein reducing machinery noise. Orperhaps we are interested in theeffects of a tube of saline solutionon the transmission of bloodpressure waveforms from anartery to a monitor.

All of these problems and manymore are examples of networkanalysis. As you can see a“network” can be any system atall. One-port network analysis isthe variation of one parameterwith respect to another, bothmeasured at the same point (port)of the network. The impedance orcompliance of the electronic ormechanical networks shown inFigure 2.7 are typical examplesof one-port network analysis.

Two-port analysis gives theresponse at a second port dueto an input at the first port.We are generally interested inthe transmission and rejectionof signals and in insuring theintegrity of signal transmission.The concept of two-port analysiscan be extended to any number ofinputs and outputs. This is calledN-port analysis, a subject we willuse in modal analysis later in thisapplication note.

We have deliberately definednetwork analysis in a very generalway. It applies to all networkswith no limitations. If we placeone condition on our network,linearity, we find that networkanalysis becomes a very powerfultool.

Figure 2.7. One-port network analysis examples

* Network Analysis is sometimes calledStimulus/Response Testing. The input is then known asthe stimulus or excitation and the output is called theresponse.

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When we say a network is linear,we mean it behaves like thenetwork in Figure 2.9. Supposeone input causes an output Aand a second input applied at thesame port causes an output B.If we apply both inputs at thesame time to a linear network,the output will be the sum of theindividual outputs, A + B.

At first glance it might seem thatall networks would behave in thisfashion. A counter example, anon-linear network, is shown inFigure 2.10. Suppose that the firstinput is a force that varies in asinusoidal manner. We pick itsamplitude to ensure that thedisplacement is small enough sothat the oscillating mass does notquite hit the stops. If we add asecond identical input, the masswould now hit the stops. Insteadof a sine wave with twice theamplitude, the output is clippedas shown in Figure 2.10b.

This spring-mass system withstops illustrates an importantprincipal: no real system iscompletely linear. A systemmay be approximately linearover a wide range of signals, buteventually the assumption oflinearity breaks down. Our spring-mass system is linear before it hitsthe stops. Likewise a linearelectronic amplifier clips whenthe output voltage approaches theinternal supply voltage. A springmay compress linearly until thecoils start pressing against eachother.

Figure 2.8. Two-port network analysis

Figure 2.9. Linear network

Figure 2.10. Non-linear system example

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Other forms of non-linearitiesare also often present. Hysteresis(or backlash) is usually present ingear trains, loosely riveted jointsand in magnetic devices. Sometimesthe non-linearities are less abruptand are smooth, but nonlinear,curves. The torque versus rpm ofan engine or the operating curvesof a transistor are two examplesthat can be considered linear overonly small portions of theiroperating regions.

The important point is not that allsystems are nonlinear; it is thatmost systems can be approximatedas linear systems. Often a largeengineering effort is spent inmaking the system as linear aspractical. This is done for tworeasons. First, it is often a designgoal for the output of a network tobe a scaled, linear version of theinput. A strip chart recorder is agood example. The electronicamplifier and pen motor mustboth be designed to ensure thatthe deflection across the paper islinear with the applied voltage.

The second reason why systemsare linearized is to reduce theproblem of nonlinear instability.One example would be thepositioning system shown inFigure 2.12. The actual position iscompared to the desired positionand the error is integrated andapplied to the motor. If the geartrain has no backlash, it is astraightforward problem todesign this system to the desiredspecifications of positioningaccuracy and response time.

However, if the gear train hasexcessive backlash, the motor will“hunt,” causing the positioningsystem to oscillate around thedesired position. The solution iseither to reduce the loop gain and

therefore reduce the overallperformance of the system, or toreduce the backlash in the geartrain. Often, reducing the backlashis the only way to meet theperformance specifications.

θ2

θ2

θ1

θ1

Figure 2.11. Examples of non-linearities

∑

Figure 2.12. A positioning system

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Analysis of Linear NetworksAs we have seen, many systemsare designed to be reasonablylinear to meet design specifications.This has a fortuitous side benefitwhen attempting to analyzenetworks*.

Recall that a real signal can beconsidered to be a sum of sinewaves. Also, recall that theresponse of a linear network isthe sum of the responses to eachcomponent of the input. Therefore,if we knew the response of thenetwork to each of the sine wavecomponents of the input spectrum,we could predict the output.

It is easy to show that the steady-state response of a linear networkto a sine wave input is a sine waveof the same frequency. As shownin Figure 2.13, the amplitude ofthe output sine wave is proportionalto the input amplitude. Its phaseis shifted by an amount thatdepends only on the frequency ofthe sine wave. As we vary thefrequency of the sine wave input,the amplitude proportionalityfactor (gain) changes, as does thephase of the output. If we dividethe output of the network by theinput, we get a normalized resultcalled the frequency response ofthe network. As shown in Figure2.14, the frequency response is thegain (or loss) and phase shift ofthe network as a function offrequency. Because the network islinear, the frequency response isindependent of the inputamplitude; the frequency responseis a property of a linear network,not dependent on the stimulus.

Figure 2.13. Linear network response to a sine wave input.

Figure 2.14. The frequency response of a network

* For a discussion of the analysis of networks that havenot been linearized, see Agilent Application Note 1405-2.

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The frequency response of anetwork will generally fall intoone of three categories; lowpass, high pass, bandpass or acombination of these. As thenames suggest, their frequencyresponses have relatively highgain in a band of frequencies,allowing these frequencies topass through the network. Otherfrequencies suffer a relativelyhigh loss and are rejected by thenetwork. To see what this meansin terms of the response of a filterto an input, let us look at thebandpass filter case.

In Figure 2.16, we put a squarewave into a bandpass filter. Werecall from Figure 2.6 that asquare wave is composed ofharmonically related sine waves.The frequency response of ourexample network is shown inFigure 2.16b. Because the filteris narrow, it will pass only onecomponent of the square wave.Therefore, the steady-stateresponse of this bandpass filteris a sine wave.

Notice how easy it is to predictthe output of any network from itsfrequency response. The spectrumof the input signal is multipliedby the frequency response ofthe network to determine thecomponents that appear in theoutput spectrum. This frequencydomain output can then betransformed back to the timedomain.

In contrast, it is very difficult tocompute in the time domain theoutput of any but the simplestnetworks. A complicated integralmust be evaluated, which oftencan be done only numerically on a

computer*. If we computed thenetwork response by bothevaluating the time domainintegral and by transforming tothe frequency domain and back,we would get the same results.

However, it is usually easierto compute the output bytransforming to the frequencydomain.

Figure 2.15. Three classes of frequency response

* This operation is called convolution.

Transient ResponseUp to this point we have onlydiscussed the steady-stateresponse to a signal. By steady-state we mean the output afterany transient responses caused byapplying the input have died out.However, the frequency responseof a network also contains all theinformation necessary to predictthe transient response of thenetwork to any signal.

Let us look qualitatively at thetransient response of a bandpassfilter. If a resonance is narrowcompared to its frequency, thenit is said to be a high-“Q”resonance.* Figure 2.17a shows ahigh-Q filter frequency response.It has a transient response thatdies out very slowly. A timeresponse that decays slowly issaid to be “lightly damped.” Figure2.17b shows a low-Q resonance.It has a transient response thatdies out quickly. This illustrates ageneral principle: signals that arebroad in one domain are narrowin the other. Narrow, selectivefilters have very long responsetimes, a fact we will findimportant in the next section.

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Figure 2.16. Bandpass filter response to a square wave input

Figure 2.17. Time response of bandpass filters

* Q is usually defined as:

Q =Center Frequency of Resonance

Frequency Width of -3 dB Points

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Just as the time domain can bemeasured with strip chart recorders,oscillographs or oscilloscopes,the frequency domain is usuallymeasured with spectrum andnetwork analyzers.

Spectrum analyzers are instrumentsthat are optimized to characterizesignals. They introduce very littledistortion and few spurioussignals. This insures that thesignals on the display are trulypart of the input signal spectrum,not signals introduced by theanalyzer.

Network analyzers are optimizedto give accurate amplitude andphase measurements over a widerange of network gains and losses.This design difference means thatthese two traditional instrumentfamilies are not interchangeable.*A spectrum analyzer cannot beused as a network analyzerbecause it does not measureamplitude accurately and cannotmeasure phase. A networkanalyzer would make a verypoor spectrum analyzer becausespurious responses limit itsdynamic range.

In this section we will discuss theproperties of several types ofanalyzers in these two categories.

The Parallel-FilterSpectrum AnalyzerAs we developed in Section 2 ofthis chapter, electronic filters canbe built which pass a narrow bandof frequencies. If we were to adda meter to the output of such abandpass filter, we could measurethe power in the portion of thespectrum passed by the filter. InFigure 3.1a we have done this fora bank of filters, each tuned to adifferent frequency. If the centerfrequencies of these filters arechosen so that the filters overlapproperly, the spectrum coveredby the filters can be completelycharacterized as in Figure 3.1b.

How many filters should we useto cover the desired spectrum?Here we have a trade-off. Wewould like to be able to seeclosely spaced spectral lines, sowe should have a large numberof filters. However, each filter isexpensive and becomes moreexpensive as it becomes narrower,so the cost of the analyzer goes upas we improve its resolution.Typical audio parallel-filteranalyzers balance these demandswith 32 filters, each covering1/3 of an octave.

Section 3: Instrumentation for the Frequency Domain

Figure 3.1. Parallel filter analyzer

* Dynamic signal analyzers are an exception to this rule.They can act as both network and spectrum analyzers.

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Swept Spectrum AnalyzerOne way to avoid the need forsuch a large number of expensivefilters is to use only one filterand sweep it slowly through thefrequency range of interest. If,as in Figure 3.2, we display theoutput of the filter versus thefrequency to which it is tuned,we have the spectrum of theinput signal. This swept analysistechnique is commonly used in RFand microwave spectrum analysis.

We have, however, assumed theinput signal hasn’t changed in thetime it takes to complete a sweepof our analyzer. If energy appearsat some frequency at a momentwhen our filter is not tuned tothat frequency, then we will notmeasure it.

One way to reduce this problemwould be to speed up the sweeptime of our analyzer. We couldstill miss an event, but the time inwhich this could happen would beshorter. Unfortunately though, wecannot make the sweep arbitrarilyfast because of the response timeof our filter.

To understand this problem, recallfrom Section 2 that a filter takes afinite time to respond to changesin its input. The narrower thefilter, the longer it takes torespond. If we sweep the filterpast a signal too quickly, the filteroutput will not have a chance torespond fully to the signal. As we

show in Figure 3.3, the spectrumdisplay will then be in error; ourestimate of the signal level willbe too low.

In a parallel-filter spectrumanalyzer we do not have thisproblem. All the filters areconnected to the input signal allthe time. Once we have waited theinitial settling time of a singlefilter, all the filters will be settledand the spectrum will be valid andnot miss any transient events.

So there is a basic trade-offbetween parallel-filter and sweptspectrum analyzers. The parallel-filter analyzer is fast, but haslimited resolution and is expensive.The swept analyzer can be cheaperand have higher resolution, butthe measurement takes longer(especially at high resolution),and it cannot analyze transientevents*.

Figure 3.2. Simplified swept spectrum analyzer

Figure 3.3. Amplitude error from sweeping too fast

* More information on the performance of sweptspectrum analyzers can be found in Agilent ApplicationNote Series 150.

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Dynamic Signal AnalyzerIn recent years, another kindof analyzer has been developedwhich offers the best featuresof the parallel-filter and sweptspectrum analyzers. Dynamicsignal analyzers are based on ahigh-speed calculation routinethat acts like a parallel filteranalyzer with hundreds of filters,yet they are cost competitive withswept spectrum analyzers. Inaddition, two-channel dynamicsignal analyzers are in many waysbetter network analyzers than theones we will introduce next.

Network AnalyzersNetwork analysis requiresmeasurements of both the inputand output, so network analyzersare generally two-channel deviceswith the capability of measuringthe amplitude ratio (gain or loss)and phase difference between thechannels. All of the analyzersdiscussed here measure frequencyresponse by using a sinusoidalinput to the network and slowlychanging its frequency. Dynamicsignal analyzers use a different,much faster technique fornetwork analysis. See AgilentApplication Note 1405-2 for moreinformation.

Gain-phase meters are broadbanddevices that measure the amplitudeand phase of the input and outputsine waves of the network. Asinusoidal source must be suppliedto stimulate the network whenusing a gain-phase meter as inFigure 3.4. The source can betuned manually and the gain-phase plots done by hand or asweeping source, and an x-yplotter can be used for automaticfrequency response plots.

The primary attraction of gain-phase meters is their low price. Ifa sinusoidal source and a plotterare already available, frequencyresponse measurements can bemade for a very low investment.However, because gain-phasemeters are broadband, theymeasure all the noise of thenetwork as well as the desiredsine wave. As the networkattenuates the input, this noiseeventually becomes a floor belowwhich the meter cannot measure.This typically becomes a problemwith attenuations of about 60 dB(1,000:1).Figure 3.4. Gain-phase meter operation

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Tuned network analyzersminimize the noise floor problemsof gain-phase meters by includinga bandpass filter which tracks thesource frequency. Figure 3.5 showshow this tracking filter virtuallyeliminates the noise and anyharmonics to allow measurementsof attenuation to 100 dB (100,000:1).

By minimizing the noise, it isalso possible for tuned networkanalyzers to make more accuratemeasurements of amplitude andphase. These improvements donot come without their price,however, as tracking filters anda dedicated source must beadded to the simpler and lesscostly gain-phase meter.

Tuned analyzers are available inthe frequency range of a few Hertzto many Gigahertz (109 Hertz).If lower frequency analysis isdesired, a frequency responseanalyzer is often used. To theoperator, it behaves exactly like atuned network analyzer. However,it is quite different inside. Itintegrates the signals in the timedomain to effectively filter thesignals at very low frequencieswhere it is not practical to makefilters by more conventionaltechniques. Frequency response islimited to a range from 1 mHz toabout 10 kHz.

Figure 3.5. Tuned network analyzer operation

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In the preceding sections wediscussed the properties of thetime and frequency domains andthe instrumentation used in thesedomains. In this section, we willdelve into the properties of anotherdomain, the modal domain. Thischange in perspective to a newdomain is particularly useful ifwe are interested in analyzing thebehavior of mechanical structures.

To understand the modal domain,let us begin by analyzing a simplemechanical structure, a tuningfork. If we strike a tuning fork, weeasily conclude from its tone thatit is primarily vibrating at a singlefrequency. We see that we haveexcited a network (tuning fork)with a force impulse (hitting thefork). The time domain view of thesound caused by the deformationof the fork is a lightly damped sinewave shown in Figure 4.1b.

In Figure 4.1c, we see in thefrequency domain that thefrequency response of the tuningfork has a major peak that is verylightly damped, which is the tonewe hear. There are also severalsmaller peaks.

Each of these peaks, large andsmall, corresponds to a “vibrationmode” of the tuning fork. Forinstance in this simple example,we might expect the major tone tobe caused by the vibration modeshown in Figure 4.2a. The secondharmonic might be caused by avibration like Figure 4.2b.

Section 4: The Modal Domain

Figure 4.1. The vibration of a tuning fork

Figure 4.2. Example vibration modes of a tuning fork

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We can express the vibrationof any structure as a sum of itsvibration modes. Just as we canrepresent a real waveform as asum of much simpler sine waves,we can represent any vibration asa sum of much simpler vibrationmodes. The task of “modal” analysisis to determine the shape andthe magnitude of the structuraldeformation in each vibrationmode. Once these are known, itusually becomes apparent howto change the overall vibration.

For instance, let us look again atour tuning fork example. Supposethat we decided that the secondharmonic tone was too loud. Howshould we change our tuning forkto reduce the harmonic? If we hadmeasured the vibration of the forkand determined that the modesof vibration were those shown inFigure 4.2, the answer becomesclear. We might apply dampingmaterial at the center of the tinesof the fork (see Figure 4.3). Thiswould greatly affect the secondmode that has maximum deflectionat the center, while only slightlyaffecting the desired vibration ofthe first mode. Other solutionsare possible, but all depend onknowing the geometry of eachmode.

Figure 4.3. Reducing the second harmonic by damping the second vibration mode

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The Relationship between theTime, Frequency and ModalDomainTo determine the total vibrationof our tuning fork or any otherstructure, we have to measurethe vibration at several points onthe structure. Figure 4.4a showssome points we might pick. If wetransformed this time domaindata to the frequency domain, wewould get results like Figure 4.4b.We measure frequency responsebecause we want to measurethe properties of the structureindependent of the stimulus.*

We see that the sharp peaks(resonances) all occur at the samefrequencies independent of wherethey are measured on the structure.Likewise we would find bymeasuring the width of eachresonance that the damping (or Q)of each resonance is independentof position. The only parameterthat varies as we move from pointto point along the structure is therelative height of resonances.**By connecting the peaks of theresonances of a given mode, wetrace out the mode shape of thatmode.

Experimentally we have to measureonly a few points on the structureto determine the mode shape.However, to clearly show themode shape in our figure, we havedrawn in the frequency responseat many more points in Figure4.5a. If we view this three-dimensional graph along thedistance axis, as in Figure 4.5b,we get a combined frequency

Figure 4.4. Modal analysis of a tuning fork

* Those who are more familiar with electronics mightnote that we have measured the frequency response of anetwork (structure) at N points and thus have performedan N-port analysis.

** The phase of each resonance is not shown for clarityof the figures but it, too, is important in the mode shape.The magnitude of the frequency response gives themagnitude of the mode shape, while the phase gives thedirection of the deflection.

Figure 4.5. The relationship between the frequency and modal domains

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response. Each resonance has apeak value corresponding to thepeak displacement in that mode.If we view the graph along thefrequency axis, as in Figure 4.5c,we can see the mode shapes ofthe structure.

We have not lost any informationby this change of perspective.Each vibration mode is character-ized by its mode shape, frequencyand damping from which we canreconstruct the frequency domainview.

However, the equivalence betweenthe modal, time and frequencydomains is not quite as strong asthat between the time and frequencydomains. Because the modaldomain portrays the properties ofthe network independent of thestimulus, transforming back tothe time domain gives the impulseresponse of the structure, nomatter what the stimulus. Amore important limitation of thisequivalence is that curve fittingis used in transforming from ourfrequency response measurementsto the modal domain to minimizethe effects of noise and smallexperimental errors. Noinformation is lost in this curvefitting, so all three domainscontain the same information,but not the same noise. Therefore,transforming from the frequency

domain to the modal domain andback again will give results likethose in Figure 4.6. The results arenot exactly the same, yet in all theimportant features, the frequencyresponses are the same. This isalso true of time domain dataderived from the modal domain.There are many ways that themodes of vibration can bedetermined. In our simple tuningfork example, we could guesswhat the modes were. In simplestructures like drums and plates itis possible to write an equationfor the modes of vibration.

However, in almost any realproblem, the solution can neitherbe guessed nor solved analyticallybecause the structure is toocomplicated. In these cases it isnecessary to measure the responseof the structure and determinethe modes.

There are two basic techniques fordetermining the modes of vibrationin complicated structures: 1)exciting only one mode at a time,and 2) computing the modes ofvibration from the total vibration.

Figure 4.6. Curve fitting removes measurement noise.

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Single-Mode ExcitationModal AnalysisTo illustrate single-modeexcitation, let us look once againat our simple tuning fork example.To excite just the first mode, weneed two shakers, driven by a sinewave and attached to the ends ofthe tines as in Figure 5.1a. Varyingthe frequency of the generatornear the first mode resonancefrequency would then give us itsfrequency, damping and modeshape.

In the second mode, the ends ofthe tines do not move, so to excitethe second mode we must movethe shakers to the center of thetines. If we anchor the ends ofthe tines, we will constrain thevibration to the second modealone.

In more realistic, three-dimensionalproblems, it is necessary to addmany more shakers to ensure thatonly one mode is excited. Thedifficulties and expense of testingwith many shakers has limitedthe application of this traditionalmodal analysis technique.

Section 5: Instrumentation for the Modal Domain

Figure 5.1. Single-mode excitation modal analysis

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Modal Analysis fromTotal VibrationTo determine the modes ofvibration from the total vibrationof the structure, we use thetechniques developed in theprevious section. Basically, wedetermine the frequency responseof the structure at several pointsand compute at each resonancethe frequency, damping and whatis called the residue (whichrepresents the height of theresonance). This is done by acurve-fitting routine to smooth outany noise or small experimentalerrors. From these measurementsand the geometry of the structure,the mode shapes are computedand drawn on a display or aplotter. You can animate thesedisplays to help you understandthe vibration mode.

From the above description, it isapparent that a modal analyzerrequires some type of networkanalyzer to measure the frequencyresponse of the structure and acomputer to convert the frequencyresponse to mode shapes. This canbe accomplished by connecting adynamic signal analyzer through adigital interface to a computerfurnished with the appropriatesoftware.

Figure 5.2. Measured mode shape

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In this chapter we have developedthe concept of looking at problemsfrom different perspectives.These perspectives are the time,frequency and modal domains.Phenomena that are confusing inthe time domain are oftenclarified by changing perspectiveto another domain. Small signalsare easily resolved in the presenceof large ones in the frequencydomain. The frequency domain isalso valuable for predicting theoutput of any kind of linearnetwork. A change to the modaldomain breaks down complicatedstructural vibration problems intosimple vibration modes.

No one domain is always the bestanswer, so the ability to easilychange domains is quite valuable.Of all the instrumentationavailable today, only dynamicsignal analyzers can work in allthree domains. See AgilentApplication Note 1405-2 for adiscussion of the properties of thisimportant class of analyzers.

Related Agilent LiteratureAgilent Application Note —Understanding Dynamic SignalAnalysis, pub. no. 1405-2

Agilent Application Note —UsingDynamic Signal Analysers, pub. no. 1405-3

Agilent Application Note —The Fourier Transform: AMathematical Background, pub. no. 1405-4

Product Overview — Agilent 35670A DynamicSignal Analyzer, pub. no. 5966-3063E

Product Overview — Agilent E1432/33/34VXI Digitizers/Source, pub. no. 5968-7086E

Product Overview — Agilent E9801B DataRecorder/Logger, pub. no. 5968-6132E

Summary

Accelerometer — A transducerwhose output is directlyproportional to acceleration.Typically uses piezoelectriccrystals to produce output.

Curve-fit — A method for creatinga mathematical model that bestfits a set of sampled data.The least square method iscommonly used.

Damping — The dissipation ofenergy with time or distance

Decibels (dB) — A logarithmicrepresentation of a ratioexpressed as 10 or 20 timesthe log of the ratio

Distortion — An undesired changein waveform

Fourier — French mathematicianJean Baptiste Joseph Fourier(1768-1830)

Fourier transform — Analgorithm used to transformtime domain data intofrequency domain

Frequency response — A ratio ofthe output over the input, bothas a function of frequency

Gain-phase meter — A two-channel instrument thatcompares the amplitude levelsand phases of two signals anddisplays the results

Linearity — The response of eachelement is proportional to theexcitation

Load cell — A transducer whoseoutput is directly proportionalto force

Modal analysis — A method forcharacterizing the dynamicbehavior of a structure in termsof natural frequencies, modeshapes and damping

Network analysis — Thegeneral engineering problem ofdetermining how a network willrespond to an input. Networkanalysis is sometimes calledstimulus/response testing.The input is then known as thestimulus or excitation, and theoutput is called the response.

Network analyzer — Aninstrument used to characterizethe frequency response ofelectronic networks

Oscilloscope — An instrumentthat displays voltage waveformsas a function of time

Q (of resonance) — A measure ofthe sharpness of resonance orfrequency selectivity of aresonant vibratory systemhaving a single degree offreedom. In a mechanicalsystem, equal to _ the reciprocalof the damping ratio.

Resonance — Resonance of asystem in forced vibrationexists when any change infrequency, however small,causes a decrease in systemresponse

Shaker — A device for subjectinga mechanical system tocontrolled and reproduciblemechanical vibration

Spectrum — A frequency domainrepresentation of the signal

Spectrum analyzer — Aninstrument for characterizingwaveforms in the frequencydomain

Steady-state — The condition thatexists after all initial transientsor fluctuating conditions havedamped out, and all currents,voltages, or fields remainessentially constant, or oscillateuniformly

Stimulus/response testing —Another name for networkanalysis, or determining how anetwork will respond to aninput. The input is known asthe stimulus or excitation, andthe output is called theresponse.

Strip chart recorders — A devicethat uses one or more pens torecord data on a strip of papermoving at a constant speed.The device provides apermanent graphic record of aparameter (i.e. displacement)vs. time.

Transient response — Thetransitional period of a system'sresponse to excitations until itreaches the steady state. Usedto characterize the dynamicbehavior of a system.

Vibration mode — Acharacteristic pattern assumedby a vibrating system in whichthe motion of every particle is asimple harmonic with the samefrequency. Two or more modesmay exist concurrently in asystem with multiple degreesof freedom.

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Glossary

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