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1 The Process of Measurement: An Overview
ThomasG Beckwith/RoyD. Marangoni/JohnH. LienhardV
2 Standards and Dimensional Units of Measurement
ThomasG.Beckwith/RoyD. Marangoni/JohnH. LienhardV
3 Assessing and Presenting Experimental Data
ThomasG. Beckwith/RoyD. Marangoni/JohnH.LienhardV
4 The Analog Measurand: Time-Dependent Characteristics
ThomasG. Beckwith/Roy D. Marangoni/John H. LienhardV...........................................
6 Sensors
ThomasG.Beckwith/RoyD. Marangoni/JohnH.LienhardV
7 Signal Conditioning
ThomasG.Beckwith/Roy D. Marangoni/JohnH. LienhardV
8 Digital Techniques in Mechanical Measurements
ThomasG.Beckwith/Roy D. Marangoni/JohnH.
9 Strain and Stress: Measurement and Analysis
ThomasG.Beckwith/RoyD. Marangoni/JohnH.Lienhard
Thomas
Thomas
15 Appendix: Theoretical Basis for Fourier Analysis
Thomas
16 Appendix: Number Systems
Thomas
17 Appendix: Some Useful
18 Appendix: Stress and Strain Relationships
19 Appendix: Statistical Tests of Least Squares Fits
Thomas
The Process of Measu rement:
An Overview
INTRODUCTION
2 THE SIGNIACANCE OF MECHANICAL MEASUREMENT
3 FUNDAMENTAL METHODS OF MEASUREMENT
4 THE GENERALIZED MEASURING SYSTEM
s TYPES OF INPUT QUANTmEs
6 MEASUREMENT STANDARDS
7 CALIBRATION
8 UNCERTAINTY: ACCURACY OF RESULTS
9 REPORTING RESULTS
10 FINAL REMARKS
INTRODUCTION
It has been said, "Whatever exists, exists in some amount." The detennination ofthe amount is what measurement is all about. If those things that exist are related 10 the practice of mechanical engineering, then the de!ermination of their amounts conslilutes the subject of mechanical measurements.1
The processorthe act ofmeasurementconsists ofobtaining aquantitative comparison between a predefined standard and a measurand. The word measurand is used to designate the particular physical parameter being observed and quanlified; that is, the input quantity to the measuring process. The act of measurement produces a result (see Fig. I).
The standard of comparison must be of the same character as the measuranid, and usually, bu! not always, is prescribed and defined by a legal or recognized agency or organization-for example, the National Institute of Standards and Technology (NIST), fonnerly called the National Bureau ofStandards (NBS), the International Organization for Standardization (ISO), or theAmerican Naiional Standards Institute (ANSI). The me:ter, for example, is a clearly defined siandard of length.
Such quantities as temperature, strain, and the parameters associated with fluid flow, acoustics, and motion, in addiiion to the fundamental quantities of mass, length, time, and so on. arc typical of those within the scope of mechanical measurements. Unavoidably,
1MeclumicalmeantremtnUarenotneeessarilyaccomplishedbymechanicalmeans: rather,iiistothemeasured quantityitselfthatthe term mtchankal isdirected. Thephrase mtasunmtnl ofmechanical quantit1its, or of paramettrJ, wouldperhapsexpressmoreoomp�lythenaningintended. Intheinterestofbrevity, however, thesubjectissimplycalledmechanicalmecuurY11Un1s.
(readout)
FIGURE I: Fundamental measuringprocess.
themeasurementofmechanicalquantitiesalsoinvolvesconsiderationofthingselectrical, sinceitisoftenconvenientornecessary to change,ortransduce, amechanicalmeasurand intoacorrespondingelectricalquantity.
2 THE SIGNIFICANCE OF MECHANICALMEASUREMENT
Measurementprovidesquantitativeinformationontheactualstateofphysicalvariablesand processesthatotherwisecouldonlybeestimated.Assuch,measurementisboththevehicle fornewunderstandingofthephysicalworldandtheultimatetestofanytheoryordesign. Measurementisthe fundamental basis forall research, design, and development, andits roleisprominentinmanyengineeringactivities.
Allmechanicaldesign ofanycomplexity involvesthreeclements: experience, the rationalelement, and theexperimentalelement. Theelementofexperienceisbasedon previousexposureto similarsystemsandon anengineer's commonsense. Therational elementreliesonquantitativeengineeringprinciples, thelawsofphysics,andsoon. The experimentalelementisbasedon measurement-that is, onmeasurementofthevarious quantitiespertainingtotheoperationandperformanceofthedeviceorprocessbeingdeveloped. Measurement provides a comparison between what was intended and what was actually achieved.
Measurement is also a fundamental element ofany control process. The concept ofcontrol requires themeasureddiscrepancybetween the actual and thedesiredperformances. Thecontrollingportionofthesystem mustknowthemagnitudeanddirectionof thedifferenceinordertoreactintelligently.
Inaddition,manydailyoperationsrequiremeasurementforproperperformance. An example isin thecentral powerstation. Temperatures, flows,pressures,and vibrational amplitudesmustbeconstantly monitoredbymeasurement toensure properperformance ofthesystem. Moreover,measurementis vital tocommerce. Costsareestablishedonthe basisofamountsofmaterials,power,expenditureoftimeandlabor,andotherconstraints.
Tobeuseful,measurementmustbereliable. Havingincorrectinformation is potentially more damaging than having no information. The situation, ofcourse, raises the questionoftheaccuracyoruncertaintyofameasurement. Arnold0.Beckman,founderof BeckmanInstruments,oncestated,"Onethingyoulearninscienceisthatthereisnoperfect
answer, noperfect measure."2 Itisquiteimportantthatengineersinterpreting theresultsof measurement havesomebasis forevaluating thelikely uncertainty. Engineersshouldnever simply read a scale or printout and blindly accept the numbers. They must carefully place realistic tolerances on each of the measured values, and not only should have a doubting mind but also should attempt to quantify their doubts. We will discuss uncertainty in more detail in Section 8.
3 FUNDAMENTAL METHODS OF MEASUREMENT
Therearetwo basic methods ofmeasurement: (I) directcomparison with either a primary orasecondary standard and(2)indirectcomparison through the useof acalibratedsystem.
3.1
Direct Comparison
How would you measure the length of a bar ofsteel? If you were to be satisfied with a measurement to within, let us say, k in. (approximately 3 mm), you would probably use a steel tapemeasure. Youwouldcomparethelengthofthebarwith astandardandwouldfind that the bar is so many inches long because that many inch-units on your standard are the same length as the bar. Thus you would havedetermined the length bydirectcomparison. The standard that you have used is called a secondary standard. No doubt you could trace its ancestry back through no more than four generations to the primary length standard, which is related to the speed oflight.
Although to measure by direct comparison is to strip the measurement process to its barest essentials, the method is not always adequate. The human senses are not equipped to make direct comparisons ofall quantities with equal facility. In many cases they arc not sensitive enough. We can make direct comparisons of small distances using a steel rule, withaprecisionofabout 1 mm(approximately 0.04 in.). Often werequiregreater accuracy. Then we must call for additional assistance from some more complex form of measuring system. Measurement by direct comparison is thus less common than is measurement by indirectcomparison.
3.2 Using a Calibrated System
Indirectcomparison makes use ofsome form oftransducing devicecoupledto a chain of connecting apparatus, which we shall call, in toto, the measuring system. This chain of devices converts the basic form of input into an analogous form, which it then processes and presents al the output as a known function of the original input. Such a conversion is often necessary so that the desired information will be intelligible. Thehuman senses arc simply not designed to detect the strain in a machine member, for instance. Assistance is required from a system that senses, converts, and finally presents an analogous output in the form ofa displacement on a scale or chart or as a digital readout.
Processing of the analogous signal may take many forms. Often it is necessary to increase an amplitude or a power through some form of amplification. Or in another case it may, be necessary to extract the de.�ired information from a mass of extraneous input by a process of filtering. A remote reading or recording may be needed, such as ground recording ofa temperature or pressure within a rocket in flight. ln this case the pressureor
In each ofthe various cases requiring amplification, or filtering, orremoterecording, electrical methods suggest themselves. In fact, the majority of transducers in use, particularlyfordynmrncmechanicalmeasurements, conven the mechanical input into an analogouselectrical fonn forprocessing.
4 THE GENERALIZED MEASURING SYSTEM
Mostmeasuringsystems fall within lhcframeworkofageneral arrangementconsisting of threephasesorstages:
Each stage consists ofadistinct component or group ofcomponents that perfonns required and definite steps in the measuremenL These are called basic elements; their scopeisdetenninedbytheirfunctionratherthanbytheirconstruction. Figure 2 andTable I outlinethe significance ofeachofthesestages.
4.1
First. or Sensor-Transducer, Stage
Theprimary functionofthefirststageis lodetect ortosensethemeasurand. Al thesame time, ideally,thisstage should be insensitivetoevery otherpossibleinput For instance, if il isapressure pickup, itshould be insensitiveto,say,acceleration; ifit isastraingage, it shouldbe insensitive to temperature; if a linearaccelerometer, it should be insensitive lo angularacceleration;andsoon. Unfonunately, ii is rare indeedlofindadetectingdevice that iscompletely selective. Unwantedsensitivity isa measuringerror, called noise when ii variesrapidlyanddriftwheniivaries veryslowly.
Frequently one findsmorethanasingletransduction(changein signalcharacter) in thefirststage,particularly ifthefirst-stageoutput iselectrical.
4.2 Second, or Signal-Conditioning, Stage
Thepurposeofthesecondstageofthegeneralsystemistomodifythetransducedinfonnation sothatitisacceptabletothethird, ortenninating, stage. Inaddition, itmay perform one or more basic operations, such asselective filtering to remove noise, integration,dif-
Modifiestransduced signal into fonnusableby final stage. Usually increasesamplitude and/orpower, depending on requirement. May also selectively filterunwantedcomponentsor convertsignal into pulsedfonn
Stage3: · Readout-Recording
Provides an indicationor recordinginformthatcan be evaluated byan unaidedhuman sense or by a controller. Records datadigitally on acomputer
Recorders: Digital printing,inked penandchart,directphot1>graphy. magnetic recording (hard disk)
ProcessorsandcompUle,,r:
Various typesorcomputing systems.eitherspecial-purpose or general. used tofeed readout/recordingdevices and/or controllingsystems
Controllers: AU types
Probablythemostcommonfunctionofthesecondstageistoincreaseeithera11Dplitude or powerofthe signal, or both, to the level required to drive the final terminating.device. Inaddition,thesecondstagemustbedesignedforpropermatchingcharacteristicsloetween thefirstandsecondand between thesecondandthirdstages.
4.3
Third, or Readout-Recording, Stage
The third stageprovides the information sought in a form comprehensible to one ofthe human sensesortoacontroller. Iftheoutputisintendedforimmediatehuman recc•gnition, itis,withrare exception,presentedinoneofthefollowing forms:
FIGURE 3: (a)Gage for measuring pressure in automobile tires. (b) Block diagram oftire-gage functions. Inthis example, thespringservesasasecondarytransducer.
2. In digital form, as presented by a counter such as an automobile odometer, or by a liquid crystal display (LCD) orlight-emitting diode (L_ED) display ason adigital voltmeter
To illustratea very simple measuring system, let us considerthe familiar tire gage used forcheckingautomobile tire pressure. Suchadevice isshown in Fig. 3(a). Itconsists ofa cylinder and piston, a spring resisting the piston movement, and a stem with scale divisions. As the air pressure bears against the piston, the resulting forcecompresses the spring until thespring andairforcesbalance. Thecalibratedstem, which remainsinplace afterthespringreturnsthepiston, indicatesthe applied pressure.
Thepiston-cylindercombinationconstitutesaforce-summingapparatus,sensingand transducing pressureto force. As a secondary transducer, thespringconverts the force to adisplacement. Finally, thetransduced input istransferred without signalconditioningto thescaleand index forreadout [seeFig. 3(b)].
TheProcessofMeasurement:AnOverview
Asanexampleofa morecomplexsystem,letussaythatavelocityistobemeasured, as showninFig. 4. Thefirst-stagedevice, theaccelerometer,providesavoltageanalogous to acceleration.3 In addition to a voltage amplifier, the second stage may also include a filter that selectively attenuates unwanted high-frequency noise components. It may also integratetheanalogsignal with respecttotime,therebyprovidingavelocity-timerelation, rather than an acceleration-time signal. Finally, the signal voltage will probably need lo be increased to the level necessary to be sensed by the third, or recording and readout, stage, which may consist ofadata-acquisition computer and printer. The final record will thenbe inthe formofacomputer-generated graph; withthe proper c;ilibration. anaccurate velocity-versus-time measurementshould be the result.
5 TYPES OF INPUT QUANTITIES
5.1 Time Dependence
Mechanical quantities, in addition to their inherent defining characteristics, alsohavedistinctive time-amplitude properties, whichmay be classified as follows:
Ofcourse.theunchanging, staticmcasurandis themosteasilymeasured. Ifthesystem is terminated by some form ofmeter-type indicator. the meter's pointer has no difficulty in eventually reaching a definite indication. The rapidly changing. dynamic measurand presentstherealmeasurementchallenge.
Two general forms ofdynamic input are possible: steady-stale periodic input and transient input. The steady-stale periodic quantity is one whose magnitude has a definite repeating time cycle, whereas the time variation of a transient quantity docs not repeat. "Sixty-cycle"linevoltageisanexampleofasteady-slate periodic signal. Soalso are many mechanical vibrations, after abalancehas been reached between aconstant inputexciting energyandenergydissipatedby damping.
Anexampleofapulsedtransientquantityistheacceleration-lime relationship accompanying an isolated mechanical impact. Themagnitude is temporary. being completed in a matter of milliseconds, with the portions of interest existing perhaps for only a few microseconds. The presence of extremely high rates of change, or wavefronts. can place seven: demands onthe measuring system.
5.2 Analog and Digital Signals
Most measurandsofinterestvary with timeinacontinuousmannerovera range ofmagnitudes. Forinstance, the speed ofan automobile. asitstarts from rest, has some magnitude
lAlthough lheaccelerometermaybesusceptibletoananalysisof"stages"within i1self,weshall forgo such ananalysisinthisex.ample.
Stage 1
Sensor-lransducer
Voltageoutput fromaccelerometer with unwanted "noise"
Stage2
Stage 3
Signal-conditioningsystem i Recording-readoutsystem I
Integrating circuit Ampllller
Signal withnoisa removed
Time-integrated 'IOltageanalogus tovelocily
4: Block diagram of a relatively complex measuring system.
FIGURE
atevery instantduringitsmotion. Asensorthatrespondstovelocitywillproduce:anoutput signal havingatimevariationanalogoustothe time change intheauto'sspeed. Werefer to such a signal as ananalog signal becauseit is analogous to a continuousphysicalprocess. An analog signal has a value at every instant in time, and it usually varies smoothly in magnitude.
Some quantities, however, may change in a stepwise manner between two distinct magnitudes: a high and low voltage«on and off, for instance. The revolutions ofa shaft couldbecountedwithacam-actuatedelectricalswitchthatis openorclosed, depending on the position ofthe cam. If the switch controls current from a battery, current either flows with a given magnitude or does not flow. The current flow varies discretely between two values, which we could represent as single digits: I (flowing) and 0 (not Hewing). The amplitude ofsuch a signal may thus be calleddigital.
Many electronic circuits store numbers assets of digits-strings of ls and Os-with each string held in a separate memory register. When digital circuits, such as those in computers, are used to record an analog signal, they do so only at discrete points in time . becausethey have only a fixed numberofmemory registers. The analogsignal, which has a valueat every instantoftime, becomesadigitalsignal.A digitalsignal isasetofdiscrete numbers, each corresponding to the value of the analog signal at a single specilfic instant of time. Clearly, the digital signal contains no information about the value oftine analog signal attimes other than sample times.
Mechanical quantities-such as temperatures, fluid-flow rates, pressure, stress, and strain-normally behave timewise in an analog manner. However, distinct advantages are often obtained in converting an analog signal to an equivalent digital signal for the purposes ofsignal conditioningand/or readout. Noiseproblemsare reducedors1Jmetimes eliminatedaltogether,anddatatransmissionissimpler. Computersaredesigned toprocess digital information, and directnumerical displayor recording is more easily accomplished by manipulating digital quantities.
6 MEASUREMENTSTANDARDS
As stated earlier, measurement is a process ofcomparison. Therefore, regardless of our measurement method, we must employ a basis of comparison-standardized units. The standardsmustbepreciselydefined,and,becausedifferentsystemsofunitsexist, themethod ofconversion from system to system must be mutually agreed upon.
Most importantly, a relationship between the standards and the readout scale of each measuring systemmust be established through aprocess known as calibration.
7 CALIBRATION
Atsomepointduringthepreparationofameasuringsystem,knownmagnitudesoftheinput quantity must be fed into the sensor-transducer, and the system's output behavior·must be observed. Suchacomparisonallowsthemagnitudeoftheoutputto becorrectlyinterpreted in terms ofthe magnitude ofthe input. This calibration procedure establishes the correct output scale for the measuring system.
By performing such a test on aninstrument, we both calibrate its scale and prove its abilityto measure reliably. Inthissense, wesometimes speakofprovinganinstrument. Of
TheProcessofMeasurement:AnOverview
course,ifthe calibration is to be meaningful, the knowninput must itself be derivedfrom a defined standard.
Ifthe output is exactly proportional to the input (output = constant x input), then a single simultaneous observation of input and output will suffice to fix the constant of proponionality. This is called single-point calibration. More often, however, multipoint calibrations areused, wherein a number of different input values are applied. Multipoint calibrationworkswhentheoutputis notsimplyproportional,and,moregenerally,improves theaccuracy ofthe calibration.
If a measuringsystemwill be usedtodetectatime-varyinginput, thenthe calibration should ideally be made using atime-dependentinputstandard. Such dynamic calibration can bedifficult,however, andastaticcalibration, using aconstantinputsignal,is frequently substituted. Naturally, thisprocedure.isnotoptimal;themorenearlythecalibrationstandard correspondstothemeasurandinallitscharacteristics,thebetter theresultingmeasurements.
Occasionally, thenature ofthe system or one ofits componentsmakes theintroduction of a sample ofthe basic input quantity difficultOI' impossible. One of the important characteristicsofthebondedresistance-typestraingageisthefactthat,throughqualitycontrol atthetime ofmanufacture, spot calibration may beapplied to acompletelotofgages. As a result, an indirect calibration ofa strain-measuring system may beprovided through the gage factorsupplied by the manufacturer. Instead ofattempting toapplya known unit strainto thegage installed onthetest structure-which, ifpossible, wouldoftenresultinan ambiguoussituation-aresistancechangeissubstituted. Through thepredetermined gage factor,thesystem'ssuainresponsemay thereby be obtained.
8 UNCERTAINTY: ACCURACY OF
RESULTS
Error may be defined as thedifference between the measured result and the true value of the quantity being measured. While we do know the measuredvalue,wedo notknow the true value, and so we do notknowthe erroreither. Ifwe estimate a likely upper bound on the magnitude of the error, that bound is called the uncertainty: We estimate, with some level ofodds, thatthe error will be nolarger than the uncertainty.
To estimate the sizeoferrors, we musthave some understanding oftheir causes and classifications. Errors can be oftwo basic types: bias, or systematic, error and precisio11, or random, error.
Should an unscrupulous butcher place a ball of putty under the scale pan, the scale readouts would be consistently in error. The scale wouldindicate a weight of product too great by the weightof the putty. This zero offset represents one type of systematicerror.
Shrink rules areused to make patterns forthecastingofmetals. Caststeelshrinksin coolingbyabout 2%; hencethepatternsusedfor preparingthe moldsare oversized by the proper percentage amounts. Thepatternmakerusesashrinkrule on which the dimensional units are increased by that amount. Should apattern maker's shrink rule for cast steel be inadvenently used for ordinary length measurements, the readouts would be consistently undersized by s\i in one (that is,by 2%). This isanexample ofscaleerror.
In each of the foregoing examples the errors are constant and of a systematic nature. Such errorscannot be uncovered by statisticalanalysis.
An inexpensivefrequencycountermayuse the 60-Hz power-line frequencyasacomparison standard. Power-linefrequencyis held veryclosetothe 60-Hz standard. Although it does wander slowly above and below the average value, over a period of time-say,
aday-the average is very closeto 60 Hz. The wandering israndom and the momentto-momenterror in the frequencymeterreadout(fromthissource)iscalled precision, or random, error.
Randomness may also be introduced by variations in the measurand itself. If a numberofhardness readings ismadeonagivensampleofsteel,arangeofreadingswill beobtained. Anaveragehardnessmaybecalculatedandpresented as theactualhardness. Singlereadings will deviatefromtheaverage, somehigherandsomelower. Ofcourse, theprimaryreasonforthevariations inthiscaseisthenonhomogeneityofthecrystalline structureofthetestspecimen. Thedeviationswillberandomandareduetovariationsin themeasurand. Randomerrormaybeestimatedbystatisticalmethods.
9 REPORTING RESULTS
Whenexperimental setups are made and time andeffort are expendedtoobtainresults, it normally follows thatsomeformofwrittenrecordorreport istobe made. The purpose ofsuchareconl will determineitsform. Infact, insomecases,severalversions will be prepared. Reportsmaybecategorizedasfollows:
1. Executivesummary
2. Laboratorynoteortechnicalmemo
3. Progressreport
4. Fulltechnicalreport
S. Technicalpaper
Very briefly, an executive summary is directed ata busy overseerwho wants only thekey featuresofthework: whatwasdoneandwhatwasconcluded,outlinedinafew paragraphs. Alaboratorynoteiswrittento be readbysomeonethoroughlyfamiliarwiththe project,suchasanimmediatesupervisorortheexperimentalisthimselforherself. A full report tellsthecompletestorytoonewhoisinterestedinthesubjectbutwhohasnotbeen indirecttouchwiththespecificwork-perhapstopofficialsofalargecompanyorareview committeeofasponsoringagency. A progressreportisjustthat-oneofpossiblyseveral interimreportsdescribing1hecurrentstatusofanongoingproject,whichwilleventuallybe incorporatedinafullreport. Ordinarily,atechnical paperisabriefsummaryofaproject, theextentofwhichmustbetailoredtofiteitheratimeallotmenlatameeting or spaceina publication.
Severalfactorsarecommontoallthevariousforms. Witheachtype,thefirstpriority istomakesurethattheproblemorproject thathas beentackled is clearlystated. Thereis nothingquitesofrustrating as readingdetailsinatechnicalreportwhileneverbeingcertain oftheraisond'etre. Itisextremelyimportanl 10 makecertain 1hatthereaderisquickly cluedinonthe why beforeoneattemptstoexplain the how andtheresults. Aclearlystated objectivecanbeconsideredthemostimportantpart of thereport. The.entirereportshould bewritteninsimplelanguage. A rulestatedbySamuelClemensisnotinappropriate: "Omit unnecessaryadverbsandadjectives."
9.1 Laboratory Note orTechnical Memo
Thelaboratorynoleiswrittenforaverylimitedaudience,possiblyevenonlyasamemory joggerfortheexperimenteror, perhaps moreoften,fortheinformationofan immediate supervisor whois thoroughly familiarwith the work. Insomecases, a singlepage may
be sufficient, including a sentence or two stating the problem. a block diagram of the experimentalsetup, and some data presented either in tabular form oras aplotteddiagram. Any pertinent observations not directly evident from the data should also be included. Sufficientinformationshouldbeincludedsothattheexperimentercanmentallyreconstruct thesituationandresults Iyearoreven 5 years hence. Adate and signatureshouldalwaysbe includedand, ifthereis apossibility ofimportant developments stemming from the work, such asa patent, asecond witnessing signature should beincludL'<I anddated.
9.2 Full Report
The full reportmust relate all the facts pertinent to the project. II is evenmore important in thiscaseto makethe purpose oftheprojectcompletely clear, for thereportwillbe read by persons not closely associated with the work. The full report should also include enough detailto allow another professional lo repeal the measurements and calculations. Oneformatthat has much merit isto make the reportproper-<he main body-short and to the point, relegating to appendices the supporting materials, such as data, detailed descriptions of equipment, review of literature, sample calculations, and so on. Frequent reference to these materials can be made throughout the report proper, but the option to peruse the details is left to the reader. This scheme also provides a good basis for the technicalpaper,should it be planned.
93 Technical Paper
A primary purpose of a technical paper is to make known (to advertise) the work of the writer. For this reason, two particularly important portions ofthe writing are theproblem statement and the results. Adequately done, these two items will attract the attention of otherworkers interested in the particular field, who can then make directcontact with the writer{s) foradditional details and discussion.
Space, number of words, limits on illustrations, and perhaps lime are all factors making the preparation of a technical paper particularly challenging. Once the problem statementandtheprimary results have been adequately established, theremaining available spacemay be used to summarize procedures, test setups, and the like.
10 FINALREMARKS
An attempthas been made in this chapterto provide an overall preview ofthe problems of mechanical measurement. In conformance with Section 9, we have tried to state the problem as fully as possible inonly a few pages.
PROBLEMS
I. Write an executive summary orthis chapter.
2. Consider a mercury-in-glass thermometer as a temperature-measuring system. Discuss the various stages ofthis measuring system indetail
3. For the thermometer of Problem 2, specify how practical single point calibration may be obtained.
4. Set uptestproceduresyouwouldusetoestimate, withtheaidonlyofyourpresentjudgment and experience, the magnitudes ofthe common quantities listed.
(a) Distance between the centerlines ortwo holes in a machined part
(b) Weighl of two smallobjects ofdifferenldensities
( c) nme intervals
(d) Temperatureof waler
( e) Frequency of pure tones
5. Consider the impac1 frame shown in Figure5. Mass M, whichtravelsalonggui�lerails, is raised to an initial height H and released from resl Discuss how you would me:asure the mass velocityjust prior to impacl wilh lhe leSIilem in order to accounl for friction between mass M and lhe guide rails.
FIGURE 5: Impact test frame for Problem 5.
Standards and Di mensional
Un its of Measurement
INTRODUCTION
2 HISTORICAL BACKGROUND OF MEASUREMENT IN THE UNITED STATES
3 THE SI SYSTEM
4
5
6
7
B
9
10 SUMMARY
1 INTRODUCTION
The basis of measurement is thecomparison between a measurand and a suitablestandard. In thisarticle, we will take a closer look at the establishment of standards.
The term dimension connotes the defining characteristics of an entity (measurand), and the dimensionalunit is the basis for quantification of the entity. For example, length is a dimension, whereas centimeter is a unit of length; time is a dimension, and the second is a unit of time. A dimension is unique; however, a particular dimension--;ay, length-may be measured in various units, such as feet, meters, inches, or miles. Systems ofunits must be established and agreed to; that is, the systems must be standardized. Because there are various systems, there must also be agreement on the basis for conversion from system to system. It is clear, then, that standards of measurement apply to units, to systems of units, and tounitary conversion between such systems.
Ingeneralterms, standards are ubiquitous. Thereare standardsgoverningfoodpreparation, marketing, professional behavior, and so on. Many are established and governed by either federal or state laws. So that we may avoid chaos, it is especially important that the basic measurement standards carry the authority of not only federal, but also international, laws.
In the following sections, we will discuss those standards, systems of units, and problems of conversion that are fundamental to mechanical measurement.
2 HISTORICAL BACKGROUND OF MEASUREMENTIN THE UNITED STATES
The legal authority to control measurement standards in the United States was assigned by the U.S. Constitution. Quoting from Article I, Section 8, Paragraph S, of the U.S.
PromMechmiicalMeasunments, Sixth Edition, Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V. Copyright02007 by Peanon Education. Inc. Publishedby Prentice HaU. All rights reserved.
StandardsandDimensionalUnitsofMeasurement
Constitution: "1be congress shall have power to...fix the standard or weights and measures." AlthoughCongress wasgiventhepower,considerabletimeelapsedbeforeanything was done about it. In 1832, the Treasury Department introduced a uniform system or weights andmeasurestoassist thecustomsservice; in 1836, thesestandardswereapproved by Congress [1). In 1866, the Revised Statutes orthe United States, Section 3569, added the stipulation that "It shall be lawful throughout the United States ofAmericato employ the weightsand measuresofthemetric system." This simply makes itclearthat the metric system may be used. In addition, this act established the following (and now obsolete) relation forconversion:
1 meter = 39.37 inches
AninternationalconventionhelpinParisin 1875 resulted inan agreementsignedfor the United States by the U.S. ambassador to France. The following is quoted therefrom: "Thehighcontractingpartiesengagetoestablishand maintain, attheirexpense, ascientific and permanent international bureau ofweights and measures, the location ofwhich shall be Paris" (2,3]. Although this established a central bureau of standards, which set up at Sevres, a suburb of Paris, it did not, ofcourse, bind the United States to make use ofor adoptsuch standards.
On April 5, 1893, in all absence of further congressional action, Superintendent Mendenhall oftheCoastand GeodeticSurvey issued the following order [2,4J:
The Office ofWeights and Measures with the approval ofthe Secretary oftheTreasury, will in the futureregard the international prototype meterandthe kilogram asfundamental standards. and the customary unils, the yard and pound, will be derived therefrom in accordan�-c with the Act of July 28, 1866.
The Mendenhall Order turned out to be a very important action. First, it recognized the meter and the kilogram as being fundamental units on which all otherunits oflength and massshouldbebased. Second,ittiedtogether themetricandEnglishsystemsoflength and mass in a definite relationship, thereby making possible international exchange on an exactbasis.
In response to requests from scientific and industrial sources, and to a great degree influencedbytheestablishmentorlike institutionsinGreatBritainandGermany, 1 Congress onMarch3, 1901 passedanactprovidingthat"TheOfficeofStandardWeightsandMeasures shall hereafter be known as 'The National Bureau ofStandards"' [5). Expanded functions ofthe new bureau were set forth and included development orstandards, research basic to standards, and thecalibration of standards and devices. The National Bureau ofStandards (NBS) wasformallyestablishedinJuly 1910, and its functionswereconsiderablyexpanded by an amendment passed in 1950. In 1988, Congress changed the name ofthe bureau to "The National InstituteofStandards andTechnology" (NIST) [6]. Commercial standards are largely regulated by state laws; to maintain uniformity, regular meetings (National Conferences on Weights and Measures) are held by officials ofNISTand officers of state governments. Essentially all state standards of weighL• and measuresarein accordance with the Conference's standards andcodes. Internationalunifomtityismaintainedthroughregularlyscheduledmeetings(heldatabout6-ycarintervals),
Theseven baseunitsoftheSISystemarelistedinTable 1. Otherdimensionscanbe derivedfromthesebaseunitsbymultiplyingordividingthem. A fewsuchderivedunits are assigned special names; othersarenot For example, the unitofforce,the newton, is obtained from the kilogram,themeter,andthesecondas
I newton = I kg/m.s2
Incontrast,areaissimplymeterssquared(m2). Workandenergyare expressed injoules (kg · m2/s). Theterm hertz isused for frequency (s-1) andtheterm pascal isused for pressure(N/m2). SomederivedunitscarryingspecialnamesarelistedinTable 2. andsome withoutspecialnamesaregiveninTable 3. Note1hat,whereasthoseassignedspecialnames thatoriginatefrompropernamesarenotcapitalized,thecorrespondingabbreviationsare capitalized.
Itshouldbe clearthatallthe variousderivedunitscanbe expressed intermsofbase units. Incertain instances whenaunitbalanceisattemptedforagivenequation.itmaybe desirable,ornecessary,toconvertallvariables to base uni1s.
TABLE1: BaseUnitsintheSISystem
Unit
Quantity
NameSymbol length meter m mass kilogram kg time second s elcc1riccurrent ampere A temperature kelvin K amountofsubstance mole mol luminous intensity candela cd
Toaccommodatethewritingofvery largeor very small values,lheSIS:ystemdefines lhe multiplying prefixes shown inTable 4. Forexample, 2,500,000 Hzmay be written as 2.5 MHz (megahertz), and 0.000000000005 farad as 5 pF(picofarad). Only one prefix shouldbeusedwithagivendimension;thus,itwouldbeincorrecttowrite2.5 lckHzinplace of2.5 MHz. Likewise, for units ofmass, I000 kgmight be written as I Mg(megagram).
In lhe following seclions, we discuss lhe SI standards of length, mass, time, and current ingreaterdetail. The standards ofluminousintensity andamountofsubslance are describedin reference[7).
TABLE 3: SomeSI-DerivedUnits
Derived Quantity Symbol area m2 acceleration mJs2 angularacceleration rad/s2 angular velocity rad/s density kg/m3 dynamicviscosity Pa. s heatflux W/m2 momentof force N·m specific heatcapacity J/kg . K velocity mis volume m3
TABLE4: Multiplying Factors
101 dcka da 10-1 deci d 1<>2 hecto h 10-2 centi c 1<>3 kilo k 10-3 milli m 11>6 mega M 10-6 micro µ, 109 giga G 10-9 nano n 1012 tera T 10-12 pico p 101s peta p 10-IS femto f 1018 exa E 10-18 atto a 1021 zetta z 10-21 zepto z 1024 yotta y 10-24 yocto y
MetricConversion in the United States
In May 1965, theUnitedStatesannounceditsintentionofadoptingtheSI system. In 1968, the passage ofPublic Law90-472 authorizedtheSecretacy ofCommercetomakea"U.S. MetricStudy" tobe reported byAugust 1971. Afterprolongeddebates,studies,andpublic pronouncements of I0-yearconversion plans, on December 23, 1975, the94th Congress approved Public Law94-168,called theMetricConversionActof 1975. Itsstatedpurpose wa�as follows: ''Todeclareanational policyofcoordinatingtheincreasinguseofthemetric system inthe United Slates, andtoestablish.a UnitedStatesMetricBoardtocoordinatethe voluntary conversion to the metric system." Note especially that theconversion was lobe volu111ary·and thatno time limit wasset. TheAct made clear that, in usingthetermmetric, the SI System ofunits was intended.
In 1981, the U.S. Metric Board reportedloCongressthatitlackedtheclearCongressional mandate needed toeffectively bring aboutnational conversion lothemetricsystem; funding forthe Board waseliminated after fiscal year 1982 [8].
Congresssubsequently amendedtheMetricConversionAct withtheOmnibusTrade and CompetitivenessActof1988, Public Law 100-418 (6). These amendments provided strong incentives for industrial conversion to SI units. The amended act declares that the metric system is "the preferred system of weights and measures for United States trade and commerce." II further requires that all federal agencies use the metric system in procurement, grants, and business-related activities; this requirement was to be met by the end of fiscal year 1992, except in cases where conversion would harm international competitiveness.
MetricationintheUnitedStateshasprogressed,especiallyintheautomotiveindustry and certain partsofthefoodanddrinkindustries. Classroomusehasincreasedtothepoint that most engineering courses rely primarily on SI units. Throughout this book, we shall use both the SI systemandthe EnglishEngineeringsystem,2 withthehopeofencouraging thecompleteconversion toSI units.3
2This term may appeartobeincongruousgiventhatthe United Kingdomhasadopted the SISystem llowe>er. this usage is so well established that the term hasoutlivedits origins.
JAttention is dilo<ted to ref..,,nce (9(. which is anexcellentguide forapplying the metric system.
4 THE STANDARD OF LENGTH
The meter was originally intended to be one ten-millionth of the earth's quadrant. ln 1889, the First General Conference on Weights and Measures defined the meter as the length ofthe InternationalPrototypeMeter, the distance between two finely scribed lines on a platinum-iridium bar when subject to certain specified conditions. On October 14, 1960, theEleventhGeneralConferenceonWeightsandMeasures adopted a new definition of the meter as 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p1o and Sds of the krypton-R6 atom. The National Bureau of Standards ofthe United States also adopted this standard, and the inch hccame 41,929.398 54 wavelengths ofthe krypton light.
As itturnedout, thewavelength ofkryptonlight could only be determined to about 4 parts per billion, limiting the accuracy ofthe meter to a similarlevel. During the 1960s and early 1970s, laser-based measurements offrequency and wavelength evolved to such accuracy that the uncertainly in the meter became the limiting uncertainty in determining the speed of light (10,11) This limitation was of serious concern in both atomic and cosmological physics, and on October 20, 1983, the SeventeenthGeneral Conference on WeightsandMeasuresredefinedthemeterdirectly in terms of thespeedoflight:
The meter is the length of the path traveUed by light in vacuum during a timeinterval of 11299,792,458 ofasecond
Thisdefinitionhastheprofoundeffectofdefi11ingthe speed oflight to be 299,792,458 mis, whichhad been the accepted experimental valuesince 1975 [12).
4.1 Relationship ofthe Metertothe Inch
The 1866 U.S. Statutehadspecified that 1 m = 39.37 inches, resulting in the relationship I in. = 2.54000508 cm (approximately)
In 1959, theNational Bureau ofStandards made a small adjustment to this relationship to ensure international agreementonthedefinition oftheinch [ 13]:
I in. = 2.54 cm (exactly)
Thissimplerrelationshiphadalreadybeen used asan approximation bye
ersforyears. Thedifferencebetween these two standards may be written as 2.5400508/2.54 - I = 0.000002 or0.0002%, which is about � in. permile.
Wegain a sense ofthe significance ofthe difference by considering the following situation. In 1959, theworkoftheUnited StatesNational Geodetic Survey wasbased on the 39.37 inJmrelationshipandacoordinate systemwith itsorigin located in Kansas. Changing the relationship from 39.37 inJm (exactly) to 2.54 in.fem (exactly) would have caused discrepanciesofalmost 16ftatadistanceof 1500miles. Onecanonly imaginetheconfusion overproperty linesifsuchachangehad been made! This problem wasresolvedbydefining separately the U.S. surveyfoot (12139.37 m) andthei11ternat(o11a/foot (12 x 2.54cm). The survey foot isstillusedwith U.S. geodetic data and U.S. statute miles [14).
5 THE STANDARD OF MASS
The kilogram is defined as the mass of the International Prototype Kilogram, a platinumiridium weight kept at the International Bureau ofWeights and Measures near Paris. Of the basic standards, this remains the only one established by a prototype (by which is meant the original model or pattern, the unique example, to which all others are referred for comparison). Various National Prototype Kilogram masses have been calibrated by comparison to the International Prototype Kilogram. These 1nasses are in tum used by the standardsagencies ofvariouscountries to calibrate otherstandard masses,and so on, until one reaches masses or weights ofday-to-daygoodsand services.
Apart from the inconvenience of maintaining this chain ofcalibration, the definition ofthe kilogram by an international prototype leads to several very fundamental problems: The prototype can bedamaged or destroyed; the mass ofthe prototype fluctuates by about one part in 108 owing togas absorption andcleaning; and the prototypeagesin an unknown manner, perhaps having resulted in 50 µ.g ofvariation during the pastcentury [15].
Inrecent years, considerableefforthas been giventodeveloping anew mass slandard that can be reproduced in any suitably equipped lab, without the use ofa prototype. One approachbeingconsideredistopreciselydetennineAvogadro'snumberbymassanddensity measurements ofsilicon crystals. This value ofAvogadro's numbercould then beused with an atomic unit of mass to define the kilogram as the mass of a specific number of atoms [15,l 6). An alternativeapproach, which promisessomewhatbetteraccuracy,usesa"moving coil wall balance" to compare the mechanical and electrical power exerted on a currentcarrying conductor that moves against gravity in a magnetic field. This technique leads to a definition ofthe kilogram in terms of fundamental physical quantities [17,18).
The pound was defined in terms of the kilogram by the Mendenhall Order of 1893. In 1959, the definition was slightly adjusted (13], giving the relationship still in use today:
I pound avoirdupois = 0.453 592 37 kilogram
6 TIME AND FREQUENCY STANDARDS
Until 1956, the second was defined as 1186,400 of the average period of revolution of the earthon itsaxis. Althoughthisseems tobearelatively simpleandstraightforwarddefinition, problems remained. There is a gradual slowing of the earth's rotation (about0.001 second per century) ( 19), and, in addition, the rotation is irregular.
Therefore, in 1956, an improved standard was agreed on; the second was defined as 1131,556,925.9747 ofthe time required by the earth toorbit the sun in the year 1900. This is called the ephemerissecond. Although the unit is defined with a high degree ofexactness, implementation of the definition was dependent on astronomical observation, which was incapable ofrealizing the implied precision.
In the 1950s, atomic research led to the observation that the frequency of electromagnetic radiation associated with certain atomic transitions may be measured with great
StandardsandDimensionalUnitsof
Measurement
repeatability. One-thehyperfinetransitionofthecesium atom-wasrelatedtotheephemeris second with anestimatedaccuracyoftwo parts in 109. On October 13, 1967, inParis, theThirteenthGeneralConference on Weights and Measures officially adopted the following definition orthe second as the unit oftime in the SJ System (7):
The second is the duration or 9.192,631.770 periods or lhe radia1ion com:sponding 10 1he lransition belween lhclwo hyperline levels of lhe ground stale orthe cesium 133 alom.
Alomic apparatuses, commonly called "atomic clocks,"are used to produce the frequency ofthe transition (20). In afountainclock (21), a gas ofcesium atoms is introduced into a vacuum chamber, where a set of laser beams is used to slow the molecular motion, pushing a group of atoms intoa ball and cooling them to a temperature near absolute zero. Anotherlaser is then used to toss the ball ofatoms upward into a microwave cavity, where some ofthe atoms areexcited tohigherenergy levels. Whenthe ball fallsagain, yet another laser is used to force the emission of radiation. This radiation is detected, yielding the desired frequency The best cesium standards reproduce the second to an accuracy better than one part in 101s .
7 TEMPERATURE STANDARDS
The basic unit of temperature, the kelvin (K), is defined as the fraction .11273.16 of the thermodynamic temperature of the triple point of water, the temperature at which the solid, liquid, andvaporphasesofwatercoexist in equilibrium. Thedegree Celsius(0C)isdefined by the relationship
I= T -273.15
where I and T represent temperatures in degrees Celsius and in kelvins, respectively.
In reality, lwo temperature scales are defined, a thennodyrlamic scale and apractical scale. The latter is 1he usual basis for measurement. The thermodynamic temperature scale is defined in terms of entropy and the properties of heat engines [22). It can be implemented directly only with specialized thermometers that use media having a precisely known equation of state (a constant volume, ideal gas thermometer, for example). Such thermometersaredifficultandtimeconsumingtouse ifaccuracy isdesired,and,asaresult,a correspondingscale which ismoreeasilyrealizableisneeded [7]. Thus, the 11hennodynamic scale is normally approximated using a so-called practical'scale.
A practical scale has two components. The first is a set of fixed reference 1emperatures, defined by specific states of matter. The second is a procedure for interpolating between those reference points, for example, by measuring a temperature-dependent electrical resistance. Using the interpolation formulae and fixed points of the practical scale, one can calibrate any other temperature measuring device.
The fixed reference temperalures must correspond to thermodynamic slates that are very accurately reproducible. Zero degrees Celsius is the temperature of equilibrium between pure iceand air-salurated pure waterat normal atmospheric pressure. However, a more precise dalum, independent of bo1h ambient pressure and possible contaminants, is lhe triple point temperature of water. As noled above, the value 273.16 K (or O.OIOO°C) is assigned to this temperature. Relatively simple apparatus can be used to reproduce 1his temperature fixed point (23).
