《工程测试与信号处理》课程教学资源（文献资料）Displacement and Area

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution904Signal Processing and Engineering Measurementschapte5DISPLACEMENTANDAREAMEASUREMENTS5.1INTRODUCTIONMany of the transducers discussed in Chap.4 represent excellent devices for mea-surement of displacement.Inthis chapter wewishtoexaminethegeneral subjectofdimensionalanddisplacementmeasurementsandindicatesomeofthetechniquesand instruments that may be utilized for such purposes, making use, where possible,of theinformationintheprecedingsectionsDimensional measurementsarecategorizedas determinationsof the sizeofanobject, while a displacementmeasurement implies themeasurement of the move-ment of a point from one position to another. An area measurement on a standardgeometricfigure isacombination ofappropriate dimensional measurements througha correct analytical relationship. The determination of areas of irregular geometricshapes usuallyinvolves amechanical,graphical,or numerical integration.Displacementmeasurementsmaybemadeunderboth steadyandtransientcon-ditions.Transient measurements fall underthegeneralclass of subjects discussed inChap.11.Thepresentchapter is concerned only with staticmeasurements.5.2DIMENSIONALMEASUREMENTSThe standard units of length were discussed in Chap.2.Alldimensional measurementsareeventuallyrelatedtothesestandards.Simpledimensionalmeasurementswithanaccuracyof±0.01 in (0.25mm)maybemadewithgraduated metalmachinist scalesor wood scales whichhaveaccurateengravedmarkings.Forlargedimensionalmea-surementsmetal tapes are used to advantage.Theprimaryerrors in such measurementdevices,otherthanreadabilityerrors,areusuallytheresultofthermal expansionorcontractionofthescale.Onlongmetaltapesusedforsurveyingpurposesthis canrepresentasubstantialerror,especiallywhenusedunderextremetemperatureconditions256

chapter 5 Displacement and Area Measurements 5.1 Introduction Many of the transducers discussed in Chap. 4 represent excellent devices for mea￾surement of displacement. In this chapter we wish to examine the general subject of dimensional and displacement measurements and indicate some of the techniques and instruments that may be utilized for such purposes, making use, where possible, of the information in the preceding sections. Dimensional measurements are categorized as determinations of the size of an object, while a displacement measurement implies the measurement of the move￾ment of a point from one position to another. An area measurement on a standard geometric figure is a combination of appropriate dimensional measurements through a correct analytical relationship. The determination of areas of irregular geometric shapes usually involves a mechanical, graphical, or numerical integration. Displacement measurements may be made under both steady and transient con￾ditions. Transient measurements fall under the general class of subjects discussed in Chap. 11. The present chapter is concerned only with static measurements. 5.2 Dimensional Measurements The standard units of length were discussed in Chap. 2.All dimensional measurements are eventually related to these standards. Simple dimensional measurements with an accuracy of ±0.01 in (0.25 mm) may be made with graduated metal machinist scales or wood scales which have accurate engraved markings. For large dimensional mea￾surements metal tapes are used to advantage. The primary errors in such measurement devices, other than readability errors, are usually the result of thermal expansion or contraction of the scale. On long metal tapes used for surveying purposes this can rep￾resent a substantial error, especially when used under extreme temperature conditions. 256 hol29303_ch05_256-277.pdf 1 ol29303_ch05_256-277.pdf 1 8/12/2011 3:23:23 PM /12/2011 3:23:23 PM 904 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-HillCreateTM ReviewCopyforInstructorNicolescu.NotfordistributionExperimental Methods for Engineers,Eighth Edition9052575.2DIMENSIONALMEASUREMENTS1111111111VerniernSliderFine-adjustment screwFixed jawMovable jawFigure 5.1Verniercaliper.3234567891++++++++++!L1++-0101520255IndexVernier scaleFigure 5.2Expanded view of vernier scale.Itmaybenoted,however,thatthermal expansioneffectsrepresentfixederrorsandmay easily be corrected when the measurement temperature is known.Verniercalipersrepresentaconvenientmodificationofthemetal scaletoimprovethereadability of the device.The caliperconstruction is shown inFig.5.1, and anexpanded viewof thevernierscaleis shown inFig.5.2.Thecaliper isplaced on theobjectto bemeasured and the fine adjustment is rotated untilthejawsfittightly againstthe workpiece.The increments alongtheprimary scale are 0.025 in.The vernier scaleshownisused toreadto0.001 in(0.025mm)so thatithas25equal increments (0.001 is of 0.025)and a total length of times the length of the primary scale graduationsConsequently, the vernier scale does not line up exactly with the primary scale, andthe ratio of the last coincident number on the vernier to thetotal vernier length willequal the fraction of a whole primary scale division indicated by the index position.Inthe example shown in Fig.5.2 the reading would be2.350+()(0.025)=2.364 in.Themicrometercalipers shown inFig.5.3representamoreprecisemeasurementdevice thanthe vernier calipers.Instead of the vernier scalearrangement,a calibratedscrew thread and circumferential scale divisions are used to indicatethefractionalpart of theprimary scaledivisions.In orderto obtain themaximumeffectiveness ofthemicrometercaremustbeexertedtoensurethataconsistentpressureismaintainedon the workpiece.The spring-loaded ratchetdeviceon thehandle enables the operatortomaintain such acondition.Whenproperlyused,themicrometercanbeemployedforthemeasurementofdimensionswithin0.0001in(0.0025mm)Dial indicators aredevices that perform a mechanical amplification of thedis-placement of a pointer or follower in order to measure displacements within about0.001 in.The construction of such indicators provides a gear rack, which is connectedto a displacement-sensing shaft.This rack engages a pinion which in turn is used toprovide a gear-train amplification of the movement. The output reading is made on acircular dial

5.2 Dimensional Measurements 257 0 Vernier Slider Fine-adjustment screw Fixed jaw Movable jaw Figure 5.1 Vernier caliper. 123456789 3 1 2 0 5 10 15 20 25 Index Vernier scale Figure 5.2 Expanded view of vernier scale. It may be noted, however, that thermal expansion effects represent fixed errors and may easily be corrected when the measurement temperature is known. Vernier calipers represent a convenient modification of the metal scale to improve the readability of the device. The caliper construction is shown in Fig. 5.1, and an expanded view of the vernier scale is shown in Fig. 5.2. The caliper is placed on the object to be measured and the fine adjustment is rotated until the jaws fit tightly against the workpiece. The increments along the primary scale are 0.025 in. The vernier scale shown is used to read to 0.001 in (0.025 mm) so that it has 25 equal increments (0.001 is 1 25 of 0.025) and a total length of 24 25 times the length of the primary scale graduations. Consequently, the vernier scale does not line up exactly with the primary scale, and the ratio of the last coincident number on the vernier to the total vernier length will equal the fraction of a whole primary scale division indicated by the index position. In the example shown in Fig. 5.2 the reading would be 2.350 + ( 14 25 )(0.025) = 2.364 in. The micrometer calipers shown in Fig. 5.3 represent a more precise measurement device than the vernier calipers. Instead of the vernier scale arrangement, a calibrated screw thread and circumferential scale divisions are used to indicate the fractional part of the primary scale divisions. In order to obtain the maximum effectiveness of the micrometer care must be exerted to ensure that a consistent pressure is maintained on the workpiece. The spring-loaded ratchet device on the handle enables the operator to maintain such a condition. When properly used, the micrometer can be employed for the measurement of dimensions within 0.0001 in (0.0025 mm). Dial indicators are devices that perform a mechanical amplification of the dis￾placement of a pointer or follower in order to measure displacements within about 0.001 in. The construction of such indicators provides a gear rack, which is connected to a displacement-sensing shaft. This rack engages a pinion which in turn is used to provide a gear-train amplification of the movement. The output reading is made on a circular dial. hol29303_ch05_256-277.pdf 2 ol29303_ch05_256-277.pdf 2 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM Experimental Methods for Engineers, Eighth Edition 905 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM ReviewCopy forInstructorNicolescu.Notfordistribution.906Signal Processing and Engineering Measurements258CHAPTER!66DISPLACEMENT AND AREA MEASUREMENTSMain scale0.025" divisionsFrameThimble scale/25divisionsRatchetFigure 5.3MicrometercalipersExample 5.1ERRORDUETO THERMAL EXPANSION.A 30-m (at 15°C) steel tape is used forsurveying work in the summer such that the tape temperature in the sun is 45°C.Ameasurementindicates 24.567± 0.001 m.The linear thermal coefficient of expansion is 11.65 × 10-6/°Cat 15°C.Calculate the true distance measurement.SolutionThe indicated tape length would be the true value if the measurement were taken at 15°C. Atthe elevated temperature the tape has expanded and consequently reads too small a distance.The actual length of the 30-m tape at 45°C isL(1 +α△) = [1 + (11.65 × 10-6)(45 15)](30) = 30.010485 mSuch a true length would be indicated as 30 m.The true reading for the above situation is thus(24.567)[1 + (11.65 ×10-6)(4515)]=24.576m5.3GAGEBLOCKSGageblocks represent industrial dimension standards.They are small steel blocksabout ×1in with highly polished parallel surfaces.The thickness of theblocks isspecifiedinaccordancewiththefollowingtolerances:Grade of BlockTolerance, μ in!2AA4A00BITolerances are for blocks less than 1 inthick; for greater thickness the sametolerances are per inch.Gageblocks areavailableinarangeofthicknessesthatmakeitpossibleto stackthem inamanner suchthat with a set of 81blocks anydimensionbetween0.100 and

258 CHAPTER 5 • Displacement and Area Measurements Frame Main scale 0.025" divisions Thimble scale 25 divisions Ratchet Figure 5.3 Micrometer calipers. Example 5.1 ERROR DUE TO THERMAL EXPANSION. A 30-m (at 15◦C) steel tape is used for surveying work in the summer such that the tape temperature in the sun is 45◦C.Ameasurement indicates 24.567 ± 0.001 m. The linear thermal coefficient of expansion is 11.65 × 10−6/◦C at 15◦C. Calculate the true distance measurement. Solution The indicated tape length would be the true value if the measurement were taken at 15◦C. At the elevated temperature the tape has expanded and consequently reads too small a distance. The actual length of the 30-m tape at 45◦C is L(1 + α T) = [1 + (11.65 × 10−6 )(45 − 15)](30) = 30.010485 m Such a true length would be indicated as 30 m. The true reading for the above situation is thus (24.567)[1 + (11.65 × 10−6 )(45 − 15)] = 24.576 m 5.3 Gage Blocks Gage blocks represent industrial dimension standards. They are small steel blocks about 3 8 × 13 8 in with highly polished parallel surfaces. The thickness of the blocks is specified in accordance with the following tolerances: Grade of Block Tolerance, μ in1 AA 2 A 4 B 8 1Tolerances are for blocks less than 1 in thick; for greater thickness the same tolerances are per inch. Gage blocks are available in a range of thicknesses that make it possible to stack them in a manner such that with a set of 81 blocks any dimension between 0.100 and hol29303_ch05_256-277.pdf 3 ol29303_ch05_256-277.pdf 3 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM 906 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionExperimental MethodsforEngineers,EighthEdition9075.4259OPTICALMETHODS8.000 in can be obtained in increments of 0.0001 in.The blocks are stacked througha process ofwringing.With surfaces thoroughly clean,the metal surfaces arebroughttogetherina slidingfashion whilea steadypressure is exerted.Thesurfaces aresufficiently flat so that when the wringing process is correctly executed, they willadhereasaresultofmolecularattraction.Theadhesiveforcemaybeasgreatas30timesatmosphericpressure.Becauseoftheirhighaccuracy,gageblocks arefrequentlyused forcalibrationofotherdimensionalmeasurementdevices.Forveryprecisemeasurementstheymaybeusedfordirectdimensional comparisontests with a machined item.A discussionof themethods of producinggage-block standards isgiven in Ref.[3].The literatureofmanufacturers of gageblocksfurnishes an excellentsource of information onthemeasurementtechniques whichareemployed inpractice.5.4OPTICALMETHODSAnoptical methodformeasuringdimensionsveryaccuratelyisbasedontheprincipleoflightinterference.The instrumentbasedon this principleiscalled an interferometerandis usedforthecalibrationofgageblocks and otherdimensional standards.Otheroptical instruments in wide use are various types of microscopes and telescopes,includingtheconventional surveyor'stransit, whichisemployed for measurementoflargedistances.Consider thetwo sets of lightbeams shown inFig.5.4.InFig.5.4athetwobeamsare in phase so that the brightness at point P is augmented when they intersect.InFig.5.4bthebeamsareoutofphasebyhalfawavelengthsothatacancellationis observed, and thelightwaves aresaid to interfere with each other.This is theessence of the interferenceprinciple.The effect ofthe cancellation isbroughtaboutbyallowingtwolightwavesfromasinglesourcetotravelalongpathsof differentlengths.When thedifference in the distance is an integral multiple of wavelengths,Beam 1Beam 1Beam 2Beam 2(a)(b)Figure 5.4Interference principle. (a) Beams in phase; (b) beams out of phase

5.4 Optical Methods 259 8.000 in can be obtained in increments of 0.0001 in. The blocks are stacked through a process of wringing. With surfaces thoroughly clean, the metal surfaces are brought together in a sliding fashion while a steady pressure is exerted. The surfaces are sufficiently flat so that when the wringing process is correctly executed, they will adhere as a result of molecular attraction. The adhesive force may be as great as 30 times atmospheric pressure. Because of their high accuracy, gage blocks are frequently used for calibration of other dimensional measurement devices. For very precise measurements they may be used for direct dimensional comparison tests with a machined item. A discussion of the methods of producing gage-block standards is given in Ref. [3]. The literature of manufacturers of gage blocks furnishes an excellent source of information on the measurement techniques which are employed in practice. 5.4 Optical Methods An optical method for measuring dimensions very accurately is based on the principle of light interference. The instrument based on this principle is called an interferometer and is used for the calibration of gage blocks and other dimensional standards. Other optical instruments in wide use are various types of microscopes and telescopes, including the conventional surveyor’s transit, which is employed for measurement of large distances. Consider the two sets of light beams shown in Fig. 5.4. In Fig. 5.4a the two beams are in phase so that the brightness at point P is augmented when they intersect. In Fig. 5.4b the beams are out of phase by half a wavelength so that a cancellation is observed, and the light waves are said to interfere with each other. This is the essence of the interference principle. The effect of the cancellation is brought about by allowing two light waves from a single source to travel along paths of different lengths. When the difference in the distance is an integral multiple of wavelengths, Beam 1 Beam 2 Beam 1 Beam 2 2 P P (a) (b) Figure 5.4 Interference principle. (a) Beams in phase; (b) beams out of phase. hol29303_ch05_256-277.pdf 4 ol29303_ch05_256-277.pdf 4 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM Experimental Methods for Engineers, Eighth Edition 907 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution.908SignalProcessingandEngineeringMeasurements260CHAPTER!56DISPLACEMENT AND AREA MEASUREMENTSIncomingScreen,Sparallel beamsA)BGlassoptically flat0+PReflectingsurfaceFigure5.5Application of interference principle.there will be a reinforcement of the waves, while there will be a cancellation whenthe difference in the distances is an odd multiple of half-wavelengths.Now,letus apply the interference principle to dimensional measurements.Con-sider the two parallel plates shown in Fig.5.5.One plate is a transparent, strain-freeglass accuratelypolished flatwithinafewmicroinches.Theotherplatehasareflectingmetal surface.Theglassplateis called anoptical flat.Parallel lightbeams AandBareprojectedontheplatesfromasuitablecollimatingsource.Theseparationdistancebetween theplates d is assumed tobequite small.Thereflected beam Aintersects the incoming beam B at point P. Since the reflected beam has traveledfartherthan beam Bby a distanceof 2d,it will create an interference atpoint Pifthisincrementaldistanceisanoddmultipleof>/2.Ifthedistance2disanevenmultipleof^/2,thereflectedbeamwill augmentbeamB.Thus,for 2d=^/2,3>/2,etc.,thescreenSwill detectnoreflected light.Now,considerthesametwoplates,butletthem be tilted slightly so that the distance between the plates is a variable.Now,ifoneviewsthereflected lightbeams,alternatelight anddarkregions will appearonthe screen, indicating the variation in the plate spacing.The dark lines or regions arecalledfringes,andthechangeintheseparationdistancebetweenthepositionsoftwofringes corresponds to入△(2d) =[5.1]2The interference principle offers a convenient means for measuring small surfacedefects and for calibrating gageblocks.The use of a tilted optical flat as in Fig.5.5isan awkwardmethodofutilizingtheprinciple,however.Forpractical purposestheinterferometer,as indicated schematically in Fig.5.6, is employed.MonochromaticlightfromthesourceiscollimatedbythelensLontothesplitterplateS2,whichisa half-silvered mirrorthat reflects halfofthelight toward theopticallyflat mirrorMandallowstransmissionoftheotherhalftowardtheworkpieceW.Bothbeamsarereflected back andrecombinedatthesplitterplateS2 andthentransmittedtothe screen.Fringesmayappearonthescreenresultingfromdifferences intheopticalpathlengthsof the twobeams.If the instrument is properly constructed, these differences will arisefromdimensional variationsoftheworkpiece.Theinterferometerisprimarilyusedfor

260 CHAPTER 5 • Displacement and Area Measurements d A B P Incoming parallel beams Screen, S Glass optically flat Reflecting surface Figure 5.5 Application of interference principle. there will be a reinforcement of the waves, while there will be a cancellation when the difference in the distances is an odd multiple of half-wavelengths. Now, let us apply the interference principle to dimensional measurements. Con￾sider the two parallel plates shown in Fig. 5.5. One plate is a transparent, strain-free glass accurately polished flat within a few microinches. The other plate has a reflect￾ing metal surface. The glass plate is called an optical flat. Parallel light beams A and B are projected on the plates from a suitable collimating source. The separation distance between the plates d is assumed to be quite small. The reflected beam A intersects the incoming beam B at point P. Since the reflected beam has traveled farther than beam B by a distance of 2d, it will create an interference at point P if this incremental distance is an odd multiple of λ/2. If the distance 2d is an even multiple of λ/2, the reflected beam will augment beam B. Thus, for 2d = λ/2, 3λ/2, etc., the screen S will detect no reflected light. Now, consider the same two plates, but let them be tilted slightly so that the distance between the plates is a variable. Now, if one views the reflected light beams, alternate light and dark regions will appear on the screen, indicating the variation in the plate spacing. The dark lines or regions are called fringes, and the change in the separation distance between the positions of two fringes corresponds to (2d) = λ 2 [5.1] The interference principle offers a convenient means for measuring small surface defects and for calibrating gage blocks. The use of a tilted optical flat as in Fig. 5.5 is an awkward method of utilizing the principle, however. For practical purposes the interferometer, as indicated schematically in Fig. 5.6, is employed. Monochromatic light from the source is collimated by the lens L onto the splitter plate S2, which is a half-silvered mirror that reflects half of the light toward the optically flat mirror M and allows transmission of the other half toward the workpiece W. Both beams are reflected back and recombined at the splitter plate S2 and then transmitted to the screen. Fringes may appear on the screen resulting from differences in the optical path lengths of the two beams. If the instrument is properly constructed, these differences will arise from dimensional variations of the workpiece. The interferometer is primarily used for hol29303_ch05_256-277.pdf 5 ol29303_ch05_256-277.pdf 5 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM 908 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM Review Copyfor lnstructor Nicolescu.Not fordistributionExperimental MethodsforEngineers,EighthEdition9092615.4OPTICAL METHODSOptically flatmirror, MLens, LAMonochromaticWork-piece, WsourceSplitterplate,S2Screen, S3Figure 5.6Schematicofinterferometer.Table 5.1MonochromaticlightsourcesHalf-wavelengthWavelength,Fringe Interval,SourceμumμmHelium0.5890.295Krypton 860.6060.3030.5460.273Mercury 1980.5980.299Sodiumcalibrationofgageblocksand otherapplicationswhereextremelypreciseabsolutedimensional measurements are required.For detailed information on experimentaltechniques used in interferometry the reader should consult Refs. [3] and [5]. The useoftheinterferometerforfluid-flowmeasurementswillbediscussed inChap.7.As shown inEq.(5.1),thewavelength of themonochromatic light sourcewillinfluence the fringe spacing.Table 5.1 lists the wavelengths of some common lightsources and thecorrespondinghalf-wavelengthfringeinterval.INTERFERENCEMEASUREMENT.A mercury light source employs a green filter suchExample5.2that the wavelength is 5460 A. This light is colliminated and directed onto two tilted surfaceslike those shown inFig.5.5.Atone endthe surfacesare in precise contact.Between the pointofcontactand a distance of3000 in five interferencefringes are observed.Calculate the separationdistance between the two surfaces and the tilt angle at this position.SolutionThefivefringe linescorrespond to入/2.3入/2..,9/2;that is,for the fifthfringeline9入2d=2

5.4 Optical Methods 261 Screen, S3 Lens, L Optically flat mirror, M Monochromatic source Splitter plate, S2 Work￾piece, W Figure 5.6 Schematic of interferometer. Table 5.1 Monochromatic light sources Half-wavelength Wavelength, Fringe Interval, Source μm μm Helium 0.589 0.295 Krypton 86 0.606 0.303 Mercury 198 0.546 0.273 Sodium 0.598 0.299 calibration of gage blocks and other applications where extremely precise absolute dimensional measurements are required. For detailed information on experimental techniques used in interferometry the reader should consult Refs. [3] and [5]. The use of the interferometer for fluid-flow measurements will be discussed in Chap. 7. As shown in Eq. (5.1), the wavelength of the monochromatic light source will influence the fringe spacing. Table 5.1 lists the wavelengths of some common light sources and the corresponding half-wavelength fringe interval. INTERFERENCE MEASUREMENT. A mercury light source employs a green filter such Example 5.2 that the wavelength is 5460 A. This light is colliminated and directed onto two tilted surfaces ˚ like those shown in Fig. 5.5. At one end the surfaces are in precise contact. Between the point of contact and a distance of 3000 in five interference fringes are observed. Calculate the separation distance between the two surfaces and the tilt angle at this position. Solution The five fringe lines correspond to λ/2, 3λ/2,., 9λ/2; that is, for the fifth fringe line 2d = 9λ 2 hol29303_ch05_256-277.pdf 6 ol29303_ch05_256-277.pdf 6 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM Experimental Methods for Engineers, Eighth Edition 909 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution910Signal Processing and Engineering Measurements262CHAPTER SDISPLACEMENT AND AREA MEASUREMENTSWe have 入 = 5460 × 10-8 cm = 2.15 × 10-5 in so thatd = (2.15 × 10-5) = 48.4 μinThe tilt angle isΦ = tan-1 48.4 × 10-648.4 × 10-6= 16.1 × 10-6 rad3.0003.000Mechanical displacement may also be measured with the aid of the electric trans-ducers discussed in Chap.4.TheLVDT,for example,canbeused tosensedisplacements as small as1 μin.Use of LVDT devices for displacement measurements isdescribed in Ref. [17]. Resistance transducers are primarily of value for measure-mentoffairlylargedisplacementsbecauseoftheirpoorresolution.Capacitanceandpiezoelectric transducers,on the other hand,provide high resolution and are suitablefordynamicmeasurements.5.5PNEUMATICDISPLACEMENTGAGEConsider the system shown in Fig.5.7. Air is supplied at a constant pressure pi.Theflowthroughthe orifice and throughtheoutlet of diameterd2isgoverned bythe separation distancexbetween the outletand theworkpiece.Thechangein flowwith x will be indicated by a change in the pressure downstream from the orifice p2.Thus,ameasurementof thispressuremaybetakenasan indicationof theseparationdistancex.Forpurposesof analysisweassumeincompressibleflow.(SeeSec.7.3foradiscussion of thevalidityofthisassumption.)ThevolumetricflowthroughanorificemayberepresentedbyQ= CAVAp[5.2]Ambient pressure≤PoOrificeoFlowdidOFigure5.7Pneumaticdisplacementdevice

262 CHAPTER 5 • Displacement and Area Measurements We have λ = 5460 × 10−8 cm = 2.15 × 10−5 in so that d = 9 4 (2.15 × 10−5 ) = 48.4μin The tilt angle is φ = tan−1 48.4 × 10−6 3.000 = 48.4 × 10−6 3.000 = 16.1 × 10−6 rad Mechanical displacement may also be measured with the aid of the electric trans￾ducers discussed in Chap. 4. The LVDT, for example, can be used to sense displace￾ments as small as 1 μin. Use of LVDT devices for displacement measurements is described in Ref. [17]. Resistance transducers are primarily of value for measure￾ment of fairly large displacements because of their poor resolution. Capacitance and piezoelectric transducers, on the other hand, provide high resolution and are suitable for dynamic measurements. 5.5 Pneumatic Displacement Gage Consider the system shown in Fig. 5.7. Air is supplied at a constant pressure p1. The flow through the orifice and through the outlet of diameter d2 is governed by the separation distance x between the outlet and the workpiece. The change in flow with x will be indicated by a change in the pressure downstream from the orifice p2. Thus, a measurement of this pressure may be taken as an indication of the separation distance x. For purposes of analysis we assume incompressible flow. (See Sec. 7.3 for a discussion of the validity of this assumption.) The volumetric flow through an orifice may be represented by Q = CAp [5.2] Flow Orifice d1 d2 x Workpiece Ambient pressure pa 1 2 Figure 5.7 Pneumatic displacement device. hol29303_ch05_256-277.pdf 7 ol29303_ch05_256-277.pdf 7 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM 910 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionExperimentalMethodsforEngineers,EighthEdition9112635.5PNEUMATIC DISPLACEMENT GAGEwhereC=discharge coefficientA=flowareaof theorificeAp=pressuredifferential across theorificeTherearetwoorifices inthesituation depicted inFig.5.7,the obvious one andthe orifice formed bytheflow restriction between theoutlet and theworkpiece.WeshalldesignatetheareaofthefirstorificeA,andthatofthesecondA2.Then,Eq.(5.2)becomes[5.3]Q= C/AIVPI-P2= C2A2VP2-Pawhere pa is the ambient pressure and is assumed constant. Equation (5.3) may berearranged to give1P2 - Pa[5.4]r=1 +(A2/Ai)2Pi-Pawhere it is assumed that the discharge coefficients Ci and C2 are equal.We may nowobserve that元dAl =[5.5]4[5.6]A2=元d2xThus, we seethe relation between the pressure ratio r and the workpiecedis-placement x. It has been shown experimentally [1] that the relation betweenr and thearea ratioA2/A,isverynearlylinearfor0.4<r<0.9and thatr = 1.10 0.5042[5.7]A1for this range. Introducing Eqs. (5.5) and (5.6), we havedsP2 - Pa=1.102.00for 0.4 < r < 0.9[5.8]r=Pi-PaThepneumatic displacement gage is mainly usedforsmall1displacementmeasurements.Example 5.3UNCERTAINTY IN PNEUMATIC DISPLACEMENT GAGE.Apneumaticdisplace-mentgagelike theone shown inFig.5.7hasd;=0.030inand d2=0.062in.Thesupplypressure is 10.0 psig, and the differential pressure p2- pa is measured with a water manometerwhich maybe read with an uncertainty of 0.05 in H,O.Calculate the displacement range forwhichEq.(5.8)applies and the uncertainty in this measurement, assuming that the supplypressureremainsconstantSolutionWe have0.062d2=68.8(0.030)2ar

5.5 Pneumatic Displacement Gage 263 where C = discharge coefficient A = flow area of the orifice p = pressure differential across the orifice There are two orifices in the situation depicted in Fig. 5.7, the obvious one and the orifice formed by the flow restriction between the outlet and the workpiece. We shall designate the area of the first orifice A1 and that of the second A2. Then, Eq. (5.2) becomes Q = C1A1 √p1 − p2 = C2A2 √p2 − pa [5.3] where pa is the ambient pressure and is assumed constant. Equation (5.3) may be rearranged to give r = p2 − pa p1 − pa = 1 1 + (A2/A1)2 [5.4] where it is assumed that the discharge coefficients C1 and C2 are equal. We may now observe that A1 = πd2 1 4 [5.5] A2 = πd2x [5.6] Thus, we see the relation between the pressure ratio r and the workpiece dis￾placement x. It has been shown experimentally [1] that the relation between r and the area ratio A2/A1 is very nearly linear for 0.4 <r< 0.9 and that r = 1.10 − 0.50 A2 A1 [5.7] for this range. Introducing Eqs. (5.5) and (5.6), we have r = p2 − pa p1 − pa = 1.10 − 2.00 d2 d2 1  x for 0.4 <r< 0.9 [5.8] The pneumatic displacement gage is mainly used for small displacement measurements. UNCERTAINTY IN PNEUMATIC DISPLACEMENT GAGE. A pneumatic displace- Example 5.3 ment gage like the one shown in Fig. 5.7 has d1 = 0.030 in and d2 = 0.062 in. The supply pressure is 10.0 psig, and the differential pressure p2 −pa is measured with a water manometer which may be read with an uncertainty of 0.05 in H2O. Calculate the displacement range for which Eq. (5.8) applies and the uncertainty in this measurement, assuming that the supply pressure remains constant. Solution We have d2 d2 1 = 0.062 (0.030)2 = 68.8 hol29303_ch05_256-277.pdf 8 ol29303_ch05_256-277.pdf 8 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM Experimental Methods for Engineers, Eighth Edition 911 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution.912Signal Processing andEngineering Measurements264CHAPTER SDISPLACEMENT AND AREA MEASUREMENTSWhen r =0.4,wehave fromEq.(5.8)1.10 0.4=0.0509 in (0.129cm)x=(2.00)(68.8)When r = 0.9, x = 0.0145 in (0.0368 cm)Utilizing Eq. (3.2) as applied to Eq. (5.8), we havedrDr=Furthermore,arWAP(2.00)(68.8) =137.6w,=axPi-PaThe uncertainty in the measurement of p2 - Pa isWAp = (0.05)(0.0361) = 1.805 × 10- psig (12.44 N/m2)Thus, the uncertainty in x is given by1.805 × 10-3= 1.313 × 106 in = 1.313 μin (0.033 μm)wx=(137.6)(10.00)CommentFrom thisexample weseethatthe pneumaticgagecanbequitesensitive,evenwithmodestpressure-measurementfacilities at hand.Inadditiontoitsapplicationasasteady-state-displacement-measurementdevicethe pneumatic gage may be employed as a dynamic sensor in conjunction with prop-erly designed fluidic circuits. By periodically interrupting the discharge, the devicemay serve as a periodic signal generatorforfluidic circuits.Different interruptiontechniquesmaybeemployed togenerate square-,triangle-,or sine-wavesignals[7]Thedevicemayalsobeemployedforfluidicreadingof coded informationonplatesor cards that pass under the outlet jet [8 and 9]. The rapidity with which such read-ingsmaybemadedependsonthedynamicresponseofthegageand itsassociatedconnecting lines and pressure transducers.Studies[1l] have shown that thedevicecanproducegoodfrequencyresponseforsignalsupto5ooHz5.6AREAMEASUREMENTSThere are many applications that require a measurement of a plane area. Graphicaldeterminations of the area of the survey plots from maps,the integration ofafunctionto determinethe area under a curve,and analyses of experimental data plots allmay rely on a measurement of a plane area.There are also many applications for themeasurementofsurfaceareas,butsuchmeasurementsareconsiderablymoredifficultto perform

264 CHAPTER 5 • Displacement and Area Measurements When r = 0.4, we have from Eq. (5.8) x = 1.10 − 0.4 (2.00)(68.8) = 0.0509 in (0.129 cm) When r = 0.9, x = 0.0145 in (0.0368 cm). Utilizing Eq. (3.2) as applied to Eq. (5.8), we have wr =  ∂r ∂x2 w2 x 1/2 = ±     ∂r ∂x     wx Furthermore, wr = wp p1 − pa ∂r ∂x = −(2.00)(68.8) = −137.6 The uncertainty in the measurement of p2 − pa is wp = (0.05)(0.0361) = 1.805 × 10−3 psig (12.44 N/m2 ) Thus, the uncertainty in x is given by wx = 1.805 × 10−3 (137.6)(10.00) = 1.313 × 10−6 in = 1.313μin (0.033μm) Comment From this example we see that the pneumatic gage can be quite sensitive, even with modest pressure-measurement facilities at hand. In addition to its application as a steady-state–displacement-measurement device, the pneumatic gage may be employed as a dynamic sensor in conjunction with prop￾erly designed fluidic circuits. By periodically interrupting the discharge, the device may serve as a periodic signal generator for fluidic circuits. Different interruption techniques may be employed to generate square-, triangle-, or sine-wave signals [7]. The device may also be employed for fluidic reading of coded information on plates or cards that pass under the outlet jet [8 and 9]. The rapidity with which such read￾ings may be made depends on the dynamic response of the gage and its associated connecting lines and pressure transducers. Studies [11] have shown that the device can produce good frequency response for signals up to 500 Hz. 5.6 Area Measurements There are many applications that require a measurement of a plane area. Graphical determinations of the area of the survey plots from maps, the integration of a function to determine the area under a curve, and analyses of experimental data plots all may rely on a measurement of a plane area. There are also many applications for the measurement of surface areas, but such measurements are considerably more difficult to perform. hol29303_ch05_256-277.pdf 9 ol29303_ch05_256-277.pdf 9 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM 912 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionExperimental Methods forEngineers, Eighth Edition9135.7265THEPLANIMETERADEVICEOFHISTORICALINTEREST5.7THEPLANIMETER,ADEVICEOFHISTORICALINTERESTThe planimeter is a mechanical integrating device that may be used for measure-ment of plane areas.We consider it here as an illustration of a novel mechanicaldevicetoperformareameasurements.Itis seldomusedtoday.Considerthe schematicrepresentation shown in Fig.5.8.Thepoint O isfixed, while the tracingpointTis moved around the peripheryof the figure whose areais to be determined.Thewheel W ismounted onthearmBT sothat it is freeto rotatewhenthearm under-goes an angular displacement. The wheel has engraved graduations and a vernierscale so that its exact number of revolutions may be determined as the tracingpoint moves around the curve.The planimeter and area are placed on a flat, rela-tively smooth surface sothat the wheel W will only slide when thearmBT under-goes an axial translational movement. Thus, the wheel registers zero angular dis.placement when an axial translational movement of arm BT is experienced.Letthelength of the tracing arm BT beL and the distance from point B to the wheelbe a.The diameter of the wheel is D.The distance OB is taken as R. Now, sup-posethearmBT is rotated an anglede and thearm OBthrough anangledpasaresult of movement of the tracing point. The area swept out by the arms BT andOBisdA=L?de+LRcosβd+R?dp[5.9]whereβ is theanglebetween thetwo arms.Similarly,thedistancetraveled bytherim of the wheel owing to rotation isds=ade+Rcosβdp[5.10]WArea, ATracing point, TPivot point, OFigure 5.8Schematic of a polar planimeter

5.7 The Planimeter, a Device of Historical Interest 265 5.7 The Planimeter, a Device of Historical Interest The planimeter is a mechanical integrating device that may be used for measure￾ment of plane areas. We consider it here as an illustration of a novel mechanical device to perform area measurements. It is seldom used today. Consider the schematic representation shown in Fig. 5.8. The point O is fixed, while the tracing point T is moved around the periphery of the figure whose area is to be determined. The wheel W is mounted on the arm BT so that it is free to rotate when the arm under￾goes an angular displacement. The wheel has engraved graduations and a vernier scale so that its exact number of revolutions may be determined as the tracing point moves around the curve. The planimeter and area are placed on a flat, rela￾tively smooth surface so that the wheel W will only slide when the arm BT under￾goes an axial translational movement. Thus, the wheel registers zero angular dis￾placement when an axial translational movement of arm BT is experienced. Let the length of the tracing arm BT be L and the distance from point B to the wheel be a. The diameter of the wheel is D. The distance OB is taken as R. Now, sup￾pose the arm BT is rotated an angle dθ and the arm OB through an angle dφ as a result of movement of the tracing point. The area swept out by the arms BT and OB is dA = 1 2L2 dθ + LR cos β dφ + 1 2R2 dφ [5.9] where β is the angle between the two arms. Similarly, the distance traveled by the rim of the wheel owing to rotation is ds = a dθ + R cos β dφ [5.10] B R Area, A a W L  Pivot point, O Tracing point, T Figure 5.8 Schematic of a polar planimeter. hol29303_ch05_256-277.pdf 10 ol29303_ch05_256-277.pdf 10 8/12/2011 3:23:24 PM /12/2011 3:23:24 PM Experimental Methods for Engineers, Eighth Edition 913 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.