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 measurement 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 movement 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 conditions. 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 measurements 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 represent 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 displacement 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. Consider 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 reflecting 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.