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

McGraw-HillCreateTM Review Copyfor InstructorNicolescu.Notfordistribution.412Signal Processing andEngineering Measurements5CHAPTERForce, Torque, andShaftPowerMeasurement5.1STANDARDSANDCALIBRATIONForce is defined by the equation F = MA; thus a standard for force depends on stan-dardsformassand acceleration.Mass is considered a fundamentalquantity,and itsstandard is a cylinder of platinum-iridium, called the International Kilogram,keptin a vaultatSevres,France.Othermasses(such asnational standards)maybecomparedwiththis standard bymeans ofanequal-armbalance,withaprecisionofa few parts in 109 for masses of about I kg. Tolerances on various classes of stan-dardmassesavailablefromNiSTmaybefoundinitspublications.!Acceleration is not a fundamental quantity,but rather is derived from lengthand time,twofundamental quantities whose standardsarediscussed in Chap.4.Theacceleration ofgravity,g,is a convenient standard which can be determined with anaccuracy of about I part in 1oby measuring the period and effective length of apendulum orby determiningthe changewithtime of the speed of a freelyfallingbody.2Theactual valueofgvarieswithlocationand also slightlywithtime (inaperiodic predictable fashion)at a given location.It also may change (slightly)unpredictablybecause of local geological activity.The so-called standard value ofgrefersto thevalueat sea level and 45°latitudeand isnumerically980.665cm/s?.Thevalueatanylatitudedegreesmaybecomputedfromg =978.049(1 +0.0052884 sin2b0.0000059 sin?2)cm/s2(5.1)IT. W. Lashof and L. B. Macurdy, “Precision Laboratory Standards of Mass and Laboratory Weights," Nanl.Bur.Std.(U.S.),Circ.547,sec.1,1954:P.E.Pontius,Massand MassValues."Natl.Bur.Std.(U.S.)Monograph133.1974.2A.Bray,G.Barbato,and R.Levi,"Theory and Practice ofForce Measurement," Academic Press, NewYork, 1990, chap. 3; W. Torge,"Gravimetry," de Gruyter, Berlin, 1989.432

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 432 5 CHAPTER Force, Torque, and Shaft Power Measurement 5.1 STANDARDS AND CALIBRATION Force is defined by the equation F
MA; thus a standard for force depends on stan￾dards for mass and acceleration. Mass is considered a fundamental quantity, and its standard is a cylinder of platinum-iridium, called the International Kilogram, kept in a vault at Sèvres, France. Other masses (such as national standards) may be compared with this standard by means of an equal-arm balance, with a precision of a few parts in 109 for masses of about 1 kg. Tolerances on various classes of stan￾dard masses available from NIST may be found in its publications.1 Acceleration is not a fundamental quantity, but rather is derived from length and time, two fundamental quantities whose standards are discussed in Chap. 4. The acceleration of gravity, g, is a convenient standard which can be determined with an accuracy of about 1 part in 106 by measuring the period and effective length of a pendulum or by determining the change with time of the speed of a freely falling body.2 The actual value of g varies with location and also slightly with time (in a periodic predictable fashion) at a given location. It also may change (slightly) unpredictably because of local geological activity. The so-called standard value of g refers to the value at sea level and 45˚ latitude and is numerically 980.665 cm/s2. The value at any latitude f degrees may be computed from g
978.049(1 0.0052884 sin2 f 0.0000059 sin2 2f) cm/s2 (5.1) 1T. W. Lashof and L. B. Macurdy, “Precision Laboratory Standards of Mass and Laboratory Weights,” Natl. Bur. Std. (U.S.), Circ. 547, sec. 1, 1954; P. E. Pontius, “Mass and Mass Values,” Natl. Bur. Std. (U.S.), Monograph 133, 1974. 2A. Bray, G. Barbato, and R. Levi, “Theory and Practice of Force Measurement,” Academic Press, New York, 1990, chap. 3; W. Torge, “Gravimetry,” de Gruyter, Berlin, 1989. 412 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-HillCreateTM ReviewCopyforInstructorNicolescu.NotfordistributionMeasurement Systems, Application and Design,Fifth Edition413433CHAPTER5Force, Torque, and Shaft Power MeasurementLKI8/64OnepartAccuracyof forcemeasurementin10542/"NewU-Wmachines10°/Smoll D-W machineo3/ LargeD-W712M-1BmachinemachineUelo5/ Multiple pravinging6Multiple load cells102Present facilitiesFuturefacilities1010210310410510610710810Pounds force110510610810210310410710Kilograms force (1kgf:9.80665Newtons)Figure 5.1Force standards.["Future facilities"are now.available.]whilethe correction for altitudeh in meters above sea level isCorrection=-(0.00030855+0.00000022cos2b)hhcm/s2+0.000072(5.2)1,000Local valuesofgalsomaybeobtainedfromtheNationalOceanSurvey,NationalOceanicandAtmosphericAdministrationWhen thenumerical value of g has been determined at aparticular locality,thegravitationalforce(weight)onaccuratelyknownstandardmassesmaybecomputedto establish a standard of force.This is the basis of the"deadweight"calibration offorce-measuringsystems.TheNationalBureauof Standards(nowNIST)capability(Fig. 5.1)3 for such calibrations is an inaccuracy of about 1 part in 5,000 for therangeof1to1millionlbf.Abovethisrange,directdeadweightcalibrationisnotpresently available.Rather,proving ringstor load cells of a capacity of 1million Ibforlessarecalibratedagainstdeadweights,andthentheunknownforceisappliedtoamultiplearrayoftheseinparallel.TherangeIto10millionlbfiscoveredbysucharrangementswithsomewhatreducedaccuracy.Atthelow-forceendofthescale,3Accuracy in Measurements and Calibrations," Natl. Bur. Std. (U.S.), Tech. Note 262, 19654Proving Rings for Calibrating Testing Machines,"Nal. Bur.Std. (U.S.).Cire. C454,1946

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 CHAPTER 5 Force, Torque, and Shaft Power Measurement 433 while the correction for altitude h in meters above sea level is Correction
(0.00030855 0.00000022 cos 2f)h 0.000072 cm/s2 (5.2) Local values of g also may be obtained from the National Ocean Survey, National Oceanic and Atmospheric Administration. When the numerical value of g has been determined at a particular locality, the gravitational force (weight) on accurately known standard masses may be computed to establish a standard of force. This is the basis of the “deadweight” calibration of force-measuring systems. The National Bureau of Standards (now NIST) capability (Fig. 5.1)3 for such calibrations is an inaccuracy of about 1 part in 5,000 for the range of 10 to 1 million lbf. Above this range, direct deadweight calibration is not presently available. Rather, proving rings4 or load cells of a capacity of 1 million lbf or less are calibrated against deadweights, and then the unknown force is applied to a multiple array of these in parallel. The range 1 to 10 million lbf is covered by such arrangements with somewhat reduced accuracy. At the low-force end of the scale, ¢ h 1,000≤ 2 Figure 5.1 Force standards. [“Future facilities” are now available.] 3“Accuracy in Measurements and Calibrations,” Natl. Bur. Std. (U.S.), Tech. Note 262, 1965. 4“Proving Rings for Calibrating Testing Machines,” Natl. Bur. Std. (U.S.), Circ. C454, 1946. Measurement Systems, Application and Design, Fifth Edition 413 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM Review Copyfor Instructor Nicolescu.Notfordistribution.414Signal Processingand Engineering Measurements434PART 2 Measuring Devicesthe accuracy of standard masses ranges from about I percent for a mass of 10-5Ibm to 0.0001 percent for the 0.1 to 10 Ibm range to 0.001 percent for a 100-1bmass.Theaccuracyofforcecalibrationsusingthesemassesmustbesomewhatlessthan the quoted figures because of error sources in the experimental procedures.6Acommerciallyavailablecalibrating machineusingdeadweights,knife edges,andleverscoverstherangeof0to10,000lbf(or0to50kN)withanaccuracyof0.005percentofappliedloadandaresolutionof±0.0062percentofapplied load.Computerized calibration systems based on strain gage load cells and hydraulic loadframes are also available.8Themeasurementof torque is intimatelyrelated toforcemeasurement; thustorque standards as sucharenot necessary,sinceforceand length are sufficient todefine torque.Atorque standard may,however,be convenient,and one was underdevelopment in 1998.9 The power transmitted by a rotating shaft is the product oftorque and angular velocity.Angular-velocity measurement was treated in Chap.4.5.2BASICMETHODSOFFORCEMEASUREMENTAn unknown forcemay be measuredbythefollowing means:1.Balancing it against theknowngravitational forceon a standard mass,eitherdirectly orthrough a system of levers2.Measuringtheaccelerationof abodyofknownmasstowhichtheunknownforceis applied3.Balancing it against amagneticforce developed byinteraction of a current-carrying coil and a magnet4.Transducing theforce to a fluid pressure and then measuring the pressure5. Applying the force to some elastic member and measuring the resultingdeflection6.Measuring the change in precession of a gyroscope caused by an appliedtorque related to the measuredforce7.Measuring the change in natural frequency of a wire tensioned by theforce5R. M. Schoonover and F. E. Jones, "Examination of Parameters That Can Cause Errors in MassDetermination," CAL LAB, Julyl/Aug. 1998, pp. 2631.6Calibration of Force-Measuring Instruments for Verifying the Load Indication of Testing Machines,"ASTM Std. E-74, 1974.7w.C.Dillon Co. (www.dillonnews.com).$Gold Standard System, Interface, Inc., Scottsdale,AZ, 800-947-5598 (www.interfaceforce.com); C.Ferreroet al.,"Main Metrological Characteristics of IMGC Six-Component Dynamometer,"RAM, vol 2,1986pp.21-28; R.Hellwig,"Precision Force Transducer for International Comparison Measurements onForce Standard Machines," RAM, vol. 3, 1987, pp. 1722; HBM, Norcross, GA, 888-816-9006(www.hbm-home.com).°F. A. Davis,"Design of the Ist UK National Standard Static Torque Calibration Machine," NationalPhycalaboraory,Queens Road,Teddington,Middesex,United KingdomW11LW,1943-694

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 434 PART 2 Measuring Devices the accuracy5 of standard masses ranges from about 1 percent for a mass of 105 lbm to 0.0001 percent for the 0.1 to 10 lbm range to 0.001 percent for a 100-lb mass. The accuracy of force calibrations using these masses must be somewhat less than the quoted figures because of error sources in the experimental procedures.6 A commercially available7 calibrating machine using deadweights, knife edges, and levers covers the range of 0 to 10,000 lbf (or 0 to 50 kN) with an accuracy of 0.005 percent of applied load and a resolution of 0.0062 percent of applied load. Computerized calibration systems based on strain gage load cells and hydraulic load frames are also available.8 The measurement of torque is intimately related to force measurement; thus torque standards as such are not necessary, since force and length are sufficient to define torque. A torque standard may, however, be convenient, and one was under development in 1998.9 The power transmitted by a rotating shaft is the product of torque and angular velocity. Angular-velocity measurement was treated in Chap. 4. 5.2 BASIC METHODS OF FORCE MEASUREMENT An unknown force may be measured by the following means: 1. Balancing it against the known gravitational force on a standard mass, either directly or through a system of levers 2. Measuring the acceleration of a body of known mass to which the unknown force is applied 3. Balancing it against a magnetic force developed by interaction of a current￾carrying coil and a magnet 4. Transducing the force to a fluid pressure and then measuring the pressure 5. Applying the force to some elastic member and measuring the resulting deflection 6. Measuring the change in precession of a gyroscope caused by an applied torque related to the measured force 7. Measuring the change in natural frequency of a wire tensioned by the force 5R. M. Schoonover and F. E. Jones, “Examination of Parameters That Can Cause Errors in Mass Determination,” CAL LAB, July/Aug. 1998, pp. 26–31. 6“Calibration of Force-Measuring Instruments for Verifying the Load Indication of Testing Machines,” ASTM Std. E-74, 1974. 7W. C. Dillon Co. (www.dillonnews.com). 8Gold Standard System, Interface, Inc., Scottsdale, AZ, 800-947-5598 (www.interfaceforce.com); C. Ferrero et al., “Main Metrological Characteristics of IMGC Six-Component Dynamometer,” RAM, vol. 2, 1986, pp. 21–28; R. Hellwig, “Precision Force Transducer for International Comparison Measurements on Force Standard Machines,” RAM, vol. 3, 1987, pp. 17–22; HBM, Norcross, GA, 888-816-9006 (www.hbm-home.com). 9F. A. Davis, “Design of the 1st UK National Standard Static Torque Calibration Machine,” National Physical Laboratory, Queens Road, Teddington, Middlesex, United Kingdom, TW 11 OLW, 0181-943-6194. 414 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM Review Copyfor lnstructor Nicolescu.Not fordistributionMeasurementSystems,ApplicationandDesign,FifthEdition415CHAPTER5435Force,Torque,and Shaft Power MeasurementLLLLLL-TapeTape4OC口T14UnknownStandardforcemassCounter-Analytical bolan.ceweightsSteel18tapesTfPendulumscaleOFCFStandordmoss 2I"Poise weight")]白 Stondord mossI("Panweighr")16;PlotformAmtof6-Platform scole(1)Accelerometer(2)Figure 5.2Basicforce-measurementmethods.InFig.5.2,method1isillustratedbytheanalyticalbalance,thependulumscale,and theplatform scale.Theanalytical balance,whilesimpleinprinciplerequires careful design and operation to realize its maximum performance.ioThebeam isdesigned sothatthe center ofmass is onlyslightly(afewthousandthsof aninch)belowtheknife-edgepivot and thus barelyin stableequilibrium.Thismakesthebeamdeflection(which insensitiveinstruments isreadwithan optical1oL.B.Macurdy,"Performance Tests for Balances," Inst.& Cont.Syst.,pp. 127-133, September 1965

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 CHAPTER 5 Force, Torque, and Shaft Power Measurement 435 In Fig. 5.2, method 1 is illustrated by the analytical balance, the pendulum scale, and the platform scale. The analytical balance, while simple in principle, requires careful design and operation to realize its maximum performance.10 The beam is designed so that the center of mass is only slightly (a few thousandths of an inch) below the knife-edge pivot and thus barely in stable equilibrium. This makes the beam deflection (which in sensitive instruments is read with an optical Figure 5.2 Basic force-measurement methods. 10L. B. Macurdy, “Performance Tests for Balances,” Inst. & Cont. Syst., pp. 127–133, September 1965. Measurement Systems, Application and Design, Fifth Edition 415 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution416Signal Processingand EngineeringMeasurements436PART2MeasuringDevices+++++++Flexiblebearings(3a)Pan?Suspension?Parallel guideOFlexiblebearing(①???OCouplingQLeverFlexible fulcrumOCoil?O①Permanent magnet@oFlux lines?8DiaphragmOpticalpositionCindicator?Temperature sensor-O④(3)@3(3b)Figure 5.2(Continued)micrometer)a very sensitive indicator of unbalance.For the low end of a particularinstrument's range, often the beam deflection is used as the output reading ratherthanattemptingtonullbyaddingmassesoradjustingthearmlengthof apoiseweight.This approachisfasterthannullingbut requiresthat thedeflection-angleunbalancerelationbeaccuratelyknownand stable.Thisrelationtendstovarywiththe load on the balance,because of deformation of knife edges, etc., but carefuldesigncankeepthistoaminimum.Forhighlyaccuratemeasurements,thebuoyantforceduetotheimmersionof thestandardmass inairmustbetaken intoaccount.Also,themost sensitivebalancesmustbe installedintemperature-controlled cham-bers andmanipulatedby remote control to reduce the effects ofthe operator's bodyheatandconvection currents.Typically,a temperaturedifferenceof1/20'Cbetweenthe two arms of a balance can cause an arm-length ratio change of Ippm, sig-nificant in some applications.Commerciallyavailable analyticalbalancesmaybeclassified asfollows:11"f, Baur, "The Analytical Balance," Ind. Res., . 64, JulyAugust 1964

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 436 PART 2 Measuring Devices micrometer) a very sensitive indicator of unbalance. For the low end of a particular instrument’s range, often the beam deflection is used as the output reading rather than attempting to null by adding masses or adjusting the arm length of a poise weight. This approach is faster than nulling but requires that the deflection-angle unbalance relation be accurately known and stable. This relation tends to vary with the load on the balance, because of deformation of knife edges, etc., but careful design can keep this to a minimum. For highly accurate measurements, the buoyant force due to the immersion of the standard mass in air must be taken into account. Also, the most sensitive balances must be installed in temperature-controlled cham￾bers and manipulated by remote control to reduce the effects of the operator’s body heat and convection currents. Typically, a temperature difference of 1/20˚C between the two arms of a balance can cause an arm-length ratio change of 1 ppm, sig￾nificant in some applications. Commercially available analytical balances may be classified as follows:11 Pan Suspension Parallel guide Flexible bearing Coupling Lever Flexible fulcrum Coil Permanent magnet Flux lines Diaphragm Optical position indicator Temperature sensor 1 1 2 2 3 3 4 4 4 3 4 5 5 6 6 7 7 8 8 9 9 10 10 11 12 11 12 13 13 Flexible bearings (3a) (3b) G Figure 5.2 (Continued) 11F. Baur, “The Analytical Balance,” Ind. Res., p. 64, July–August 1964. 416 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM Review Copy forInstructorNicolescu.NotfordistributionMeasurement Systems,Application and Design,Fifth Edition417437CHAPTER5Force, Torque, and Shaft Power MeasurementOperating principleElectromagneticbalanceof themagneticsuspension balanceAAThe controlled electromagnet exerts a magnetic forceon the permanent magnet throughthe nonmagneticvessel wall, supporting the sample weight. This force58Control systemis measured by the electromagnetic balance-ElectromagnetSetpointcontrollerCoupling housingPermanent magnetOCYSensorcoreSensor coilPosifion transducerMany versions of this basic balanceare available for different pressureMeasuringand temperature ranges andLload decouplingLmeasuring fasksSample(3c)Figure 5.2(Continued)DescriptionRange,gResolution,g10-4200-1,000Macro analytical10-550-100Semimicro analytical10-61020Micro analytical10~6Microbalanceless than 110~7Ultramicrobalanceless than 0.01Thependulum scaleis a deflection-typeinstrumentinwhichtheunknownforceis converted to a torque that is then balanced by the torque of a fixed standard massarranged as a pendulum.Thepractical version of this principleutilizes speciallyshaped sectors and steel tapestolinearize the inherentlynonlinear torque-angle rela-tion of a pendulum. The unknown force F, may be applied directly as in Fig.5.2 orthrougha system oflevers,such asthat shownfortheplatform scale,toextend therange.An electrical signal proportional to force is easily obtained from any angular-displacement transducer attachedtomeasuretheangle.The platform scaleutilizes a system of levers to allow measurement of largeforces in terms of much smaller standard weights.The beam is broughtto null byaproper combination of pan weights and adjustment of the poise-weight lever arm

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 CHAPTER 5 Force, Torque, and Shaft Power Measurement 437 Description Range, g Resolution, g Macro analytical 200–1,000 104 Semimicro analytical 50–100 105 Micro analytical 10–20 106 Micro balance less than 1 106 Ultramicro balance less than 0.01 107 The pendulum scale is a deflection-type instrument in which the unknown force is converted to a torque that is then balanced by the torque of a fixed standard mass arranged as a pendulum. The practical version of this principle utilizes specially shaped sectors and steel tapes to linearize the inherently nonlinear torque-angle rela￾tion of a pendulum. The unknown force Fi may be applied directly as in Fig. 5.2 or through a system of levers, such as that shown for the platform scale, to extend the range. An electrical signal proportional to force is easily obtained from any angular￾displacement transducer attached to measure the angle uo . The platform scale utilizes a system of levers to allow measurement of large forces in terms of much smaller standard weights. The beam is brought to null by a proper combination of pan weights and adjustment of the poise-weight lever arm Electromagnetic balance Electromagnet Coupling housing Permanent magnet Sensor core Sensor coil Measuring load decoupling Sample Control system Set point controller Position transducer Operating principle of the magnetic suspension balance The controlled electromagnet exerts a magnetic force on the permanent magnet through the nonmagnetic vessel wall, supporting the sample weight. This force is measured by the electromagnetic balance Many versions of this basic balance are available for different pressure and temperature ranges and measuring tasks PID controller (3c ) Figure 5.2 (Continued) Measurement Systems, Application and Design, Fifth Edition 417 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-HillCreateTM ReviewCopyforInstructorNicolescu.Notfordistribution418Signal Processing and Engineering Measurements438PART2MeasuringDevices00Loading heaHardened steel ballSofftrubberClampringFlexible-AibootsupplyBox-Clampp,StayplalePreloodGaugingspringholeBridge ringDiaphrogmCasingHordened steelHydraulicringinsertsffuidPneumatic load cellHydrostatic load cell(4)Column compressionIAElasticCforce-to-deflectiontransducersBinocular bending beam+A- Shear webParollelogramflexures-beam(5)Figure 5.2(Concluded)along its calibrated scale. The scale can be made self-balancing by adding an elec-trical displacementpickupfornulldetectionandanamplifier-motorsystemtoposi-tion thepoiseweighttoachieve null.Another interestingfeature is that ifalb=cld,thereadingofthescaleisindependentofthelocationofF:ontheplatform.Sincethisis quiteconvenient,most commercial scalesprovidethisfeaturebyuse ofthesuspension system shown orothersthatallowsimilarresults.While analytical balances are used almost exclusively for“"weighing"(reallydeterminingthemassof)objectsorchemicalsamples,platformandpendulumscales areemployed alsoforforcemeasurements,such asthose involved in shaftpower determinations with dynamometers.All threeinstruments areintendedmainlyforstaticforcemeasurements

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 438 PART 2 Measuring Devices along its calibrated scale. The scale can be made self-balancing by adding an elec￾trical displacement pickup for null detection and an amplifier-motor system to posi￾tion the poise weight to achieve null. Another interesting feature is that if a/b
c/d , the reading of the scale is independent of the location of Fi on the platform. Since this is quite convenient, most commercial scales provide this feature by use of the suspension system shown or others that allow similar results. While analytical balances are used almost exclusively for “weighing” (really determining the mass of) objects or chemical samples, platform and pendulum scales are employed also for force measurements, such as those involved in shaft power determinations with dynamometers. All three instruments are intended mainly for static force measurements. A P P A Column compression Hydrostatic load cell Binocular bending beam Shear web s-beam C T T C C T T C (5) Figure 5.2 (Concluded) 418 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM Review Copyfor lnstructor Nicolescu.Not fordistributionMeasurementSystems,ApplicationandDesign,FifthEdition419CHAPTER55Force, Torque, and Shaft Power Measurement439Method 2,the use of an accelerometerforforce measurement, is of somewhatlimited application since the force determined is the resultantforce on the mass.Often several unknown forces are acting,and they cannot be separately measuredby this method.The electromagnetic balancel2 (method 3)utilizes a photoelectric (or otherdisplacement sensor) null detector, an amplifier, and a torquing coil in a servo-system to balancethedifferencebetween the unknown force F,and the gravityforceon a standard mass.Its advantages relative tomechanical balances are ease of useless sensitivityto environment,fasterresponse,smaller size,andeaseofremoteoperation.Also,the electric output signal is convenient for continuous recordingand/or automatic-control applications.Balances with built-in microprocessors13allow evengreater convenience,versatility,and speed ofuse byautomating manyroutine procedures and providing features not formerly feasible. Automatic tare-weight systems subtract containerweightfromtotal weighttogivenetweight whenmaterial isplaced in the container.Statisticalroutines allowimmediatecalculationofmean and standard deviationfora series of weighings."Counting"of small partsby weighing is speeded by programming the microprocessor to read out the partscount directly,rather than the weight. Accurate weighing of live laboratory animals(difficulton an ordinary balancebecause of animal motion)isfacilitated by averag-ing scale readings over a preselected time.Interfacing the balance to (external orbuilt-in)printersforpermanent recording alsois eased bythe microprocessor.Figure 5.2,part 3a,14 shows a design available in range from 22 to 405 grams, withresolutions from 2to 100 μg.Part 3a shows schematically the parallelogram flex-ure system that guides the motion produced by an applied force (weight), whilepart 3b shows details of thecompletesystem (exceptforthe servoand readout elec-tronics).A flexure-pivot lever system (up to 15:1) puts large input forces within therange of a relatively small magnetic force coil.The signal from the opticaldisplacementsensoristheerrorsignal inthe servo system,whichprovides a coilcurrent (and thus magnetic force) to balance the unknown input force and restorethedeflectiontonearzero.Allmotions areconstrained withflexurebearings (ratherthanrolling or slidingbearings)togive thenearlyfrictionlessperformance requiredfor resolutions as small as 2 μg.Temperature effects (observed mainly in themagneticfield strength)arecompensated in software;thetemperatureismeasuredusing the signal from the temperature sensor.The seven-digit readout testifiestotheextremeresolution of these instruments.Figure5.2,part3c,15showsa version thatallowstheweighed sampletobeimmersed inanatmosphereofcontrolledtemper-ature,pressure,and fluid composition,completely sealed off from theweighingbalance,forsensitivedensity,sorbtion,andchemical studies2L.CaEetmaeWeighingInm.ConSyst7eptembr1962;CahnInrumDiv. (www.thermocahn.com).13B. Ludewig,"Microprocessor Balance," Am. Lab., Pp. 8183, May 197914Mettler-Toledo, Inc., Hightstown, NJ, 800-638-8537 (www.mico.mt.com).15Rubotherm GMBH,S.Natick, MA, 508-655-3950 (www.rubotherm.com)

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 CHAPTER 5 Force, Torque, and Shaft Power Measurement 439 Method 2, the use of an accelerometer for force measurement, is of somewhat limited application since the force determined is the resultant force on the mass. Often several unknown forces are acting, and they cannot be separately measured by this method. The electromagnetic balance12 (method 3) utilizes a photoelectric (or other displacement sensor) null detector, an amplifier, and a torquing coil in a servo￾system to balance the difference between the unknown force Fi and the gravity force on a standard mass. Its advantages relative to mechanical balances are ease of use, less sensitivity to environment, faster response, smaller size, and ease of remote operation. Also, the electric output signal is convenient for continuous recording and/or automatic-control applications. Balances with built-in microprocessors13 allow even greater convenience, versatility, and speed of use by automating many routine procedures and providing features not formerly feasible. Automatic tare￾weight systems subtract container weight from total weight to give net weight when material is placed in the container. Statistical routines allow immediate calculation of mean and standard deviation for a series of weighings. “Counting” of small parts by weighing is speeded by programming the microprocessor to read out the parts count directly, rather than the weight. Accurate weighing of live laboratory animals (difficult on an ordinary balance because of animal motion) is facilitated by averag￾ing scale readings over a preselected time. Interfacing the balance to (external or built-in) printers for permanent recording also is eased by the microprocessor. Figure 5.2, part 3a,14 shows a design available in range from 22 to 405 grams, with resolutions from 2 to 100 mg. Part 3a shows schematically the parallelogram flex￾ure system that guides the motion produced by an applied force (weight), while part 3b shows details of the complete system (except for the servo and readout elec￾tronics). A flexure-pivot lever system (up to 15:1) puts large input forces within the range of a relatively small magnetic force coil. The signal from the optical displacement sensor is the error signal in the servo system, which provides a coil current (and thus magnetic force) to balance the unknown input force and restore the deflection to near zero. All motions are constrained with flexure bearings (rather than rolling or sliding bearings) to give the nearly frictionless performance required for resolutions as small as 2 mg. Temperature effects (observed mainly in the magnetic field strength) are compensated in software; the temperature is measured using the signal from the temperature sensor. The seven-digit readout testifies to the extreme resolution of these instruments. Figure 5.2, part 3c,15 shows a version that allows the weighed sample to be immersed in an atmosphere of controlled temper￾ature, pressure, and fluid composition, completely sealed off from the weighing balance, for sensitive density, sorbtion, and chemical studies. 12L. Cahn, “Electromagnetic Weighing,” Instrum. Contr. Syst., p. 107, September 1962; Cahn Instrument Div. (www.thermocahn.com). 13B. Ludewig, “Microprocessor Balance,” Am. Lab., pp. 81–83, May 1979. 14Mettler-Toledo, Inc., Hightstown, NJ, 800-638-8537 (www.mico.mt.com). 15Rubotherm GMBH, S. Natick, MA, 508-655-3950 (www.rubotherm.com). Measurement Systems, Application and Design, Fifth Edition 419 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution

McGraw-Hill CreateTM Review Copyfor Instructor Nicolescu.Not fordistribution.420Signal Processing and Engineering Measurements440PART2MeasuringDevicesMethod 4 is illustrated in Fig. 5.2 by hydrostaticl6 and pneumatic load cells.Hydraulic cells are completely filled with oil and usually have a preload pressure oftheorderof 30Ib/in?.Application of load increasestheoilpressure,whichisreadon an accurategage.Electrical pressure transducers canbe used toobtain an elec-trical signal.The cells arevery stiff,deflecting onlyafewthousandths of an inchunderfull load.Capacitiesto100,000lbf areavailableas standard while specialunits up to 10 million Ibf are obtainable. Accuracy is of the order of 0.1 percentof full scale; resolution is about 0.02 percent. A hydraulic totalizerl7 is availableto produce a single pressure equal to the sum of up to 10 individual pressures inmultiple-cell systems used for tank weighing, etc. (see Chap. 10).The pneumatic load cell shown uses a nozzle-flapper transducer as a high-gainamplifierinaservoloop.ApplicationofforceF,causesadiaphragmdeflectionx,which in turn causes an increase in pressure P。since the nozzle is more nearly shutoff. This increase in pressure acting on the diaphragm area A produces an effectiveforceF,that tends to return thediaphragmto its formerposition.For any constantFr,the system will come to equilibrium at a specific nozzle opening and correspon-ding pressure Po.The static behavior is given by(5.3)(F, -P。A)KaK, = PoK. diaphragm compliance,in/lbf(5.4)whereKnozzleflappergain,(lb/in2)/in(5.5)Solving for Po we getF(5.6)P。= 1(K,K.)+ANow K, is not strictly constant, but varies somewhat with x,leading to a non-linearity between x and p.However,in practice,theproductK,K, isverylarge,sothat 1/(KK,)ismade negligible compared withA, which givesFi(5.7)Po"Awhich is linear since A is constant. As in any feedback system, dynamic instabilitylimitstheamountofgainthat actuallycanbeused.Atypical supplypressurep,is60Ib/in?,and sincethemaximumvalueofPocannotexceedps,thislimitsF,tosomewhat less than 60 A.A line of commercial pneumatic weighing systems18 usingsimilar principles (combined with lever/knife-edge methods)is available in standardranges to110,000lbf.While allthe previously described force-measuring devices are intended mainlyforstaticor slowlyvaryingloads,theelasticdeflectiontransducersofmethod5are16A. H. Emery Co. (www.emerywinslow.com).17Tbid.1$*An Introduction to the Darenth Gnu-Weigh Pneumatic Weighing System," Darenth Americas,Bridgeville, DE, 1980. (A google search in 2002 could not find this company.)

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 440 PART 2 Measuring Devices Method 4 is illustrated in Fig. 5.2 by hydrostatic16 and pneumatic load cells. Hydraulic cells are completely filled with oil and usually have a preload pressure of the order of 30 lb/in2. Application of load increases the oil pressure, which is read on an accurate gage. Electrical pressure transducers can be used to obtain an elec￾trical signal. The cells are very stiff, deflecting only a few thousandths of an inch under full load. Capacities to 100,000 lbf are available as standard while special units up to 10 million lbf are obtainable. Accuracy is of the order of 0.1 percent of full scale; resolution is about 0.02 percent. A hydraulic totalizer17 is available to produce a single pressure equal to the sum of up to 10 individual pressures in multiple-cell systems used for tank weighing, etc. (see Chap. 10). The pneumatic load cell shown uses a nozzle-flapper transducer as a high-gain amplifier in a servoloop. Application of force Fi causes a diaphragm deflection x , which in turn causes an increase in pressure po since the nozzle is more nearly shut off. This increase in pressure acting on the diaphragm area A produces an effective force Fp that tends to return the diaphragm to its former position. For any constant Fi , the system will come to equilibrium at a specific nozzle opening and correspon￾ding pressure po . The static behavior is given by (Fi po A)Kd Kn
po (5.3) where Kd

diaphragm compliance, in/lbf (5.4) Kn

nozzle-flapper gain, (lb/in2)/in (5.5) Solving for po , we get po
(5.6) Now Kn is not strictly constant, but varies somewhat with x , leading to a non￾linearity between x and po . However, in practice, the product Kd Kn is very large, so that 1/(Kd Kn) is made negligible compared with A, which gives po
(5.7) which is linear since A is constant. As in any feedback system, dynamic instability limits the amount of gain that actually can be used. A typical supply pressure ps is 60 lb/in2, and since the maximum value of p0 cannot exceed ps , this limits Fi to somewhat less than 60 A. A line of commercial pneumatic weighing systems18 using similar principles (combined with lever/knife-edge methods) is available in standard ranges to 110,000 lbf. While all the previously described force-measuring devices are intended mainly for static or slowly varying loads, the elastic deflection transducers of method 5 are Fi A Fi 1/(KdKn) A 16A. H. Emery Co. (www.emerywinslow.com). 17Ibid. 18“An Introduction to the Darenth Gnu-Weigh Pneumatic Weighing System,” Darenth Americas, Bridgeville, DE, 1980. (A google search in 2002 could not find this company.) 420 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.

McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionMeasurementSystems,ApplicationandDesign,FifthEdition421CHAPTER5Force, Torque,and Shaft Power Measurement441widely used for both static and dynamic loads of frequency content up to manythousand hertz.Whileall areessentiallyspring-mass systemswith(intentional orunintentional)damping,they differ mainly inthe geometric form of"springemployed and in the displacement transducer used to obtain an electrical signal.Thedisplacement sensedmaybe a gross motion, or strain gages may be judiciouslylocatedtosenseforceintermsofstrain.Bonded straingageshavebeenfoundparticularly useful in force measurements with elastic elements.In addition to serv-ingasforce-to-deflectiontransducers,someelasticelementsperformthefunctionofresolvingvectorforces ormoments intorectangularcomponents.Asan example,theparallelogramflexureof Fig.5.2(part5)is extremely rigid (insensitive)toallappliedforces andmoments except inthedirection shownbythearrow.Adisplace-ment transducer arranged to measure motion in the sensitive direction thus willmeasure only that component of an applied vector force which lies along thesensitiveaxis.Perhaps theaction of thisflexuremaybemost easilyvisualizedbyconsidering it as a four-bar linkage withflexure hinges at the thin sections a, b, c,and d.Because of the importance of elastic force transducers in modern dynamicmeasurements, we devote a considerable portion of this chapter to their considera-tion.Although they may differ widely in detail construction, their dynamic-responseform isgenerallythe same,and so wetreat an idealized modelrepresentative of allsuch transducers in the next section.Discussion of methods 6 and 7is deferred totheend ofthechaptersincetheyarenotascommonasmethodsI through55.3CHARACTERISTICSOFELASTICFORCETRANSDUCERSFigure 5.3 shows an idealized model of an elastic force transducer.Therelationshipbetween inputforceand output displacement is easily established as a simplesecond-orderform:(5.8)F,-K,x。-Bx=Mx。K曾(D)=(5.9)FiD2/+2D/+1区o,4(5.10)whereVMB14(5.11)2VK,M1K4(5.12)K.Notethatdevicesof thistypearealso(unintentional)accelerometersandproduceaspurious outputinresponsetobase vibration inputs (seeProb.5.1).For transducers that do not measure a gross displacement but rather use straingages bonded to the"spring,"the output strain may be substituted for xif K

Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 CHAPTER 5 Force, Torque, and Shaft Power Measurement 441 widely used for both static and dynamic loads of frequency content up to many thousand hertz. While all are essentially spring-mass systems with (intentional or unintentional) damping, they differ mainly in the geometric form of “spring” employed and in the displacement transducer used to obtain an electrical signal. The displacement sensed may be a gross motion, or strain gages may be judiciously located to sense force in terms of strain. Bonded strain gages have been found particularly useful in force measurements with elastic elements. In addition to serv￾ing as force-to-deflection transducers, some elastic elements perform the function of resolving vector forces or moments into rectangular components. As an example, the parallelogram flexure of Fig. 5.2 (part 5) is extremely rigid (insensitive) to all applied forces and moments except in the direction shown by the arrow. A displace￾ment transducer arranged to measure motion in the sensitive direction thus will measure only that component of an applied vector force which lies along the sensitive axis. Perhaps the action of this flexure may be most easily visualized by considering it as a four-bar linkage with flexure hinges at the thin sections a, b, c, and d. Because of the importance of elastic force transducers in modern dynamic measurements, we devote a considerable portion of this chapter to their considera￾tion. Although they may differ widely in detail construction, their dynamic-response form is generally the same, and so we treat an idealized model representative of all such transducers in the next section. Discussion of methods 6 and 7 is deferred to the end of the chapter since they are not as common as methods 1 through 5. 5.3 CHARACTERISTICS OF ELASTIC FORCE TRANSDUCERS Figure 5.3 shows an idealized model of an elastic force transducer. The relationship between input force and output displacement is easily established as a simple second-order form: Fi Ks xo B˙xo
M¨xo (5.8) (D)
(5.9) where vn

(5.10) 

(5.11) K

(5.12) Note that devices of this type are also (unintentional) accelerometers and produce a spurious output in response to base vibration inputs (see Prob. 5.1). For transducers that do not measure a gross displacement but rather use strain gages bonded to the “spring,” the output strain e may be substituted for xo if Ks 1 Ks B 2
Ks M B Ks M K D2 /2 n 2D/n 1 xo Fi Measurement Systems, Application and Design, Fifth Edition 421 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.