McGraw-Hill CreateTM Review Copyfor Instructor Nicolescu.Not fordistribution.422Signal Processingand Engineering Measurements442PART 2MeasuringDevicesFigure 5.3Elasticforcetransducer.is reinterpreted as force per unit strain rather than force per unit deflection.Manyelasticelements havebeen analyzed, and the results"cataloged,"for stress(strain),19 deflection,20 and natural frequency.21 These results are in the formof formulasinvolvingmaterialpropertiesanddimensions,and soareusefulforpreliminary design of transducers. For unusual shapes, or to check details such asstress concentrations, standard finite-element software can be used, but recall herethat this approach does not provide anyformulas,only specific numerical resultsforspecific dimensions.Mostforcetransducers (withfewexceptions)22have no inten-tional damping;isdue entirelyto parasitic effects and is not possibletopredicttheoretically.While theoretical and finite-element results are often adequateformachine design, since transducers must be accurate toI percent or better,they mustalways be calibrated after construction and before use; this calibration will easilyget the actual damping ratio.Since the dynamic response of second-order instruments has been fully dis-cussed,we concentrate mainly on details peculiar to specific force transducers.Fortransducers based on strain gages,gage manufacturers provide helpful handbooks.23Measurements Group Inc.also sells software calledTRANSCALC"that facilitatesdesignfor 14 different load-cell geometries and diaphragm pressure transducers.Thissoftwarecan bedownloadedfromtheInternetforafree10-daytrialWhile simple geometries such as the cantilever beam can be easily designedusingavailableformulasfromtexts,suchformulas arenotavailableforsomeofthemore popular forms, such as the“binocular."The TRANSCALC software includesthisform,buttheformulas are saidtobeproprietary.An analysis of thisform inthe19w.C. Young and R. G. Budynas, “Roark's Formulas for Stress and Strain," McGraw-Hill, New York,2001.This is also available as a software product.20Tbid.2'R. D. Blevins, “Formulas for Natural Frequency and Mode Shape," Krieger, Malabar, FL, 1995.22Tedea Huntleigh, Canoga Park, CA, models 240, 1410, and 9010, 818-716-0593(www.tedeahuntleigh.com).23The Route to Measurement Transducers, HBM Inc., Marlboro, MA, 888-235-4243(www.hbminc.com/gageland); "Strain Gage Based Transducers," Measurement Group Inc., Raleigh, NC,919-365-3800 (www.measurementsgroup.com)
Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 442 PART 2 Measuring Devices is reinterpreted as force per unit strain rather than force per unit deflection. Many elastic elements have been analyzed, and the results “cataloged,” for stress (strain),19 deflection,20 and natural frequency.21 These results are in the form of formulas involving material properties and dimensions, and so are useful for preliminary design of transducers. For unusual shapes, or to check details such as stress concentrations, standard finite-element software can be used, but recall here that this approach does not provide any formulas, only specific numerical results for specific dimensions. Most force transducers (with few exceptions)22 have no intentional damping; is due entirely to parasitic effects and is not possible to predict theoretically. While theoretical and finite-element results are often adequate for machine design, since transducers must be accurate to 1 percent or better, they must always be calibrated after construction and before use; this calibration will easily get the actual damping ratio. Since the dynamic response of second-order instruments has been fully discussed, we concentrate mainly on details peculiar to specific force transducers. For transducers based on strain gages, gage manufacturers provide helpful handbooks.23 Measurements Group Inc. also sells software called “TRANSCALC” that facilitates design for 14 different load-cell geometries and diaphragm pressure transducers. This software can be downloaded from the Internet for a free 10-day trial. While simple geometries such as the cantilever beam can be easily designed using available formulas from texts, such formulas are not available for some of the more popular forms, such as the “binocular.” The TRANSCALC software includes this form, but the formulas are said to be proprietary. An analysis of this form in the Figure 5.3 Elastic force transducer. 19W. C. Young and R. G. Budynas, “Roark’s Formulas for Stress and Strain,” McGraw-Hill, New York, 2001. This is also available as a software product. 20Ibid. 21R. D. Blevins, “Formulas for Natural Frequency and Mode Shape,” Krieger, Malabar, FL, 1995. 22Tedea Huntleigh, Canoga Park, CA, models 240, 1410, and 9010, 818-716-0593 (www.tedeahuntleigh.com). 23“The Route to Measurement Transducers, HBM Inc., Marlboro, MA, 888-235-4243 (www.hbminc.com/gageland); “Strain Gage Based Transducers,” Measurement Group Inc., Raleigh, NC, 919-365-3800 (www.measurementsgroup.com). 422 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution
McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionMeasurementSystems,ApplicationandDesign,FifthEdition423CHAPTER55Force, Torque, and Shaft Power Measurement443openliteraturedoes not appeartobeavailable.Of course,anycomplexgeometrycan be subjected to finite-element software for analysis, but such studies do notprovidedesign formulas, only numerical results for a given set of dimensions andmaterial properties.I decided to try astrength of materialstype of analysis myselfand,aftersomestudy,cameup withasimpleanalysisbasedon certainassumptions.To check this analysis,I then also did a finite-element analysis, and finally actuallybuilt one unit for laboratory testing.Since these various steps mightbe part of anyload-cell designprocess,thefollowingexample,whichbrieflysummarizestheprocedure, is providedFigure 5.4a shows the meshed finite-element model but will also serve as ananalysis sketch"for the strength of materials study.The vertical support at the leftis not an inherent part of such load cells; it was included to allow space under theload cellfor a deflection-measuring dial indicator.Thefirst assumption made wasto think of the deviceas a four-bar linkage,wheretheusual pin joints are replacedby elastic hinges at thefour thin sections.If this assumption is essentially correct,then the"linkat the right (wherethevertical load Fis applied)would haveasimple (mainly vertical) translation x. The symmetry of the device suggests thateach of thefour hinges contributes an equal elastic resisting moment, tobalance theapplied loading.(Figure 5.4b shows the finite-element deflection results that werelaterobtained andwhich seemtovalidatetheseassumptions.)I next tried a"direct"approach from strength of materials beam theory,butwas unableto evaluatetheneeded bendingmomentsor convincemyself of theinsensitivity to load application point that is claimed for this design. Finally I triedan energy-based method which led to useful results.The assumption here was thattheapplied force would moveverticallythrougha certain deflectiony,therebydoing a definite amount of work Fy/2.This work would show up as an equalamount of storedelastic energy in thefour hinges,which would eachfeel a torqueT and rotate through the same angle .Since the angle is sure to be small,tan y/L 9, where L is the distance between hole center-lines (2.0 inches in theexample).Equating the energies gives the result Fy/2 = 4T0/2, and thus the bend-ingmomentateachhinge isT=FL/4.This result notonlyallowsus to computestresses and strains,butalsoverifies the claimthat the load cellgives essentiallythesameoutput irrespectiveof thehorizontal location of theappliedforce.(This claimreally applies only to locations somewhat to the right of the right-most gage.)Using standardbeambendingformulas nowgives96T3FL8-9(5.13)EbtE2btEwhere t is the thickness at the thinnest point and b is the depth, (0.04 in and 0.50 inintheexample).Thisformulaisuseful forchoosingdimensions whenthefullscale load F and design strain (often about 1500 μe)are given (our example hasF=5.0lb).TheTRANSCALCsoftwaredoesnotcomputetheloadcellstiffness;the company told me that their customers rarely request this bit of information.Itwould ofcourse beofinterestfor anydynamic application since it,together withthe
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 443 open literature does not appear to be available. Of course, any complex geometry can be subjected to finite-element software for analysis, but such studies do not provide design formulas, only numerical results for a given set of dimensions and material properties. I decided to try a “strength of materials” type of analysis myself and, after some study, came up with a simple analysis based on certain assumptions. To check this analysis, I then also did a finite-element analysis, and finally actually built one unit for laboratory testing. Since these various steps might be part of any load-cell design process, the following example, which briefly summarizes the procedure, is provided. Figure 5.4a shows the meshed finite-element model but will also serve as an “analysis sketch” for the strength of materials study. The vertical support at the left is not an inherent part of such load cells; it was included to allow space under the load cell for a deflection-measuring dial indicator. The first assumption made was to think of the device as a four-bar linkage, where the usual pin joints are replaced by elastic hinges at the four thin sections. If this assumption is essentially correct, then the “link” at the right (where the vertical load F is applied) would have a simple (mainly vertical) translation x. The symmetry of the device suggests that each of the four hinges contributes an equal elastic resisting moment, to balance the applied loading. (Figure 5.4b shows the finite-element deflection results that were later obtained and which seem to validate these assumptions.) I next tried a “direct” approach from strength of materials beam theory, but was unable to evaluate the needed bending moments or convince myself of the insensitivity to load application point that is claimed for this design. Finally I tried an energy-based method which led to useful results. The assumption here was that the applied force would move vertically through a certain deflection y, thereby doing a definite amount of work Fy /2. This work would show up as an equal amount of stored elastic energy in the four hinges, which would each feel a torque T and rotate through the same angle u. Since the angle is sure to be small, tan u y/L u, where L is the distance between hole center-lines (2.0 inches in the example). Equating the energies gives the result Fy/2 4Tu/2, and thus the bending moment at each hinge is T FL /4. This result not only allows us to compute stresses and strains, but also verifies the claim that the load cell gives essentially the same output irrespective of the horizontal location of the applied force. (This claim really applies only to locations somewhat to the right of the right-most gage.) Using standard beam bending formulas now gives e (5.13) where t is the thickness at the thinnest point and b is the depth, (0.04 in and 0.50 in in the example). This formula is useful for choosing dimensions when the full scale load F and design strain (often about 1500 me) are given (our example has F 5.0 lb). The TRANSCALC software does not compute the load cell stiffness; the company told me that their customers rarely request this bit of information. It would of course be of interest for any dynamic application since it, together with the E 6T bt2 E 3FL 2bt2 E Measurement Systems, Application and Design, Fifth Edition 423 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.
McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution.424SignalProcessingandEngineeringMeasurements444PART 2MeasuringDevices+ 1.5 in.1 ;LBSLBS2.0in4.0 in.5.25 inThinnestsectionare0.040 in.Semicirclesare0.750in.diametet5-pound load shown at two locations7.0 in.(a)DX-0.000e+0DD DY=1.890e-004DZ--3.968e-002(b)Figure 5.4Finite-element model of binocularforce transducer
Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 444 PART 2 Measuring Devices (a) Thinnest section are 0.040 in. Semicircles are 0.750 in. diameter 5-pound load shown at two locations 1.5 in. 5.25 in. 4.0 in. 2.0 in. 7.0 in. Figure 5.4 Finite-element model of binocular force transducer. (b) 424 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution
McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.NotfordistributionMeasurementSystems,ApplicationandDesign,FifthEditior425445CHAPTER5Force,Torque,andShaft PowerMeasurement588.0002(rad/time)Mode1:0Erequencys93.5831 (cycles/tine2(c)Figure 5.4(Concluded)attachedmass,determinesthenaturalfrequency.With a fewmore assumptions andsomeadditional analysis,wecan estimatethestiffnessbycomputingtheangle.Theformulafordeflectionangleofabeamis(MA0=Ejdr(5.14)where the integral is taken over the length of the beam and M is the bendingmoment.Wewilltake thebendingmomentas the constantTbut express theareamoment of inertia in terms of x since the beam thickness I varies with x. Note thatincomputing stress and strain atthe gagelocations,we treated tas a constantequalto the thickness at the thinnest point. Strain gages have a finite gage length, so themeasured strainwillbesomewhatlessthanthepeakvaluethatwehaveestimatedsince the beam gets thicker away from the midpoint. We cannot use this approachfor the deflection.Expressingt(and then I)as a function of xgives12T(RdxA0 =(5.15)bEJ-R(R+IVR2-x33
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 445 attached mass, determines the natural frequency. With a few more assumptions and some additional analysis, we can estimate the stiffness by computing the angle u. The formula for deflection angle of a beam is u dx (5.14) where the integral is taken over the length of the beam and M is the bending moment. We will take the bending moment as the constant T but express the area moment of inertia in terms of x since the beam thickness t varies with x. Note that in computing stress and strain at the gage locations, we treated t as a constant equal to the thickness at the thinnest point. Strain gages have a finite gage length, so the measured strain will be somewhat less than the peak value that we have estimated since the beam gets thicker away from the midpoint. We cannot use this approach for the deflection. Expressing t (and then I) as a function of x gives u (5.15) 12T bE R R dx (R tR2 x 2 ) 3 M EI Figure 5.4 (Concluded) (c) Measurement Systems, Application and Design, Fifth Edition 425 McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution.
McGraw-Hill CreateTM ReviewCopyforInstructorNicolescu.Notfordistribution426SignalProcessingandEngineeringMeasurements446PART2MeasuringDeviceswhere t is now the constant equal to the thickness at the thinnest point. I couldnot find an analytical formula for this integral so it was computed numericallyfortheassumed dimensions,givingthenumerical value310l;thentheangleis0.01879 rad, the deflection under the 5-pound load is 0.03759 in. and the stiffnessis 133lbf/in.Theblockof aluminumunder theload isthemajor movingmass;it is about 1.22 in3 and weighs 0.122 Ibf. Using n = VK,/M, we get a naturalfrequency of 100 Hz.The actual frequency will probably be somewhat lower sincethe two linksthat rotate will add somemass.UsingtheAlgorfinite-element software(www.algor.com),thefirstnaturalfrequencywasfound tobe93.6Hz,with themode shape shown inFig.5.4c.Thismode shapelooks much likethestatic deflection curve; firstmodesoftendo.Thesecond and third modes were 626 and2588Hz,respectively,with more complexmode shapes. For transducer design, only the first mode is generally of interestbecause we can get good dynamic accuracy onlyfor frequencies below the firstresonant peak.Algor also gave the deflection under the load as 0.0397 in, close toour estimateof 0.0376in.Figure5.5shows strainsintheneighborhood ofoneofthegage locations; all four were very similar,justifying our symmetry assumption.The peak values of 0.00193 are close to our estimates of 0.00189.For the laboratory-tested transducer the natural frequency observed was 97 Hzand the stiffness was 121lbf/in.Gagestrains atthe5-lb load were0.002073.0.001838,0.001827,and0.001971,which averageto0.00193.It appears that thedesignformulasdevelopedareusefulforthistypeoftransducer,whether itis imple-mented with strain gages or some gross-deflection sensor such as an LVDT.Bonded-Strain-GageTransducersAtypical construction for a strain-gage load cell for measuring compressive forcesis showninFig.5.6.(Cellstomeasurebothtensionand compressionrequiremerelythe addition of suitable mechanical fittings at the ends.)The load-sensing memberisshortenoughtopreventcolumnbucklingundertheratedloadandisproportionedtodevelop about1,500μ atfull-scaleload (typical design valueforall forms offoil gage transducers). Materials used include SAE 4340 steel, 17-4 PH stainlesssteel,and 2024-T4 aluminum alloy,with thelast beingquite popular for"home-made"transducers.Foil-type metal gages are bonded on all four sides;gages0HHHHLLUULHA.8.3e-042.8e043e-04-8e-040.00130.0019Display Range : Min=0.00192166, Max=0.00192637Figure5.5Finite-element analysis results for strain near gage locations
Doebelin: Measurement Systems, Application and Design, Fifth Edition II. Measurement Devices 5. Force, Torque, and Shaft Power Measurement © The McGraw−Hill Companies, 2004 446 PART 2 Measuring Devices where t is now the constant equal to the thickness at the thinnest point. I could not find an analytical formula for this integral so it was computed numerically for the assumed dimensions, giving the numerical value 3101; then the angle is 0.01879 rad, the deflection under the 5-pound load is 0.03759 in. and the stiffness is 133 lbf/in. The “block” of aluminum under the load is the major moving mass; it is about 1.22 in3 and weighs 0.122 lbf. Using vn , we get a natural frequency of 100 Hz. The actual frequency will probably be somewhat lower since the two links that rotate will add some mass. Using the Algor finite-element software (www.algor.com), the first natural frequency was found to be 93.6 Hz, with the mode shape shown in Fig. 5.4c. This mode shape looks much like the static deflection curve; first modes often do. The second and third modes were 626 and 2588 Hz, respectively, with more complex mode shapes. For transducer design, only the first mode is generally of interest because we can get good dynamic accuracy only for frequencies below the first resonant peak. Algor also gave the deflection under the load as 0.0397 in, close to our estimate of 0.0376 in. Figure 5.5 shows strains in the neighborhood of one of the gage locations; all four were very similar, justifying our symmetry assumption. The peak values of 0.00193 are close to our estimates of 0.00189. For the laboratory-tested transducer the natural frequency observed was 97 Hz and the stiffness was 121 lbf/in. Gage strains at the 5-lb load were 0.002073, 0.001838, 0.001827, and 0.001971, which average to 0.00193. It appears that the design formulas developed are useful for this type of transducer, whether it is implemented with strain gages or some gross-deflection sensor such as an LVDT. Bonded-Strain-Gage Transducers A typical construction for a strain-gage load cell for measuring compressive forces is shown in Fig. 5.6. (Cells to measure both tension and compression require merely the addition of suitable mechanical fittings at the ends.) The load-sensing member is short enough to prevent column buckling under the rated load and is proportioned to develop about 1,500 me at full-scale load (typical design value for all forms of foil gage transducers). Materials used include SAE 4340 steel, 17-4 PH stainless steel, and 2024–T4 aluminum alloy, with the last being quite popular for “homemade” transducers. Foil-type metal gages are bonded on all four sides; gages Ks /M 0.00193 0.00138 8.3e–04 2.8e–04 –3e–04 –8e–04 –0.0013 –0.0019 Display Range : Min=–0.00192166, Max=0.00192637 Figure 5.5 Finite-element analysis results for strain near gage locations. 426 Signal Processing and Engineering Measurements McGraw-Hill Create™ Review Copy for Instructor Nicolescu. Not for distribution