F subscript a, and defined as σ

Transcription

1 Stre, Strin, nd Strin Gge, Pge 1 Stre, Strin, nd Strin Gge Authr: Jhn M Cimbl, Penn Stte Univerity Ltet reviin: 15 Octber 2009 Intrductin Stre nd trin re imprtnt pect f Mechnicl Engineering, epecilly in tructurl deign In thi lerning mdule, we dicu tre nd trin nd their reltinhip, nd hw t meure them Definitin Stre When mteril i lded with frce, the tre t me lctin in the mteril i defined the pplied frce per unit f cr-ectinl re r exmple, cnider wire r cylinder, nchred t the tp, nd hnging dwn Sme frce (fr exmple, frm hnging weight) pull t the bttm, ketched, where A i the riginl cr-ectinl re f the wire, nd L i the riginl wire length A L In thi itutin, the mteril experience tre, clled n xil tre, dented by the ubcript, nd defined σ = A Ntice tht the dimenin f tre re the me the f preure frce per unit re Strin L L δl In the bve imple exmple, the wire tretche verticlly reult f the frce Strin i defined the rti f incree in length t riginl length δl Specificlly, when frce i pplied t the wire, it length L incree by mll increment δl, while it cr-ectinl re A decree, ketched In the xil directin (the directin f the pplied frce), xil trin i defined δ L = L The dimenin f trin re unity trin i nndimeninl quntity Hke lw It turn ut tht fr eltic mteril, tre i linerly prprtinl t trin Mthemticlly, thi i expreed by Hke lw, which tte σ = E, where E = Yung mdulu, l clled the mdulu f elticity Yung mdulu i umed t be cntnt fr given mteril σ Yield Hke lw brek dwn when the trin get t high On typicl tre tre-trin digrm, Hke lw pplie nly in the eltic tre Eltic tre regin, in which the lding i reverible Beynd the eltic limit (r prprtinl limit), the mteril trt t behve irreveribly in the pltic defrmtin regin, in which the tre v trin curve devite frm liner, nd Hke lw n lnger hld, ketched regin Eltic limit In thi lerning mdule, nly the eltic tre regin i cnidered Wire reitnce ρl The electricl reitnce R f wire f length L nd cr-ectinl re A i given by R =, where ρ i A the reitivity f the wire mteril (D nt cnfue ρ with denity, fr which the me ymbl i ued) The electricl reitnce f the wire chnge with trin: A trin incree, the wire length L incree, which incree R A trin incree, the wire cr-ectinl re A decree, which incree R r mt mteril, trin incree, the wire reitivity ρ l incree, which further incree R The bttm line i tht wire reitnce incree with trin In fct, it turn ut tht t cntnt temperture, wire reitnce incree linerly with trin δ R δ R / R Mthemticlly, = S, where S i the trin gge fctr, defined S = R S i typiclly rund 20 fr cmmercilly vilble trin gge S i dimeninle

2 Strin gge The principle dicued bve, nmely tht wire reitnce incree with trin, i key t undertnding hw trin gge wrk A trin gge cnit f mll dimeter wire (ctully n etched metl fil) tht i ttched t bcking mteril (uully mde f pltic) ketched The wire i lped bck nd frth everl time t crete n effectively lnger wire The lnger the wire, the lrger the reitnce, nd the lrger the chnge in reitnce with trin Here, fur lp f metl fil re hwn, prviding n effective ttl fil length L tht i eight time greter thn if ingle wire, rther thn lping pttern, were ued Cmmercilly vilble trin gge hve even mre lp thn thi The ne ued in ur lb hve ix lp The directin f the pplied trin i indicted n the ketch The cnnecting wire r led g t n electrnic circuit (dicued belw) tht meure the chnge in reitnce Cnider bem underging xil trin; the trin i t be meured Stre, Strin, nd Strin Gge, Pge 2 Directin f trin Slder terminl Cnnecting wire (led) Etched metl fil Bcking mteril A trin gge i glued t the urfce f the bem, with the lng ectin f the etched metl fil ligned with the pplied xil trin ketched belw left (the trin gge i munted n the frnt fce f the bem) A the urfce tretche (trin), the trin gge tretche lng with it The reitnce f the trin gge therefre incree with pplied trin Auming the chnge in reitnce cn be meured, the trin gge prvide methd fr meuring trin Other prcticl pplictin re hwn belw trin gge glued (rther lppily) nt cylindricl rd, nd trin gge munted n re-br, which i then enced in cncrete, ued t meure hrinkge nd t mnitr the trin n tructurl cmpnent in bridge, building, etc Bem Strin gge Typicl trin gge vlue Here re me typicl vlue fr reitnce, trin gge fctr, nd trin, lng with the predicted vlue f chnge in reitnce: The electricl reitnce R f cmmercil trin gge (with n pplied trin) i typiclly either 120 Ω r 350 Ω The mt widely ued cmmercilly vilble trin gge hve R = 120 Ω The trin gge fctr S f the metl fil ued in trin gge i typiclly rund 20 In typicl engineering pplictin with metl bem, the rnge f xil trin i 10-6 < < 10-3 Uing thee limit nd the bve equtin fr chnge in reitnce functin f trin nd trin gge fctr, δ R = RS, nd the typicl rnge f δr i ( )( )( ) δ R ( 120 )( 20)( 10 3 ) Ω < < Ω, r Ω < δr < 024 Ω Ntice hw mll δr i! r typicl 120 Ω trin gge, the rnge f frctinl chnge in reitnce i < δr/r < Thi i the min prblem when wrking with trin gge: We cnnt ue imple hm meter t meure the chnge in reitnce, becue δr/r i mll Mt hm meter d nt hve ufficient relutin t meure chnge in reitnce tht re 3 t 6 rder f mgnitude mller thn the reitnce itelf

3 Stre, Strin, nd Strin Gge, Pge 3 Strin gge electrnic Since δr/r i very mll nd difficult t meure directly, electrnic circuit mut be deigned t meure the chnge in reitnce rther thn the reitnce itelf rtuntely, there re circuit vilble t d jut tht The Whettne bridge A clever circuit t meure very mll chnge in reitnce i clled Whettne bridge A chemtic digrm f imple Whettne bridge circuit i hwn t the right A een in the ketch, upply vltge i upplied (tp t bttm) cr the bridge, which cntin fur reitr (tw prllel leg f tw V = upply reitr ech in erie) vltge The utput vltge i meured cr the leg in the middle f the bridge In the nlyi here, it i umed tht the meuring device (vltmeter, cillcpe, cmputerized digitl dt cquiitin ytem, etc) ued t meure utput vltge h n infinite input impednce, nd therefre h n effect n the circuit Output vltge = V V RR 3 1 RR 4 2 i clculted by nlyzing the circuit Nmely, V = V ( R2 R3)( R1 R4) [Thi equtin i exct n pprximtin f mll chnge in reitnce were mde in it derivtin] Hw de the Whettne bridge wrk? Well, if ll fur reitr re identicl (R 1 = R 2 = R 3 = R 4 ), the bridge i blnced ince the me current flw thrugh the left leg nd the right leg f the bridge r blnced bridge, = 0 Mre generlly ( cn be een frm the bve equtin), Whettne bridge cn be blnced even if the reitr d nt ll hve the me vlue, lng the numertr in the bve equtin i zer, ie, if R1 R4 RR 3 1 = RR 4 2 Or, expreed rti, the bridge i blnced if = R2 R3 In prctice, the bridge will nt be blnced utmticlly, ince identicl reitr re nt ctully identicl, with reitnce vrying by up t everl percent Thu, ptentimeter (vrible reitr) i metime pplied in plce f ne f the reitr in the bridge tht minr djutment cn be mde in rder t blnce the bridge An rrw thrugh the reitr indicte tht it reitnce cn vry, ketched t the right In thi circuit, reitr R 2 w rbitrrily chen t be replced by ptentimeter, but ny f the fur reitr culd hve been ued inted Qurter bridge circuit T meure trin, ne f the reitr, in thi ce R 3, i replced by the trin gge, ketched t the right (Nte tht ne f the ther reitr my till be ptentimeter rther thn fixed reitr, but tht will nt be indicted n the circuit digrm t fllw) Agin, n rrw thrugh the reitr indicte tht it reitnce cn vry thi time becue R 3 i n ctive trin gge, nt ptentimeter With nly ne ut f the fur vilble reitr ubtituted by trin gge, in the bve chemtic, the circuit i clled qurter bridge circuit V = upply vltge V = upply vltge Pt R 3 = trin gge RR 3 1 RR 4 2 The utput vltge i clculted frm Ohm lw, previuly, V = V ( R 2 R 3)( R 1 R 4) Let R 1 = R 2 = R 4 = 120 Ω, nd let the initil reitnce f the trin gge (with n ld) be R 3,initil = 120 Ω The bridge i therefre initilly blnced when R 3 = R 3,initil, ince R 3,initil R 1 R 4 R 2 = 0, nd i thu zer

4 Stre, Strin, nd Strin Gge, Pge 4 Unblnced qurter bridge circuit - t meure trin In nrml pertin, the Whettne bridge i initilly blnced bve Nw uppe trin i pplied t the trin gge, uch tht it reitnce chnge by me mll munt δr 3 In ther wrd, R 3 chnge frm R 3,initil t R 3,initil δr 3 Under thee cnditin the bridge i unblnced, nd the reulting utput vltge i nt zer, but cn be ( R3,initil δ R3 ) R1 R4R2 clculted V = V R R δ R R R ( 2 3,initil 3 )( 1 4 ) We implify the numertr by pplying the initil blnce equtin, R 3,initil R 1 R 4 R 2 = 0, yielding V V δ R3 R1 = [Thi equtin i exct nly if the bridge i initilly blnced] R R δ R R R ( 2 3,initil 3 )( 1 4 ) We implify the denmintr by recgnizing, pinted ut previuly, tht the chnge in reitnce f δ R3 R1 trin gge i very mll; in ther wrd, δr 3 /R 3,initil << 1 Thi yield V V R R R R ( 2 3,initil )( 1 4 ) We pply the reltinhip derived erlier fr chnge in reitnce f trin gge functin f xil trin, ( ) 2 V 1 R2 R3,initil reitnce, nd trin gge fctr, nmely, δ R3 = R3,initilS After me lgebr, V S R R V 1 S urthermre, ince R 2 = R 3,initil (eg, bth re 120 Ω), thi reduce t 4 r V V V S 4 2 3,initil The ignificnce f thi reult i thi: r cntnt upply vltge V nd cntnt trin gge fctr S, xil trin t the lctin f the trin gge i liner functin f the utput vltge frm the Whettne bridge circuit Even mre ignificntly: r knwn vlue f S nd V, the ctul vlue f the trin cn be clculted frm the bve equtin fter meurement f utput vltge Exmple: Given: A tndrd trin gge i ued in qurter bridge circuit t meure the trin f bem in tenin The trin gge fctr i S = 20, nd the upply vltge t the Whettne bridge i V = 500 V The bridge i blnced when n ld i pplied Aume ll reitr re equl when the trin gge circuit i initilly blnced with n ld r certin nn-zer ld, the meured utput vltge i = 113 mv T d: Clculte the xil trin n the bem Slutin: We pply the bve equtin fr xil trin fr qurter bridge circuit, yielding V mv 1 1 V 4 = 4 = V S 500 V mv Since trin i uch mll number, it i cmmn t reprt trin in unit f micrtrin (μtrin), defined the trin time 10 6 Nte tht trin i dimeninle, micrtrin i dimeninle unit The unit cnverin between trin nd micrtrin, expreed dimeninle rti, i (10 6 micrtrin/trin) Thu, 6 10 μtrin Bem = = 452 μtrin trin Rer inlly, keeping t tw ignificnt digit (ince S i given t nly tw digit), trin gge = 450 μtrin rnt Dicuin: It i l crrect t give the finl nwer = trin Hlf bridge circuit gge Suppe we munt tw ctive trin gge n the bem, ne t the frnt nd ne t the bck ketched t the right Al uppe tht bth trin gge re put int the Whettne bridge circuit, hwn in the circuit digrm belw, nting tht reitr R 1 nd R 3 hve been

5 Stre, Strin, nd Strin Gge, Pge 5 replced by the tw trin gge Since hlf f the fur vilble reitr in the bridge re trin gge, thi i clled hlf bridge circuit After me lgebr, uming tht bth trin gge reitnce chnge identiclly the trin i pplied, it ( ) 2 V 1 R2 R3,initil cn be hwn tht V 2S R R 2 3,initil urthermre, ince R 2 = R 3,initil = 120 Ω, the bve equtin reduce t R 1 = trin gge V 1 S 2 r V V R 1 R 2 V S 2 Cmpred t the qurter bridge circuit, the hlf bridge circuit yield V = upply twice the utput vltge fr given trin We y tht the enitivity f vltge the circuit h imprved by fctr f tw Yu might k why R 1 (rther thn R 2 r R 4 ) w chen the reitr t replce with the ecnd trin gge It turn ut tht R 1 i ued fr the ecnd trin gge if it trin i f the me ign tht f R 3 R 3 = trin gge T prve the bve ttement, uppe ll fur reitr re trin gge with initil vlue R 1,initil, R 2,initil, etc The crrepnding chnge in reitnce due t pplied trin re δr 1, δr 2, etc It cn be hwn (vi pplictin f Ohm lw, nd neglecting higher-rder term previuly) tht V R 2,initilR 3,initil δr1 δr2 δ R3 δr 4 the utput vltge vrie 2 V R R R1,initil R2,initil R [Thi equtin i 3,initil R4,initil ( 2,initil 3,initil ) pprximte ume initilly blnced bridge nd mll chnge in reitnce] A cn be een, the term with δr 1 nd δr 3 re f pitive ign, nd therefre cntribute t pitive utput vltge when the pplied trin i pitive (trin gge in tenin) Hwever, the term with δr 2 nd δr 4 re f negtive ign, nd therefre cntribute t negtive utput vltge when the pplied trin i pitive (trin gge in tenin) In the bve bem exmple, in which bth trin gge meure the me trin, it i pprprite t che R 1 fr the ecnd trin gge If R 2 r R 4 hd been chen inted, the utput vltge wuld nt chnge t ll trin i increed, becue f the ign in the bve equtin (The chnge in reitnce f the tw trin gge wuld cncel ech ther ut!) Clmp Exmple cntilever bem experiment A n exmple, cnider imple lb experiment A cntilevered bem i clmped t the lb bench, Bench nd weight i pplied t the end f the bem ketched t the right A trin gge i ttched n the tp urfce f the bem, nd nther i ttched t the bttm urfce, hwn Cntilevered bem Strin gge (n tp nd bttm) A the bem i trined due t the pplied frce, the tp trin gge i tretched (pitive xil trin), but the bttm trin gge i cmpreed (negtive xil trin) If the bem i ymmetric in cr ectin, nd if the tw trin gge re identicl, the tw trin gge hve pprximtely the me mgnitude f chnge in reitnce, but ppite ign, ie, δr bttm = δr tp In thi ce, if R 1 nd R 3 were chen fr the tw trin gge in the bridge circuit, the Whettne bridge wuld remin blnced fr ny pplied ld, ince δr 1 nd δr 3 wuld cncel ech ther ut In thi exmple, the hlf bridge circuit huld be cntructed with pir f reitr tht hve ppite ign in the bve generl equtin the chice re R 1 nd R 2, R 1 nd R 4, R 2 nd R 3, r R 3 nd R 4 the tw reitr t be ubtituted by the trin gge An exmple circuit fr thi imple experiment ue R 3 fr the tp trin gge nd R 4 fr the bttm trin gge, with the Whettne bridge circuit wired ketched t the right V = upply vltge R 4 = bttm trin gge R 3 = tp trin gge

6 V 1 S Circuit nlyi fr thi ce yield 2 r V V V S 2 Stre, Strin, nd Strin Gge, Pge 6 Cmpred t the qurter bridge circuit, the vltge utput f thi hlf bridge circuit (with tw ctive trin gge) i twice tht f the qurter bridge circuit (with nly ne ctive trin gge), ll ele being equl In generl, fr ny ytem, enitivity i defined the rti f utput t input In thi ce, the utput i the vltge, nd the input i the xil trin being meured Thu, we cnclude: The enitivity f hlf bridge Whettne bridge R circuit i twice tht f qurter bridge Whettne bridge circuit 1 R 2 V = upply vltge ull bridge circuit If we ubtitute trin gge fr ll fur reitr in Whettne bridge, the reult i clled full bridge circuit, ketched t the right Wrning: Yu need t be very creful with the ign when wiring full bridge circuit! If the wiring i dne prperly (eg, R 1 nd R 3 hve pitive trin, while R 2 nd R 4 hve negtive trin), the V 1 enitivity f the full bridge circuit i fur time tht f qurter bridge circuit, r V SV V S In generl, we define n the number f ctive gge in the Whettne bridge: n = 1 fr qurter bridge n = 2 fr hlf bridge n = 4 fr full bridge 4V 1 n Then the trin cn be generlized t r V SV nv S 4 One cutinry nte: In derivtin f the bve equtin, it i umed tht pitive trin gge (R 1 nd R 3 ) re chen fr pitive trin (tenin), nd negtive trin gge (R 2 nd R 4 ) re chen fr negtive trin (cmprein) If inted we were t wire the circuit uch tht the pitive gge re in cmprein nd the negtive gge re in tenin, negtive ign wuld pper in the bve equtin On finl nte, it i nt lwy necery t initilly blnce the bridge In ther wrd, uppe there i me initil nn-zer vlue f bridge utput vltge, nmely,reference 0 Thi vltge repreent the reference utput vltge t me initil cnditin f the experiment, which my nt necerily even be zer trin We cn till clculte the trin by uing the utput vltge difference rther thn the utput vltge itelf, 4( V V,reference ) 1 n V S r n V V,reference SV 4 In the lb, we ue vltmeter with REL buttn, which tnd fr reltive vltge [On me vltmeter, the REL buttn i indicted by tringulr ymbl Δ inted Under cnditin f zer trin with lightly unblnced bridge, the reference utput vltge i nt zer (,reference 0) Hwever, puhing the REL buttn cue the vltmeter t red ll ubequent vltge reltive t,reference In ther wrd, the vltmeter red,reference inted f itelf