Definition of Strain. Tutorial

Transcription

1 Defit f Stra Tutrial

2 Part 1. Defit f Stra Stra is the parameter used t quantify the defrmat f an bject. In igure 1B and 1C, ppsg frces are applied t each end f a rd with an rigal length f L. The applied frces will cause the rd t defrm s that L f = L + ΔL, (1) where L f is the length f the rd when the ppsg frces are applied and Δ L = L f L. (2) Unstraed Rd Tensile Stra Cmpressive Stra L f L L L L f ΔL ΔL igure 1: Schematic drawgs f a rd with n stra (A), tensile stra (B), and cmpressive stra (C). The stra parameter ε is then defed as the rati f the rd s change length t the rd s rigal length. ΔL ε. (3) L When ΔL is psitive, the rd is undergg tensile stra, which is als referred t as psitive stra. When ΔL is negative, the rd is undergg cmpressive stra, which is als referred t as negative stra. Althugh the stra parameter is a dimensless quantity, it is ften expressed as a rati f length dimenss such as / r mm/mm. Sce mst rigid bjects d nt macrscpically cmpress r stretch when subjected t frces, the magnitude f ε is usually small, and as a result, stra is typically reprted as micrstra (με = ε x 10-6 ) D03, 02/07/2007 Page 2

3 Part 2. The Metallic Stra Gauge Metallic stra gauges are ne f many devices, alng with piez resistrs and devices based n terfermetric techniques, that have been develped t measure micrstra. Invented by Edward E. Simmns 1938, the metallic stra gauge cnsists f a fe wire r metallic fil with an electrical resistance (R ) adhered t a flat rigid substrate. R typically varies frm tens t thusands f hms and the substrate is ften referred t as the carrier. The wire r metallic fil usually has a serpente cnductg path rder t maximize its length parallel t the axis that stra is beg measured, while mimizg the surface area f the metallic stra gauge, see igure 2. Slder Tabs Axis Alng Which Stra is Measured Carrier igure 2: Schematic drawg f a metallic stra gauge The stra gauge is then bnded t the bject beg measured. Any expans r cntract f the bject alng the axis f the stra gauge will als ccur the cnductg path, and as a result, will cause a prprtal change the resistance f the stra gauge. The prprtality factr, r gauge factr (G ), fr metallic stra gauges is usually tw. ΔR ΔR R R G = ΔL ε, (4) L where ΔR is the change the resistance f the stra gauge when it is straed D03, 02/07/2007 Page 3

4 Part 3. Measurg Stra The PZS001 actuatr has an L f 20 mm. The full bridge stra gauge bnded t the surface is made frm fur dividual stra gauges, each with a G f tw, and an R f 350 Ω. r a sgle stra gauge f the type used the PZS001, a 5 nm change f length the actuatr crrespnds t a stra f 2.5x10-7. As a result, the length f the actuatr can nly be cntrlled if a change f resistance the stra gauge equal t 1.75x10-4 Ω can be measured. Tryg t measure the abslute resistance f the stra gauge t less than a mω is difficult; hwever, makg a precise measurement is nt the nly prblem. Changes the temperature f the gauge will cause significant, n the millihm scale, changes the resistance f the gauge. Bth f these issues can be addressed by placg tw active and tw dummy stra gauges, see igure 3, a Wheatstne bridge cnfigurat as shwn igure 4. A passive stra gauge is riented rder t mimize the length f cnductg wire parallel t the axis alng which the stra is beg measured. Passive Stra Gauge Active Stra Gauge Axis Alng Which rce is Applied igure 3: Passive stra gauge versus active stra gauge rientat In this rientat, the passive stra gauge is unaffected by the applied stra; hwever, the resistance f the stra gauge is still dependent n the temperature the same way as the active stra gauge. Active Gauge R 1 R 2 Passive Gauge ut Output ltage Input ltage R 3 R 4 Active Gauge igure 4: Wheatstne bridge circuit with tw active and tw passive stra gauges D03, 02/07/2007 Page 4

5 The Wheatstne bridge, r full bridge cnfigurat, f stra gauges shwn igure 4 allws fr the precise determat f the stra gauge resistance by measurg the change the utput vltage ( ut ), while a fixed put vltage ( ) is applied t the bridge. Ideally, the bridge shuld be built s that R 1 =R 3 and R 2 =R 4, which makes the ut equal t zer, dependent f the, fr an unstraed bject. R 1, R 2, R 3, and R 4 are used t symblize the unstraed resistance f each stra gauge. Sce there is a passive stra gauge each branch f the Wheatstne bridge, the ut will be dependent f temperature, but dependent n the stra experienced by the bject. In general, the rati f the ut t the can be written as: ut R + ΔR R + ΔR + R 1 4 =. (5) 1 3 R + ΔR R + ΔR + R 4 2 Nte that if ΔR=0, n stra, then ut =0 even if the values f R 1 and R 4 change due t temperature sce R 2 and R 3 will change by the same amunt sce they are dummy stra gauges. Sce each branch f the Wheatstne bridge is a vltage divider, typically, the full bridge stra gauge is built s that the unstraed resistance f each stra gauge is equal: R = R 1 = R 2 = R 3 = R 4. (6) Replacg R 1, R 2, R 3, and R 4 equat 5 with R yields: ut ΔR =. (7) 2R + ΔR Usg equat 4, ΔR equat 7 can be replaced t create equat 8: ut R G ε G ε = =. (8) 2R + R G ε 2 + G ε Slvg equat 8 fr ε yields: ut ε. (9) = G 2 ( ) ut If the is 10 vlts, the ut must be measured with micrvlt precis rder t cntrl the length f the actuatr n the nanmeter scale fr the PZS D03, 02/07/2007 Page 5

6 Part 4. Practical Cnsiderats When Measurg Stra When measurg the ut f a full bridge stra gauge n a micrvlt scale, several factrs must be cntrlled rder t get an accurate measurement f the stra. The must be a lw-nise AC r DC vltage surce. Obviusly, if the is creased, the ut will crease prprtally. Hwever, there is a practical limit t the magnitude, because as that creases, the amunt f heat generated by the circuit creases as well, which can lead t temperaturerelated errrs. Typically a f ten f vlts r less is used. Because it is difficult t build a circuit which the resistance f each stra gauge is exactly the same, an stalled full bridge stra gauge is unlikely t have a zer utput vltage readg when there is n stra. An equality the resistance f the leads, which shuld be kept as similar as pssible, t the circuit will als create a nn-zer ut the unstraed circuit. T cmpensate fr small nn-zer vltages fr unstraed circuits, an external ffset circuit, r the APT Sftware System that cmes with Thrlabs piez-electric drivers, can be used. Cmpensatg fr larger vltages with this sftware, hwever, reduces the dynamic range f the measurement due t limits the amplifier ga that can be applied t ut. The stra gauge circuit shuld be lcated as clse as pssible t the driver, because the length f the leads is prprtal t their resistance. The resistance f the leads adds t the ttal resistance f the circuit. This creases the denmatr f the right side f equat 7, which reduces ut fr a given amunt f stra. The leads t the stra gauge circuit shuld als be kept shrt t mimize the effects f transmiss le nise. A lw-pass filter can be added t the circuit t remve high-frequency nise prevalent mst wrk envirnments. The stra gauge circuit can be calibrated by mnitrg the ut fr knwn amunts f stra, r by simulatg stra. Stra can be simulated by placg a large knwn resistr parallel t each active resistr the full bridge stra gauge. The large resistr creates a knwn theretical ΔR, used t calculate a value fr ut that can be used t calibrate the measured ut D03, 02/07/2007 Page 6