Understanding Instrument Compliance Correction in Oscillation

Transcription

1 Understndin Instruent Coplince Correction in Oscilltion by A.J. Frnc, TA Instruents

2 2 Instruent Rdil Coplince Correction durin Dynic echnicl Testin Coplex odulus [P] Fiure 1: Poly-isobutylene esured in oscilltion in prllel plte eoetry t 20 o C Scope es. γ with 8 pltes es. γ with 25 pltes with 8 pltes with 25 pltes Frequency ω [rd/s] Liner viscoelstic esureents re often ten for rnted. ny users re not wre tht systetic errors such s instruent coplince y sinificntly corrupt the test results, especilly t hih frequency when the sple becoes stiff in coprison to the instruent itself. Instruent coplince effects cn be deonstrted, when runnin hih odulus terils in different size plte eoetries. Fiure 1 shows the coplex odulus of PIB esured t -20 C usin 8 nd 25 pltes (1). With incresin frequency, the coplex odulus of the test run with 25 plte eoetry devites sinificntly fro the result obtined with the 8 plte eoetry. In the se rph, the esured sple defortion is plotted versus frequency. While the strin plitude for the test perfored with 8 prllel pltes is virtully constnt nd 2 1 Strin Aplitude γ [%] close to the 2% cond vlue, does the ctul sple strin for the tests perfored with 25 pltes, decrese with frequency. This shows tht for these tests, the defortion plitude pplied by the ctutor is not fully trnsferred onto the sple. The sller the sple strin in reltion to the cond strin, the hiher the coplince effect of the instruent. Wht is instruent coplince? The true sher strin pplied to the sple is lwys lower thn the cond (otor) strin becuse the test fixtures nd the torque trnsducer re lso defored by the stress, required to sher the sple. If the sple/ eoetry confiurtion is stiff copred to the instruent, instruent coplince effects becoe sinificnt nd need to be corrected (2). Fiure 2 exhibits the typicl confiurtion for rheoeter with seprte torque trnsducer. Since the torque esureent syste hs finite stiffness, the nulr displceent ϕ T (t) is not zero. Siilr does the test fixture itself defor slihtly by the ount ϕ (t), due to the pplied torque. The finl sple defortion is ϕ s (t)=ϕ (t)-(ϕ T (t)+ ϕ (t)) with the test fixture coplince defined s C = ϕ (t)/ (t) nd the trnsducer coplince C T =ϕ T (t)/ (t). Both coplince contributions becoe iportnt when the sple is stiff copred to the instruent itself. APN013e

3 Understndin Instruent Coplince Correction Stiffness S T ϕ T (t) ϕ (t) es. torque: T (t) = (t) Since the test fixtures re usully de of steel nd defor very little under the iposed stress, they behve purely elstic durin the esureent nd C is constnt, rel nuber. The FRT trnsducer in the ARES hs response tie controlled by the servo loop nd therefore the coplince chnes with frequency nd is coplex nuber. Fiure 2: Scheticl representtion of rheoeter with seprte torque trnsducer ϕ (t) Note tht the FRT lso hs n xil coplince. The xil coplince does only rinlly ffect the odulus, but hs sinificnt influence on the trnsient norl force esureent. D = 1 C ϕ s = D ϕ Tble 1: Coplince correction for the dynic oduli, nd loss tnδ (3) Fiure 3: Vector representtion of the tool coplince C " 1 C = D " " = D tβ tδ = C t (1 β ) β 2 ϕ + C δ " 2 ϕ s =ϕ -ϕ " ϕ (1) (2) (3) For both rheoeters, AR nd ARES coplince corrections re necessry, when stiff sples re esured. Since the AR hs no trnsducer, only the test fixture coplince needs to be corrected. Test fixture coplince in AR & ARES rheoeters For the AR rheoeter nd the ARES, ssuin qusi-infinite stiffness for the trnsducer thus trnsducer deflection is neliible, the nulr displceent of the otor nd the sple torque cn be represented s vectors in the coplex plne s shown in fiure 3. The totl otor nulr displceent isϕ, the test fixture displceent due to the tool coplince C isϕ. The true nulr displceent for the sple ϕ s is the vector su. ϕ = ϕ ϕ s (1) Since C is rel nuber,ϕ is in phse with the sple torque nd not the nulr displceent ϕ. Nov-06 V1

4 4 try constnt. Fiure 4 shows the rel nd iinry torque contributions of the pprent odulus. The true dynic odulus is iven by: = ( / ) ϕ s (3) Fiure 4: Decoposition of the torque ) in reference to the otor strin vector nd b) in reference to the sple strin vector APN013e δ Fiure 5: Vector representtion of the FRT trnsducer coplince β ϕ Since ϕ = C = S δ ϕ s ϕ = ( ( ϕ = C / ) ϕ / (2) β ϕ T ϕ s = ϕ ϕ T ϕ ) ϕ S is the pprent stiffness nd the pprent dynic odulus; the eoe When insertin eqs. (2) nd (3) into (1), the true odulus cn be written s: = (1 C ( / (4) The equtions for the true store nd loss odulus re clculted fro = +i nd shown in tble 1. The rel nd iinry torque contribution of the true odulus re shown in fiure 4. It should be noted, tht the tool coplince, lso rel nuber does ffect the odulus nd the phse of the sple dynic odulus. Trnsducer coplince for the ARES rheoeter The FRT Force reblnce Trnsducer uses servo control to drive the upper plte bc to its zero position, when torque is pplied (4). The FRT trnsducer therefore cn be considered to be qusi-infinite stiff. However durin hih frequency testin of stiff sples, the servo will not correct instntneously (5), the coplince of the trnsducer increses nd cnnot be nelected nyore. For the se of siplicity, the test fixture coplince is oitted in the followin. The true sple defortion cn be represented by the difference of the otor nd trnsducer nulr displceent (Fiure 5): ϕ = ϕ s ϕ T ) (5)

5 Understndin Instruent Coplince Correction Torque Anulr displceent Fiure 6: Rel tie coplince correction in the tie doin Torque S [-c] Fiure 7: 1KFRTN1 trnsducer coplince s function of the test frequency β ϕ ϕ F ϕ s Tie δ withϕ T =C T ; ϕ T is the trnsducer nulr displceent. In contrst to the tool coplince C, is the trnsducer coplince C T coplex nuber nd not in phse with the torque nyore. Since the phse offset β is not nown nd vries with frequency nd sple stiffness, it is preferble to deterine the sple defortion ϕ s directly in the tie doin by subtrctin the trnsducer ϕ T (t) nulr displceents fro the rw otor ϕ (t) displceent Trnsducer displceent X-ducer coplince rd/-c s c coplince only function of frequency Frequency ω [rd/s] 6cP 500% 1000cP 50% 1000cP 10% X/ducer displceent [rd], Coplince [rd/ c] (Fiure 6). This is referred to s rel tie coplince correction. The dvnte of this pproch is tht the instruent coplince is eliinted durin the rw dt splin. Since it is virtully ipossible to esure the defortion of the sple directly, the correction of the test fixture coplince hs to be ipleented s discussed in the previous section. The coplince of the FRT trnsducer s function of frequency is chrcteristic of the trnsducer itself. Fiure 7 shows the nulr displceent ϕ T of the 1KFRTN1 trnsducer s function of frequency for 3 different test scenrios. The trnsducer displceent scles with the esured torque nd bottos out round 20 nrd. This is the nulr position resolution of the servo encoder. The trnsducer coplince clculted fro the trnsducer nulr displceent nd the torque, is only function of the oscilltion frequency nd increses linerly with frequency nd levels off round 100 rd/s. At 8 rd/s, the coplince of the 1KFRTN1 trnsducer is pproxitely 10 rd/n. Correction of test fixture coplince The test fixture coplince correction fctor cn be defined s the rtio of the pprent sple stiffness / nd the tool stiffness 1/C. If ccurtely nown, the tool coplince cn be redily corrected with the equtions in tble 1. Usully the coplince correction is sll, but cn becoe sinificnt when the product of sple stiffness nd test fixture coplince is lre. In fiure 8 the loss odulus is severely underestited if the Nov-06 V1

6 6 APN013e odulus, [P] Fiure 8: nd of polyisobutylene esured t 20 o C before coplince correction odulus, [P] Fiure 9: nd of polyisobutylen esured t 20 o C fter coplince correction Frequency ω [rd/s].(1-k θ.(b. )) = with 8 pltes with 8 pltes with 25 pltes with 25 pltes coplince effects re not ten into ccount. The error for the store odulus is reltively sll since the nuertor of eq.1 in tble 1 is lwys sller thn nd the denointor D is sller thn 1. Note, tht the correction for the loss fctor tnδ is non liner in respect to the pprent loss fctor. Since the sple stiffness depends on the dynic odulus nd the sple eoetry, the eoetry cn be chosen, such s to eliinte with 8 pltes with 8 pltes Corrected with 25 pltes Corrected with 25 pltes Frequency ω [rd/s]) instruent coplince. If the sple odulus is below 0.4 P, the coplince correction for the 1KFRTN1 trnsducer, when usin 25 pltes cn be oitted. The plteu odulus en for ost polyers however is hiher. With 8 pltes, the odulus liit is bout 100 ties hiher i.e. 40 P, becuse the sple stiffness is proportionl to the 4 th power of the plte rdius: πr S s = 2 h (6) As such, the dt obtined with 8 pltes in fiure 1, represent the true store nd loss odulus nd cn be used to clibrte the coplince of the 25 tool fixture. With eqs.1&2 fro tble 1, the tool coplince C cn esily be clculted usin the pprent oduli obtined with the 25 pltes nd the dt obtined with the 8 pltes s true oduli. A tool coplince of 7 rd/n is obtined for the 25 plte test fixture. The corrected 25 plte dt tch very well the 8 plte dt, shown in fiure 9. The correction of the 8 plte dt shows no chne which iplies the vlidtion of the FRT trnsducer correction. Conclusion 4 Nelectin instruent coplince cn led to sinificnt errors in liner viscoelstic esureents in the plteu nd trnsition reion. Without tool coplince correction, the upper liit for the coplex odulus is 0.4 P when usin 25 pltes, 1 p for the ARES. Tool coplince is coon proble for ll rheoeters, with or without seprte torque trnsducer, when stiff sples re bein tested. Fiure 10 shows the

7 Understndin Instruent Coplince Correction conference New techniques in Experientl Rheoloy, Septeber 1985, Redin, UK cy,.e.; Hlley, P.J.; J.Rheol.35, 1609 (1991) 80 Rtio of Strins [%] esured/conded Strin FRT Corrected/esured Strin Tools Frequency ω [rd/s] Fiure 10: Rtio of esured/ cond strin nd corrected/ esured strin rtio of esured nd cond strin (FRT coplince) nd the rtio of corrected nd esured strin (Tool coplince) s function of frequency for the PIB t -20 C usin 25 pltes, 1 p. The rtio representin the FRT coplince is only slihtly hiher thn the rtio for the tool coplince. This ens for the ARES rheoeter, tht when coplince effects re detected nd the esured strin is sinificnt sller thn the cond strin, the tool coplince correction needs to be ctive in order to obtin the correct dynic oduli. The Orchestrtor softwre llows the correction of test fixture coplince in ddition to trnsducer coplince fro softwre version 7.00 on. References 1. Liu, C.Y.; Billy, C.; Yo,.L.; rritno, R.; Frnc, A. presented t the SOR in October 2006, Portlnd, ine 2. ottlieb,.; coso, C.W.; Rheol. Act 21, 90 (1982) 3. Sternstein, S.S.; Adv. Che. Series 203, 123 (1983) 4. Frnc, A.; presented t the Nov-06 V1

8 Keywords: rdil coplince, coplince correction, FRT trnsducer, plteu odulus, coplince Copyriht TA Instruents