Errors Due to Misalignment of Strain Gages

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1 VISHAY MICO-MEASUEMENTS Strai Gages ad Istrumets Errors Due to Misaligmet of Strai Gages Sigle Gage i a Uiform Biaxial Strai Field Whe a gage is boded to a test surface at a small agular error with resect to the iteded axis of strai measuremet, the idicated strai will also be i error due to the gage misaligmet. I geeral, for a sigle gage i a uiform biaxial strai field, the magitude of the misaligmet error deeds uo three factors (igorig trasverse sesitivity): 1. The ratio of the algebraic maximum to the algebraic miimum ricial strai, /.. The agle betwee the maximum ricial strai axis ad the iteded axis of strai measuremets. 3. The agular moutig error,, betwee the gage axis after bodig ad the iteded axis of strai measuremet. These uatities are defied i Figures 1 ad for the articular but commo case of the uiaxial stress field. Figure 1 is a olar diagram of strai at the oit i uestio, ad Figure gives the cocetric Mohr s circles for stress ad strai for the same oit. I Figure 1, the distace to the boudary of the diagram alog ay radial lie is roortioal to the ormal strai alog the same lie. The small lobes alog the Y axis i the diagram rereset the egative Poisso strai for this case. It ca be see ualitatively from Figure 1 that whe is 0 or 90, a small agular misaligmet of the gage will roduce a very small error i the strai idicatio, sice the olar strai diagram is relatively flat ad assig through zerosloe at these oits. However, for agles betwee 0 ad 90, Figure 1 shows that the error i idicated strai due to a small agular misaligmet ca be surrisigly large because the sloe of the olar strai diagram is very stee i these regios. More secifically, it ca be oticed from Figure (o age 108), whe = 45, or = 90, that the same small agular misaligmet will roduce the maximum error i idicated strai, sice is chagig most raidly with agle at this oit. The same result could be obtaied by writig the aalytical exressio for the olar strai diagram, ad settig the secod derivative eual to zero to solve for the agle at which the maximum sloe occurs. I fact, the geeral statemet ca be made that i ay uiform biaxial strai field the error due to gage misaligmet is always greatest whe measurig strai at 45 to a ricial axis, ad is always least whe measurig the ricial strais.* The error i strai idicatio due to agular misaligmet * The excetio to this statemet is the sigular case whe, as o the surface of a ressurized shere. I this istace, the strai is everywhere the same ad ideedet of directios. Figure 1 Polar Strai distributio corresodig to uiaxial stress, illustratig the error i idicated strai whe a gage is misaliged by ± from the iteded agle. 107

2 Vishay Micro-Measuremets Errors Due to Misaligmet of Strai Gages cos cos 1 cos 1 (4) STAIN CICLE STESS CICLE However, from Euatio (3) it ca be see that becomes umeaigfully large for small values of, ad ifiite whe vaishes. I order to better illustrate the order of magitude of the error due to gage misaligmet, Euatio () will be evaluated for a more-or-less tyical case. I a uiaxial stress field, =. Ad, for steel, = Assume = 0μ The, = 85μ Ad, = 64.5 [cos(± ) cos] (5) Figure Mohr s circles of stress ad strai for uiaxial stress, a alterative reresetatio of the misaligmet errors. of the gage ca be exressed as follows: Or, 108 (1) = Error, μ = Strai alog axis of iteded measuremet at agle from ricial axis, μ ( ± ) = Strai alog gage axis with agular moutig error of ±, μ cos cos (), = Maximum ad miimum ricial strais, resectively The error ca also be exressed as a ercetage of the iteded strai measuremet, : (3) Figure 3 Error i idicated strai due to gage misaligmet for the secial case of a uiaxial stress field i steel. = 0μ = 85μ

3 Errors Due to Misaligmet of Strai Gages Vishay Micro-Measuremets Euatio (5) is lotted i Figure 3 over a rage of from 0 to 90, ad over a rage of moutig errors from 1 to 10. I order to correct for a kow misaligmet by readig the value of from Figure 3, it is oly ecessary to solve Euatio (1) for ad substitute the value of ; icludig the sig as give by Figure 3. This figure is give oly as a examle, ad alies oly to the case i which = 0.85 (uiaxial stress i steel). Euatio () ca be used to develo similar error curves for ay biaxial strai state. Two-Gage ectagular osette While the above aalysis of the errors due to misaligmet of a sigle gage may hel i uderstadig the ature of such errors, the 90-degree, two-gage rosette is of cosiderably greater ractical iterest. A two-gage rectagular rosette is ordiarily used by stress aalysts for the urose of determiig the ricial stresses whe the directio of the ricial axes are kow from other sources. I this case, the rosette should be boded i lace with the gage axes coicidet with the ricial axes. Whether there is a error i orietatio of the rosette with resect to the ricial axes, or i the locatios of the ricial axes themselves, there will be a corresodig error i the ricial stresses as calculated from the strai readigs. I Figure 4, a geeral biaxial strai field is show, with the axes of a two-gage rosette, misaliged by the agle, suerimosed. The ercetage errors i the ricial stresses ad maximum shear stress due to the misaligmet are: ˆ x (6) 1 v 1 1cos x (7) v ˆ x (8) 1 1v 1 cos x (9) 1v ˆ x (10) 1 cos x (11) ˆ, ˆ, ˆ are the ricial stresses ad maximum shear stress iferred from the idicated strais whe the rosette is misaliged by the agle. = /, the ratio of the algebraic maximum to the algebraic miimum ricial strai, as before. Figure 4 Biaxial strai field with rosette axes misaliged by the agle from the ricial axes. 109

4 Vishay Micro-Measuremets Errors Due to Misaligmet of Strai Gages Whe the ricial strai ratio is relaced by the ricial stress ratio, 8 1 = (1) 7 6 /=.0 Or, P 1 1 = 1 (13) 1 - cos x (14) 1 - cos x (15) Euatios (11), (14), ad (15) will ow be alied to a examle i order to demostrate the magitudes of the errors ecoutered. Cosider first a thi-walled cylidrical ressure vessel. I this case, the hoo stress or circumferetial stress is twice the logitudial stress, ad of the same sig. Thus, Ad Euatios (11), (14), ad (15) become: P 1 1 cos x (11a) 1/ 4 1 cos x (14a) 1/ 1 cos x (15a) STESS EO % Figure 5 Percetage erros i ricial stresses ad maximum shear stress for a biaxial stress field with / =.0 of from Figure 5, icludig the sig. That is, ˆ OSETTE MOUNTING EO (16) 1 Euatios (11a), (14a), ad (15a) are lotted i Figure 5. From the figure, it ca be see that the errors itroduced by rosette misaligmet i this istace are uite small. For examle, with a 5 moutig error,,, ad are i error by oly 1.5%, 0.38%, ad 0.75%, resectively. I order to correct for a kow misaligmet by readig the value of from Figure 5, or ay similar grah derived from the basic error euatios [Euatios (7), (9), (11), (14), (15)], it is oly ecessary to solve Euatios (6), (8), ad (10) for,, ad, resectively, ad substitute the value 110 ˆ 1 (17) ˆ (18) M 1 AX

5 Errors Due to Misaligmet of Strai Gages Vishay Micro-Measuremets ˆ = maximum ricial stress as calculated from gage readigs ˆ = miimum ricial stress as calculated from gage readigs ˆ = maximum shear stress as calculated from iterally ressurized cylider with a axial comressive load alied exterally to the eds. If, for examle, the load were 0.8 r, where r is the iside radius of the cylider, ad is the iteral ressure, the ricial stress ratio would become, =10 ˆ ˆ P ˆ Euatios (14) ad (15) become: = 0.45(1 cos) x = 4.5(1 cos) x (14b) (15b) While the errors i the above case were very small, this is ot true for stress fields ivolvig extremes of. I geeral, becomes very large for <<1.0, as does for >>1.0. The error i shear stress is ideedet of the stress state. The above geeralities ca be demostrated by extedig the revious case of the ressurized cylider. Cosider a For this case, a 5 error i moutig the rosette roduces a 0.68% error i ad a 6.75% error i. The errors defied ad evaluated i the foregoig occur, i each case, due to misaligmet of a sigle strai gage or of a etire rosette. The effect of misaligmet amog the idividual gages withi a rosette is the subject of a searate study. 111

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a)

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a) alacig NOT COMPLETE Rotatig Compoets Examples of rotatig compoets i a mechaism or a machie. Figure 1: Examples of rotatig compoets: camshaft; crakshaft Sigle-Plae (Static) alace Cosider a rotatig shaft