Lesson 3. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)

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1 Lesso 3 Thermomechacal Measuremets for Eergy Systems (MENR) Measuremets for Mechacal Systems ad Producto (MMER) A.Y Zaccara (Ro ) Del Prete

2 ACCURACY (Precso) Atttude of a measurg strumet to gve the «true value» a v of a physcal quatty! Egeers ad techcas always wshed the strumets eg ale to perform precse measuremets (wth hgh accuracy). However t s uavodale to make errors durg measuremet processes, so we must defe a coveet way to epress how good or how ad our measuremets a are! A frst smple way could e epressg the dstace etwee the measuremet a ad the true value a v ε = a - a v whch s the asolute error of the measuremet. However, there s a more coveet way egeerg to epress the accuracy of a measuremet: a a v % 00 a whch s the relatve error of the measuremet (epressed %), ad gves a mmedate ad tutve dea of the measuremet accuracy! If we kew the true value a v, we were ascally all set ad we could go ahead to aalyze the et characterstcs. Ufortuately, the true value of a physcal quatty s geerally ukow ad ukowale!

3 To get EXACT INFORMATION aout the measurad we should have avalale a eemplar measuremet method! A measuremet s therefore a techcal procedure that strves to get as close as possle to the truth of the atural world ad/or of the techology! But, t ca ever succeed We ca oly mmze the uavodale ucertates ε a : A a a U tryg to get ε a 0 There are two ma types of error that ca e doe durg a measuremet:. Systematc (BIAS) errors. Radom (PRECISION) errors Measuremet accuracy results from a comato of these two errors! Bas errors are ofte macroscopc errors ecause they have a «techcal org» (strumet malfucto, ad codtos of use, strog eteral ose that the strumet ca ot reect, strumet out of calrato). Bas errors CAN e dscovered y a calrato procedure ad, oce hghlghted, they MUST e elmated!

4 Radom errors are geerally small scatter errors, that ca ot e avoded durg a measuremet ad for whch s mpossle to fd a eact reaso! Radom errors ca e studed after the as error has ee corrected for, or (etter) elmated. Therefore the sum of all radom errors s, sometmes, also called resdual error. The resdual error always depeds o ukow small radom reasos, therefore t s much etter referred wth the word ucertaty! Accordg to the way we evaluatg t, there are oly two dfferet types of UNCERTAINTY, whch are classfed y the teratoal «Gude to the epresso of ucertaty measuremet» ed. JCGM 00: 008 (GUM) Type A ucertates: evaluated y a statstcal aalyss of seres of oservatos (measuremets). Type B ucertates: evaluated y a pool of comparatvely relale formato

5 The ucertaty of the result of a measuremet reflects the lack of eact kowledge of the measurad value. The result of a measuremet, after correcto for recogzed systematc effects, s stll oly a estmate of the value of the measurad ecause of the ucertaty arsg from radom effects ad from mperfect correcto of the result for systematc effects. I practce, there are may possle sources of ucertaty a measuremet, cludg: a) complete defto of the measurad; ) mperfect realzato of the defto of the measurad; c) o-represetatve samplg the sample measured may ot represet the defed measurad; d) adequate kowledge of the effects of evrometal codtos o the measuremet or mperfect measuremet of evrometal codtos; e) persoal as readg aalogue strumets; f) fte strumet resoluto or dscrmato threshold; g) eact values of measuremet stadards ad referece materals; h) eact values of costats ad other parameters otaed from eteral sources ad used the datareducto algorthm; ) appromatos ad assumptos corporated the measuremet method ad procedure; ) varatos repeated oservatos of the measurad uder apparetly detcal codtos. These sources are ot ecessarly depedet, ad some of sources a) to ) may cotrute to source ). Of course, a urecogzed systematc effect caot e take to accout the evaluato of the ucertaty of the result of a measuremet ut cotrutes to ts error!

6 For smplcty, we start the study of ucertaty estmato wth the type B ucertaty: For a estmate of a put quatty X that has ot ee otaed from repeated oservatos, the assocated estmated varace u ( ) or the stadard ucertaty u( ) s evaluated y scetfc udgemet ased o all of the avalale formato o the possle varalty of X. The pool of formato may clude : prevous measuremet data; eperece wth or geeral kowledge of the ehavour ad propertes of relevat materals ad strumets; maufacturer's specfcatos; data provded calrato ad other certfcates; ucertates assged to referece data take from hadooks. Eamples of typcal errors mechacal measuremets that ca cotrute to type B ucertates : p = d tgφ Readg error for aalogc strumets

7 Molty error for mechacal strumets Hysteress error for mechacal strumets Fdelty error: due to the dsturg ad ose factors as temperature, humdty, electromagetc felds, mechacal vratos, atmospherc pressure, o-ertal referece frame... It quatfes the sestvty of the strumet to eteral terfereces, ad t has to e assessed usg multple repeated measuremets over tme, y chagg the posto ad the codtos of use of the strumet whle keepg the put varale strctly costat Zero shft error: due to loss of calrato of the mechacal compoets (recordg sprgs) or electrcal compoets (trmmer), or to the agg of the electroc compoets (resstors ad capactors)

8 Calrato ad stadard refereces errors: due to the errors that are doe whle carryg out a calrato procedure or due to the heret ucertates of the referece stadards used for the calrato! ±α are the stadard refereces errors ±β are the errors doe whle tracg the calrato curve The total error s a quadrature sum: t represeted wth the ellpsod show the fgure. ad ca e A older way of epressg the type B ucertates [ %] was wth the precso class parameter: where ε s each error or ucertaty source oe s ale to solate ad estmate full _ scale Eample: a class 0.5 dyamometer wth a full scale of 00 N makes a asolute error of 0.5N (0.5% relatve error) whe measurg 00N the class 0.5 dyamometer makes the same asolute error of 0.5N whe measurg 5 N, whch ow s a relatve error of 0.5N / 5N = 0. = 0 %. A advce: ever use a strumet for whch a precso class s declared at the egg of the spa!

9 We ow proceed to epla how to estmate the type A ucertaty : Type A ucertaty estmato has a completely dfferet approach t does NOT eve care aout detfyg error sources! The method s ased o a statstcal data post processg ad, therefore, we eed a suffcet umer of repeated measuremets wth the measurad X held strctly costat! X IN INSTRUMENT s the th measuremet of the measurad X The fal goal s NOT eve to fd the true value v ut a certa rage of values wth whch the true value could reasoaly e foud! Ths rage could e wrtte wth where s the est represetato of the true value ( est) ad δ s a wdth parameter that mght e represetatve of the ucertaty! measuremet ( ) Because our measuremets, wll e very smlar each other ut ot equal etwee them, whch of them could possly e the est represetato of our measuremets?

10 The mea value of our data: the measuremets! So far, there s NO theoretcal ustfcato for our choce; ts oly a very reasoale choce, so reasoale that I wll try to apply t also to fd a quattatve epresso for the wdth parameter δ : d It s the devato of the geerc measuremet from the mea Ca we calculate the mea value for ths devato? d d 0 ) ( NO! ecause the devatos are equdstat from the mea value! 0 d Therefore, the Stadard Devato (of the populato) has ee proposed! or, wth oly a few measuremet pots, the Stadard Devato (of the sample)! whch produces σ = 0/0 stead of σ = 0 whe

11 measuremets? ( ) σ Now we ca epress our measuremet wth the est represetatve ad wth the wdth parameter σ! We have ow to ask ourselves: How much do we trust these choces? Is every measuremet fallg sde the rage we ust determed? Or, etter, what CONFIDENCE do we place ths rage of values? To aswer these questo we have frst to costruct the fgure o the left, where we dvded the rage where the measuremets fell k small tervals Δ k wth at least oe measuremet sde each terval Δ k. The more measuremets we have avalale the more (ad smaller) tervals we ca choose. The heght f k of each rectagle s proportoal to the umer of measuremets sde each terval Δ k. Ths fgure s called the Frequecy Hstogram.

12 FREQUENCY HISTOGRAM e k umer of measuremets (oservatos) that fall to the terval Δ k f k k frequecy of the oservatos (measuremets) that fall to the Δ k terval Ovously: k ad f k k The frequecy hstogram s ormalzed y defto! f The area represet the measuremet fracto that falls k k wth the Δ k terval. k Note that t s possle to epress the mea value of all the measuremets as: where k s the est represetato of sde the Δ k terval k k k k k f k We ow thk deally to rg the umer of measuremets ꝏ ths stuato we ca thk of a eormous umer of tervals Δ k whch are also gog to e smaller ad smaller The stepwse curve of the frequecy hstogram ecomes a smooth lmt dstruto curve f()

13 Where: ꝏ f()d s the measuremet fracto that falls wth the ftesmal d terval Δ k d f k f() a f ( ) d s the measuremet fracto that falls wth the fte terval (-a) Because t s f ( ) d the lmt dstruto curve s also ormalzed! f ( ) p( Oly for ꝏ ) But, ow, whch curve descres at est the lmt dstruto curve f()? Frequecy (epermetal results) Proalty (theoretcal model ) It s the well-kow Normal dstruto curve or Gaussa curve or ell curve f X ( ) e

14 f X ( ) e where X s the mea or the true value of ( fact we have ow ꝏ measuremets) ad σ s a wdth parameter of the fucto Applyg the ormalzg codto: f ( ) d we get f X X, ( ) e f X, ( ) d If we calculate the mea value for the fucto f X,σ () : we got the mea! If we calculate the mea squared dfferece for f X,σ () : f X, ( ) d we got the varace! whch s the squared stadard devato X Therefore, theoretcally the mea X s the true value of the measuremet ad the wdth σ s the stadard devato, calculated for the deal case of ꝏ measuremets!!!

15 Fally, f we refer to the mathematcal model f X,σ (), we ca epress the measuremet wth : ( ) = X -σ X +σ We stll have to uderstad what cofdece we gve to the wdth parameter σ The good ews s that wth a mathematcal model we ca calculate the tegrals etwee two specfed lmts: p f X ( ) d p p, f ( d 3 3 X, ) 3 f ( d X, ) Ad they wll represet the proalty oe measuremet has (oe of the ꝏ we have at dsposal) to fall wth the rage ± σ, or ± σ, or ± 3σ aroud the true value X p 68,7% k k 95,45% k 99,73% k 3 The factor k s amed the GUM as coverage factor ad leads to the defto of epaded ucertaty

16 X Therefore, wrtg meas there s a 68.3 % proalty to fd a measuremet wth the ucertaty rage ad. I other words, we gaed a cofdece of 68.3% of fdg our measuremets wth the rage. The geometrcal represetato of the proalty p dscussed aove, s show left o the gaussa dstruto curve, also for coverage factors of ad 3, ad s the area uderlyg the curve. It s very mportat to uderstad the two followg cocepts: X s a proalty ad ca e calculated efore dog a sgle measuremet (havg the fucto)! s a statstcs ad ca calculated oly f we have data avalale (after dog the measuremets)!

17 Keep always md that the cofdece tervals ad ther proalty values have ee determed from a theoretcal model or cosderg ꝏ measuremets avalale, whch NEVER happes the real world, where we ALWAYS have oly a lmted umer of measuremets avalale! Before applyg the results of the proalty theory, we have to e reasoaly sure that the dstruto of our measuremet s actually a Gaussa oe (χ test) otherwse, we should rather try applyg other statstcal dstrutos (t Studet) Because, y creasg the umer of the measuremets the stadard devato σ does ot chage much, s a good way of epressg the ucertaty of the strumet or of the measuremet method. Oservg the shape of the measuremet dstrutos, t s therefore mmedate estalshg or comparg the accuracy of dfferet measurg strumets!! Beg X we stll have to uderstad f t s possle to get from our measuremets ay formato aout the accuracy of the measuremet???

18 To aswer to ths amtous questo we have to make a step ack ad get a few asc cocepts of the Error Propagato Theory Cosder a physcal quatty q = + y that ca e epressed y the sum of two other prmary physcal quattes: ad y y y We wsh to fd: q q q -δq q +δq ( ) q wth q y y y y y whch mght represet the upper lmt of the wdth δq whch mght represet the lower lmt of the wdth δq We mght, therefore, thk that q y however, f the errors that lead to δ ad δy are depedet, the choce q y s a overestmato of the ucertaty for q

19 +δ ( ) y +δy ( ) y q y I fact, to actually have we should always uderestmate or overestmate at the same tme the measuremet of ad y, whch would mply a uderlg low or a correlato etwee the two varales, makg them somehow depedet o each other! If ths s the case, the q y ca e a correct choce otherwse, whe the measuremets ad the errors of ad y are depedet, there s always a partal mutual deleto of the ucertates, the algerac sum of the ad y errors s the a overestmato of the q ucertates ad t s much more reasoale to add the errors for ad y quadrature : Smlarly, whe we have a measuremet epressed as a product or a quotet of two prmary quattes q y the geeral rule s to add the relatve errors quadrature : q y y q q y y

20 I geeral, whe the varale of terest q s a fucto of a measurale physcal quatty : q = q() (for e. q se) It s always possle to measure ad calculate q q wth the fucto relatoshp! But how are we gog to calculate δq? If δ s small ad due oly to radom errors, q m ad q ma are almost equdstat from q of a dstace δq, regardless of the fucto type! I ths hypothess we ca wrte: q q whch for δ small, ca also e wrtte as: q lm lm 0 0 q q dq d q q We reach the the mportat relatoshp: q dq d whch s the dervatve of the fucto q() calculated

21 To clude also the commo cases whe the fucto q() s decreasg ad has a egatve dervatve the pot : dq d 0 we should rather adust the result ad cosder the asolute value of the dervatve q dq d I geeral, whe we wsh to kow the testy of a physcal quatty q that ca e epressed wth a fucto of two or more other physcal varales q = q(,y), ad we measure these prmary varales wth ther ucertates: ad y y y ; the we ca always use the fucto relatoshp to calculate the est represetatve of q : q q, y whle for δq we mght cosder to apply the supermposto of the effects ad use the algerac sum: q q q y y

22 However, aga, f the measuremets of ad y ad ther errors are depedet, t s qute reasoale to cosder that there wll e a partal mutual deleto of the ucertates ad, to epress the geeral ucertaty δq t s much more relale to add the ucertates of ad y quadrature: y y q q q Please, ote that the wdth parameters we used for the measuremets ad y are othg more tha the Stadard Devatos prevously calculated: ad y y Everythg sad so far ca e geeralzed for the case of a physcal quatty q = q(,, ) whch s a fucto of measurale prmary depedet varales: Whch s the fudametal statstcal law of the Ucertaty Propagato ad represets also the Comed Stadard Ucertaty for ucorrelated put quattes reported the GUM JCGM 00: q q q q

23 We are ow ready to aswer our last questo: Ca we actually estmate somehow the accuracy of the measuremet tself wth oly a lmted umer N of measuremets N?? Ths goal ca e approached y estmatg how well the mea value represets the true value X or, tryg to calculate the ucertaty of whe the mea value s the est represetatve of the true value X. To do so, we start dvdg our N measuremets m groups of measuremet each : ' ''... (m) (m) (m) Who s ow the est represetatve of the m mea values?? It s the mea value of the m mea values : (m) Now we have N = m measuremets ad, for each group of m measuremets, we ca calculate the mea value =, m. If X s the true value of the m measuremets, t s also the true value for the m meas ecause they come from the same measuremets! m m the mea of the meas

24 If the m measuremets are affected oly y small radom errors, each dstruto curve for the m groups of measuremets wll e a ormal (Gaussa) dstruto curve p (). Therefore, the m mea values wll also dstrute wth a ormal (Gaussa) curve p aroud the mea of the meas. Ths happes ecause each mea value s a fucto of the measuremets : f The wdth parameter mea values wll e : for the dstruto of the m... the Stadard Devato of the mea The wdth parameters δ are here the Stadard Devatos calculated wth the measuremets of the th measuremets group. Sce are all measuremets of the same quatty doe wth the same strumet

25 for a reasoale umer of measuremets, they wll e almost cocdet etwee them :... From the defto of mea value: t s easly otaed:... Ad susttutg all these postos the ma equato of the wdth parameter t results:... Whch s the Stadard Error of the mea ad gves the ucertaty wth whch the mea represets the true value! Please, carefully ote: the Stadard Error ca e calculated oly wth the measuremets of oe sgle group ut, ecause of the way t was defed, t keeps the powerful meag of UNCERTAINTY of the MEASUREMENT!! f we make more measuemets ( g) the Stadard Error decreases whle the Stadard Devato stays aout the same! makg more acqustos makes a etter measuremet, NOT a etter strumet!

26 Dstruto curve of the m mea values Dstruto curve of the measuremets Stadard Devato epresses the strumet accuracy! Stadard Error epresses the measuremet accuracy! For the same set o measuremets, the Stadard Error s always smaller tha the Stadard Devato!!

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg