The determination and analysis of the total measurement uncertainty of roundness deviation

Transkrypt

1 Scietific Jourals Maritime Uiversity of Szczeci Zeszyty Nauowe ademia Morsa w Szczeciie 010, 1(9) , 1(9) s The determiatio ad aalysis of the total measuremet ucertaity of roudess deviatio Wyzaczaie i aaliza złożoej ieewości omiarowej odchyłi orągłości Krzysztof Nozdrzyowsi Maritime Uiversity of Szczeci, Marie Egieerig aculty Deartmet of Machiery Costructio ad Oeratio asics ademia Morsa w Szczeciie, Wydział Mechaiczy, Załad Podstaw udowy i Esloatacji Maszy Szczeci, ul. Podgóra 51/5, .ozdrzyowsi@am.szczeci.l Key words: aalysis, measuremet ucertaity, roudess deviatio bstract This author resets the results of a aalysis of the effect of directly measured quatities ad their measuremet (stadard) deviatios o the value of the total measuremet ucertaity of roudess deviatio. The methods described refer to the determiatio of roudess deviatio based o the mea square elemet ad basic mathematical trasformatios ad relatioshis that rovided a basis for the aalysis. Its results are give i the form of diagrams followed by fial coclusios. Słowa luczowe: aaliza, ieewość omiarowa, odchyła orągłości bstrat W artyule zarezetowae zostały wyii aalizy wływu wielości mierzoych bezośredio oraz ich ieewości omiarowych (stadardowych) a wartość złożoej ieewości omiarowej odchyłi orągłości. Przedstawioa została metodya wyzaczaia odchyłi orągłości według elemetu średiowadratowego oraz odstawowe rzeształceia i zależości matematycze, w oarciu o tóre dooao wsomiaej aalizy. Wyii aalizy zaisao w ostaci wyresów i sformułowaych wiosów ońcowych. Itroductio s the result of uavoidable errors made i the techological machiig rocess, the desiged ideal geometrical shae of maufactured machie comoets is oly aroximately similar to the omial shae assumed by the desiger. Deviatios from the omial shae rofile are critical for the life ad correct co-oeratio of movig ad matig surfaces of the comoets. However, if these errors do ot exceed the limits defied by relevat stadards [1,, ], the the roduct ca be regarded as accetable. The set of requiremets secifyig the geometry of a object (or a assembly of comoets) is ow as the Geometrical Product Secificatio (GPS). Whe GPS requiremets are satisfied, the roduct has essetial characteristics, such as fuctioality, safety, reliability ad most imortatly, iterchageability. ccordig to GPS, a umber of essetial arameters has to be stated i order to uequivocally describe the geometrical shae of machie comoets from the viewoit of their use. Oe of these arameters is the allowable deviatio from the ideal shae, referred to as the shae deviatio. The defiitio of shae deviatio maes use of the so called fittig elemets or referece elemets. or the roudess deviatio of cylidrical machie comoets this elemet ca be a circle defied as fittig circle or referece circle (ig. 1) [4]. The determiatio of roudess deviatio relative to a roerly defied circle cosists i idetifyig the greatest distace ΔR of oits i the actual rofile of the circle. Due to a umber of advatages, the 7 Scietific Jourals 1(9)

2 The determiatio ad aalysis of the total measuremet ucertaity of roudess deviatio commoly adoted referece circle is the least squares mea circle [4, 5]. Determiatio of the roudess deviatio relative to the mea square circle Holdig objects i vee blocs is the most frequet method of suortig large cylidrical machie arts. However, the correct geometrical assessmet of shae deviatios deeds o two factors: referece elemets roerly defied i the course of rocessig the results of measuremets, ad revious careful aalysis of errors of the measuremet method used. Vee bloc measuremets belog to the so called referece methods of rofile measuremets, i which a vee bloc based elemet is rotated. The exact defiitio of referece feature i this case is rather difficult, ad the least squares mea circle is the most roer for the evaluatio of roudess deviatio [6]. Practical measuremet of roudess deviatio ΔR requires that the greatest ad smallest idicatios of the measurig istrumet are idetified durig the rotatio of the object measured. s the examied object based i vee blocs is beig rotated, the measured Δ differs more or less from its real value ΔR. ccordig to commoly acceted theory of mathematical descritio of machie comoet cylidrical rofile roudess (usig the harmoic aalysis of shae rofile), ay rofile ca be develoed ito the trigoometric ourier series [4]. Dislacemet legths of the sesor i measurig a give shae rofile as the fuctio of the rotatio agle of the object suorted by two vee blocs ca be mathematically described as follows [7]: C cos (1) 1 where: C harmoic comoet amlitude; hase dislacemet of articular harmoic comoets; umber of harmoic comoet. The fuctio described by the relatioshi (1) is eriodic ad has miimum ad imum values, their differece beig the measured roudess deviatio Δ : mi () Develoig the fuctio (1), which is a mathematical descritio of the measured roudess deviatio, we ca write: 1 Δ cos si () 1 where, are comoets of the amlitudes C of articular harmoics of the measured roudess rofile ad are exressed by these relatioshis: C cos (4) C si (5) The the hase shift of the subsequet harmoic is defied by this equatio: tg (6) I umerical calculatios the relatioshis (4) ad (5) corresod to: r 1 cos (7) r 1 or these equatios: si (8) 1 π y cos (9) 1 π y si (10) where: y discretized values of the fuctio Δ (φ), umber of itervals adoted for discretizatio, umber of the subsequet harmoic. The above equatios are used for discretized y values of aalogue sigals, obtaied durig measuremets with cotiuous radius r recordig (y deviatio of -th oit of the roudess rofile from the adoted referece circle). The coordiates of the least squares mea referece circle cetre have the values x s =, y s = calculated from the relatioshis (7) ad (8) for the harmoic = 1. To calculate the value of the least squares mea circle radius this relatioshi ca be used [8]: r s r 1 (11) Relatioshis (1 11) allow to do ecessary calculatios i order to determie the measured roudess deviatio Δ, to reset the measured rofile i the form of a trigoometric ourier series (so called fiite cosiusoidal trasform) ad to deict a rofile i the grahical form i the olar or Zeszyty Nauowe 1(9) 7

3 Krzysztof Nozdrzyowsi Cartesia coordiate system. The referece elemet for the olar system will be a least squares mea circle with a radius r s, while for the Cartesia system a mea lie m [4]. Determiatio ad aalysis of the total measuremet ucertaity of roudess deviatio ccordig to the riciles of determiig comlex measuremet ucertaity [9, 10, 11], the measuremet ucertaity of the measured deviatio Δ ca be writte i this geeral form: ) ( ) r 1 1 ( r (1) y substitutios (equatios ( 8)) ad trasformatios, we ca formulate the fial formula defiig the measuremet ucertaity of the measured roudess deviatio exressed with directly measured quatities r, φ ad their stadard ucertaities Δr ad Δφ : si si si mi r 1 1 cos cos cos 1si cos cos 1cos si si 1 (1) π where: or measuremets where the radius r is cotiuously recorded, the measuremet ucertaity of roudess deviatio Δ Δ ca be writte as: 1 ( ) r r mi mi mi (14) ad taig accout of (9) ad (10): 1 si cos π π si si r cos cos mi mi 1 (15) O the basis of the relatioshi (1) the direct ifluece of stadard ucertaities Δr, Δφ of measured quatities o the total comlex ucertaity value Δ Δ was examied. Such aalysis allows to secifically affect the fial value of the calculated total measuremet ucertaity (taig ito accout the ecoomic factor resultig from icreasig or decreasig stadard deviatio of direct measuremets). Examles of simulated calculatios results of the total ucertaity value of the measured roudess deviatio based o the relatioshi (1) for a secific set of iut data are show i figures 1 ad. This set of iut data cosisted of discretized values y of the fuctio described by the relatioshi (1) (writte dow with a set of = 15 harmoics obtaied from measuremets of a real object), determied at oits φ corresodig to boudary oits of equal itervals as resultig from the fuctio give i (1). The values of agles φ ad φ mi ad roudess deviatio Δ were also determied for these data (agles φ ad φ mi are agles at which the fuctio (1) assumes the imum Δ (φ ) ad miimum Δ (φ mi ) values). I figures 1 the total measuremet ucertaity of roudess deviatio is exressed i the form relative to the roudess deviatio Δ as: w (16) Chages of Δ were examied i itervals of stadard ucertaity chages of directly measured quatities: for Δr 0 0,1, for Δφ 0 0,, resectively. This aalysis too also accout of the imact of the umber of itervals tae for discretizatio o the value w Δ. igure 1 resets the chage i w Δ deedig o Δr ad Δφ for = 60, while figure shows the chage i w Δ deedig o Δr ad Δφ for = 60. The aalysis results have show that the value of the total measuremet ucertaity Δ Δ is much 74 Scietific Jourals 1(9)

4 The determiatio ad aalysis of the total measuremet ucertaity of roudess deviatio more affected by measuremet ucertaity of Δr tha that of Δφ. I the assumed boudaries of Δr ad Δφ chages, the relative chage i the value w Δ is, resectively, for the grah w Δ = f(δr, Δφ ) with = 60 (ig. 1) , while for the grah w Δ = f(δr, Δφ ) with = 60 (ig. ) y icreasig the umber of oits describig a circular rofile that mae u a basis for the assessmet of roudess deviatio Δ we obtai a more favourable result, i.e. reduced error w Δ. w 10 The character of chages i measuremet ucertaity wδ Δ deedet o measuremet ucertaity Δr ad the umber of oits assumed for the descritio of roudess rofile based o the relatio (15) is show i figure. Therefore, the measuremet methods used should assure a accurate divisio of object rotatio agle as well as the caability of dividig the rotatio ito ossibly large umber. s it was emhasized before, for a give measurig tas ad a desired measuremet ucertaity of roudess deviatio Δ we ca select measuremet systems that will guaratee aroriate measuremet ucertaities of directly measured quatities Δr, Δφ ad to defie a roer umber of oits describig a circular rofile for assessmet. w 10 4 r ig. 1. Chage i the relative value of comlex measuremet ucertaity of roudess deviatio w Δ deedig o stadard ucertaities of the radius Δr ad agle of rotatio Δφ for = 60 Rys. 1. Zmiaa względej wartości złożoej ieewości omiarowej odchyłi orągłości w Δ w zależości od ieewości stadardowych romieia Δr i ąta rotacji Δφ dla = 60 w 10 r ig.. Chage i the relative value of comlex measuremet ucertaity of roudess deviatio w Δ deedig o stadard ucertaities of the radius Δr ad agle of rotatio Δφ for = 60 Rys.. Zmiaa względej wartości złożoej ieewości omiarowej odchyłi orągłości w Δ w zależości od ieewości stadardowych romieia Δr i ąta rotacji Δφ dla = 60 ig.. Chage i the relative value of total measuremet ucertaity of roudess deviatio w Δ deedig o stadard ucertaities of the radius Δr ad the umber of oits assumed for the descritio of roudess rofile Rys.. Zmiaa względej wartości całowitej ieewości omiarowej odchyłi orągłości w Δ w zależości od ieewości stadardowych romieia Δr i liczby utów rzyjętych do oisu rofilu orągłości Summary r The reseted results of a aalysis of the imact of directly measured quatities ad their stadard measuremet ucertaities have show uequivocally that from the viewoit of the cosidered comlex measuremet ucertaity of the roudess deviatio, the quatities affectig the measuremet ucertaity are the measuremet ucertaity of the radius Δr, ad of the rotatio agle iterval Δφ ad the umber of oits adoted for descritio of a circular rofile. Comarig searately the ifluece of each sigle arameter we ca state that the effect of ucertaity Δr ad is more sigificat tha that of ucertaity Δφ. Therefore, the selectio of these quatities should Zeszyty Nauowe 1(9) 75

5 Krzysztof Nozdrzyowsi be adjusted to the desired measuremet ucertaity Δ Δ ad characteristic dimesios of the object beig measured that result from a secific measuremet tas. Refereces 1. HUMIENNY Z.: Secyfiacje geometrii wyrobów (GPS) wyład dla uczeli techiczych. Oficya Wydawicza Politechii Warszawsiej, Warszawa MLINOWSKI J.: Pomiary długości i ąta. WNT, Warszawa PN-87/M-017. Toleracje ształtu i ołożeia. Nazwy i oreśleia. 4. DMCZK S.: Odiesieiowe metody omiaru zarysów orągłości części maszy. Moografie, Studia, Rozrawy. Politechia Świętorzysa, Kielce IT S.: aliza błędów metod omiaru ształtu rzedmiotu o rzeroju ołowym [rozrawa dotorsa]. Politechia Wrocławsa, Wrocław NOZDRZYKOWSKI K.: Metodya omiarów geometryczych błędów uładu łożysowaia wałów orbowych siliów orętowych. [rozrawa dotorsa]. Politechia Szczecińsa, Szczeci NOZDRZYKOWSKI K.: Trasformacja mierzoego metodą odiesieiową roudess rofile a zarys rzeczywisty rzy omiarach w ryzmach. IV Międzyarodowa Koferecja Nauowo-Techicza Obsługiwaie Maszy i Urządzeń OMNiU 008, Świoujście Koehaga PN-9/M-046. Metody ocey odchyłe orągłości. Pomiary zmia i romiei. 9. SZYDŁOWSKI H.: Międzyarodowe ormy ocey ieewości omiarowych. Postęy izyi, 000, 51,, DMCZK S., MKIEŁ W.: Metrologia w budowie maszy. Zadaia z rozwiązaiami. WNT, Warszawa ZIĘ.: Natura rachuu ieewości omiarowych a jego odyfiacja. Postęy izyi 001, 5, 5, Recezet: dr hab. iż. Tadeusz Iglatowicz, rof. ZUT Zachodioomorsi Uiwersytet Techologiczy w Szczeciie 76 Scietific Jourals 1(9)