Process Quality Evaluation based on Maximum Entropy Principle. Yuhong Wang, Chuanliang Zhang, Wei Dai a and Yu Zhao

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1 Appled Mechacs ad Materals Submtted: ISSN: , Vols , pp Accepted: do:0.4028/ Ole: Tras Tech Publcatos, Swtzerlad Process Qualty Evaluato based o Maxmum Etropy Prcple Yuhog Wag, Chualag Zhag, We Da a ad Yu Zhao School of relablty ad systems egeerg, Behag Uversty, Cha a dw@buaa.edu.c Keywords: Process qualty evaluato; equpmet relablty; Physcs of falure aalyss; Maxmum etropy. Abstract. Process qualty evaluato plays a sgfcat role esurg product qualty ad mprovg customer satsfacto. Curretly, the research o process qualty evaluato lacks cosderato of teractve couplg betwee product processg parameters ad equpmet relablty. Wth usg the aalyss of physcs of falure, the relato model betwee the products qualty ad the processg equpmet relablty was establshed. Besdes, based o Maxmum Etropy Prcple (MEP), a evaluato method of process qualty was provded. Fally, we verfed the feasblty of the method through the combato a applcato case. Itroducto About 60% problems of products exposed durg usg are caused by the process factors of maufacturg, so fault mechasm ad the process qualty evaluato model are of mportat egeerg applcato value to esure product qualty ad mprove customer satsfacto. Tradtoally, qualty evaluato s to use the formato geerated by the expermets each stage durg product lfe cycle, thus the partcular qualty characterstc parameter was estmated through the method of probablty ad statstcs. Nowadays research process qualty evaluato cludes process stablty ad process qualty. The former emphaszed o the output qualty of maufacturg process []. The later emphaszed o the processg equpmet relablty of maufacturg process [2]. However, these two aspects lack cosderato of teractve couplg betwee the product qualty ad equpmet relablty. Through aalyzg the physcal of equpmet falure, the relatoshp betwee the process parameters ad product qualty was establshed, ad equpmet process qualty was evaluated. Meawhle, o the bass of the Maxmum Etropy Prcple (MEP), the evaluato method of process qualty was preseted, ad feasblty of the method was verfed through coectg wth a actual cocrete applcato example. Process Qualty evaluato Process qualty refers to the ablty of producto at requred qualty ad relablty level wth certa producto codto ad lmted producto cycle. Whle aalytcal method of Physcs-of-Falure maly takes relablty egeerg as the theoretcal bass ad troduces physcal ad chemcal methods to perform mcro-mechasm aalyss o teral ad exteral factors of process qualty. It revealed the er causes prompted by process mperfecto ad ts evoluto rule, whch through researchg the correlato amog defectve method wth materal property, dosage of process, techologcal process etc. Based o the Physcs-of-Falure, the process qualty evaluato s as follows: ) Cofrmato of crtcal parameters of product ad process. Crtcal product parameter usually ca adopt FMEA to esure, ad t ca be carred out at ay stage of the product desg ad product All rghts reserved. No part of cotets of ths paper may be reproduced or trasmtted ay form or by ay meas wthout the wrtte permsso of Tras Tech Publcatos, (ID: , Pesylvaa State Uversty, Uversty Park, USA-7/05/6,8:02:37)

2 626 Mechacal Compoets ad Cotrol Egeerg III use, by the way of fdg out the hdde dagers, faults or other problems that shorte the product lfe, to detfy the relevat product features as the key oes. Take the 4ME of the process to cosderato, carry out the course ad effect aalyss ad determe the 4ME of the safety ad fault to be the crtcal to process. 2) Relatoshp of process procedure ad product characterstcs. Suppose the process characterstc to be X, X 2,..., X, qualty characterstc to be Y, Y 2,... Y. Use the aalytcal method of Physcs-of-Falure to obta the fuctoal relatoshp of process characterstc ad qualty characterstc as y = f (x, x 2,..., x ), meawhle =, 2,...,. 3) Evaluato of the process qualty. Process qualty refers to the probablty to produce products of requred qualty ad relablty level wth certa producto codto ad lmted producto cycle. Suppose the task demads the value of qualty characterstc to be y *, y 2 *,..., y *. Process qualty thus ca be expressed as R=P(y <y *), whle =, 2,...,. Evaluato based o Maxmum Etropy Prcple Recetly, the evaluato for process qualty based o falure physcs cosders the process characterstcs as varables of some kow dstrbutos. However, the process characterstcs are usually ucerta practcal machg processes, ad oly a few part of formato could be obtaed. To decrease the fluece of subjectve factors ad make the result realstc [3,4], the process qualty evaluato method s put forward based o MEP. Basc theory of MEP Evaluato based o MEP s a techque that apples the physcal prcple of etropy, whch states that, wthout exteral terferece, the etropy whch measures the dsorder always teds to the maxmum. Etropy has a drect relatoshp to formato theory ad, a sese, measures the amout of ucertaty the probablty dstrbuto [5]. Assume that the radom varable x follows a cotuous dstrbuto wth pdf f(x), the maxmum etropy s gve as Θ The crtera formula (2) ad (3) must be satsfed. max H ( x) = x l dx () x Θ dx = x Θ g ( x) dx = E Where Θ s the doma of x, g (x) are a group of kow fuctos [6]. Defe the momet costrat ad let R m (x) = x m, hece, the maxmum etropy model could be expressed as M = exp λ + ( λmx m= m 0 ) The coeffcets λ (=0,,, M) could be estmated by solvg the equatos (2) ad (3). The research [7] states that the obtaed fucto bascally satsfes the egeerg requremets f the frst four momets are take as costrats. (2) (3) (4)

3 Appled Mechacs ad Materals Vols Process qualty evaluato method based o MEP The relatoshp of process characterstcs ad product characterstcs s defed as y = f (x, x 2,, x ) as metoed above. x, x 2,, x are radom varables, hece, y s a radom varable. Here, let y be a varable wth pdf f(y ). The frst four momets of x, x 2,, x are avalable by statstcs practcal machg process. Let Z=y, the Z could be expaded as secod-order Taylor seres the mea value of each radom varable [8]. Thus, Z = Z( X ) = Z ( X 2 Z ( X X )( X X ) j X j ) = j= j (5) Assume that the radom varables are depedet, the frst four momets of Z could be approxmately calculated out based o formula (5). Thus, the product relablty s gve as R = P( Z < z * ) = * z exp( 4 = 0 λ z ) dz Where, z * =y *. So the assessmet method of process qualty ca be as follows: () Cofrm crtcal product parameter ad crtcal to process; (2) Use the aalytcal method of Physcs-of-Falure to esure the fuctoal relatoshp of process characterstc ad qualty characterstc as y = f (x, x 2,..., x ), meawhle =, 2,..., ; (3) The mea value, varace, thrd momet ad the fourth momet of radom varable X, X 2,..., X ca be calculated accordg to the statstcal data; (4) The mea value, varace, thrd momet ad the fourth momet of Y ca be calculated accordg to formula (5) to (6); (5) Get the probablty desty fucto expresso f(y ); (6) Calculate the process qualty accordg to expresso f(y ). (6) Applcato Case There s a Factory S maly produces alumum alloy box bodes. Now three processes are avalable. To guaratee the qualty of the product, processes of hgh degree of relablty are eeded, so Factory S carres out evaluato of the process qualty. Aalyzed by FMEA, surface roughess of alumum alloy box bodes s determed as the key characterstc. Whle the chopper s speed of ma shaft v, the back egagemet of cuttg edge a p, the logtudal feeds f are the major factors that affect surface roughess of alumum alloy box bodes. Therefore the three are regarded as process characterstcs. By the aalytcal method of Physcs-of-Falure, the relatoshp betwee the chopper s speed of ma shaft v, the back egagemet of cuttg edge a p, the logtudal feeds f ad surface roughess Ra of choppers of ths kd are as follows: Ra = 5.784v a p f = v a 2.37 p f (7) Accordg to the statstcal data the actual producto, three process characterstcs three processes are detfed, ad the relevat frst four order momets are as Table. Take Process as

4 628 Mechacal Compoets ad Cotrol Egeerg III a example, Make use of the frst four order momets of the three process characterstcs. The actual producto dcates that R a * should be valued 2.5µm. Makg use of formula (2), the process qualty of Process s By the same toke, the process qualty of Process 2 ad Process 3 are respectvely 0.953ad 0.85.To satsfy the actual producto demads, Factory S chooses the process of hgher relablty, Process 2. Table The frst four order momets of process characterstcs v ap f Process Process 2 Process Cocluso Wth usg the method of physcs of falure aalyss, the artcle establshed the relato model betwee the qualty of the products ad equpmet process parameters. Ad based o the MEP, a evaluato method of process qualty was provded. Fally, the feasblty of the method was verfed through the applcato a case study. Refereces [] MacGregor J. F, Kourt T. Statstcal process cotrol of multvarate processes. Cotrol Egeerg Practce, 995, 3(3): p [2] Deg C., Wu J., Mao K., et al. Process qualty evaluato for large-szed NC mache tools. Computer Itegrated Maufacturg Systems, 200, 6(0): p [3] Yag M., L Z. Relablty predcto based o smlarty betwee the products of mechacal products. Mache Tool & Hydraulcs, 2003(): p.63. [4] L C., Wag J. Relablty aalyss based o maxmum etropy prcple ad fuzzy approach very small sample products. Lftg ad trasport machery. 2005(2): p [5] Gzyl H. The Method of Maxmum Etropy.World scetfc.994 [6] Spaos P.D. ARMA algorthms for ocea ware modelg. Eergy Resources Tech, 983, 05: p [7] Slv G. Iformato theory wth appl-cato. Mc GRAWHLLCO, (977), p [8] We Z., Ye J. Maxmum etropy method of relablty theory egeerg. Vbrato ad Shock, 2007(26): p.46.

5 Mechacal Compoets ad Cotrol Egeerg III / Process Qualty Evaluato Based o Maxmum Etropy Prcple /