5h Inernaonal onference on Advanced Desgn and Manufacurng Engneerng (IADME 5 The Falure Rae Expermenal Sudy of Specal N Machne Tool hunshan He, a, *, La Pan,b and Bng Hu 3,c,,3 ollege of Mechancal and Vehcle Engneerng hangchun Unversy,hangchun, hna a emal: he3cs@6.com, b emal: panl636@6.com, c emal: 875754@qq.com Keywords: One-sample, Specal N Machne ool, Falure Rae, Relably of Specal N Machne Tool. Absrac. Based on one-sample specal N machne ool, he falure dsrbuon funcon and falure rae funcon, whose falure daa have been analyzed by expermen, are deermned n hs paper. The specal N machne ool me of whch are n early falure or haphazard falure can be judged n erms of he falure rae funcon of, and ha he relably of specal N machne ool s mproved by elmnang falures. Inroducon Based on machne falure dsrbuon funcon and relably funcon, a specal N machne ool falure rae of he sngle sample was gven. How o esablsh he specal N machne ool falure dsrbuon funcon s he key. In he paper, he me dsrbuon model beween falures has been deermned akng he fng expermenal mehod, s correcness has been verfed, and he relably of he machne ool has been evaluaed. Theory dsrbuon funcon of faul nerval me could be defned as: F ( = P( T < ( In he formula, T was nerval me of falure n general; was nerval me of any falure. Assumng ha,, k were observed values of nerval me of falure, by hs group of observed values, order sasc of nerval me of falure were obaned such as (, (, (k. Then he emprcal dsrbuon funcon of nerval me of N machne ool falure nerval me was followng as:, < ( F( k ( = / k, ( < (, =,,LL k ( +, ( k By he emprcal dsrbuon funcon F (k (, heory dsrbuon funcon F( could be esmaed. And by F( shape, he shape of a probably densy funcon f( of falure nerval me could be judged. So by he F (k ( shape, he f( shape could be prelmnary judged. Graphcs F (k ( was sepped lne char o f he graphcs n a row, o smplfy he ype ( as follows: F k = / k =,,LL k (3 ( ( Accordng o he ype (3, F (k ( has been fed, so observed values of falure nerval me [and] have been dvded no n groups. The abscssa was medan of each group me, he ordnae was cumulave frequency of each group, hus make a F (k ( scaer plo. Accordng o he scaer plo, he emprcal dsrbuon funcon F (k ( of falure nerval me could be deermned o convex or concave. And hen judge ha he dsrbuon of he falure nerval me was normal dsrbuon or lognormal dsrbuon, exponenal dsrbuon or Webull dsrbuon. Suppose he falure nerval me obeyed a ceran dsrbuon, and hen by lnear regresson analyss, he parameers of he dsrbuon have been esmaed. For fng effec of he dsrbuon, he lnear correlaon es has been adoped. If he lnear correlaon was esablshed, would nroduce ha he assumng dsrbuon se up []. Fnally dsrbuon funcon falure nerval me was d esed, because he d es was beer han heχ es. The d es s ha k es daa are arranged accordng o he order from small o large; accordng o he hypohess, calculae he dsrbuon F (x of each correspondng daa, and compared wh he emprcal dsrbuon funcon F k (x o deermne he larges absolue value of dfference whch s he 5. The auhors - Publshed by Alans Press 88
observed value D k of es sascs. ompare D k wh he crcal value D k,a. Sasfy he followng condons, hen accep he null hypohess, or rejec he null hypohess: Dk = sup Fk ( x F ( x = max{ d } Dk, (4 < x<+ In he formula, F (x was The orgnal hypohess dsrbuon funcon; F k (x was Emprcal dsrbuon funcon; D k,a was he crcal value. d = max F ( x, F ( x (5 k k Afer nspecon, probably densy funcon f( and dsrbuon funcon F( of falure nerval me could be deermned. And hen deermne relably funcon and falure rae funcon of he machne ool. Falure Rae Funcon of he Sngle Sample Specal N Machne Tool If T was connuous random varable, sad machne rouble-free workng me, so he probably of relable work a he me was followng: R(=P(T> (6 The probably of machne ool falure was: F(=P(T (7 d f ( = F( F( was a random varable dsrbuon funcon, he densy funcon was: d (8 F( was uncondonal probably densy; λ( was known as he falure rae funcon, was he condonal probably densy: P[( p T + IT f ] f( f( λ( = lm P( <T + T > = lm P ( T f = lm = R ( R ( (9 The formula (9 was consdered from he lfe of he un [], so was n conformy wh he acual requremens ha he falure rae funcon of he sngle sample specal N machne ool was usedλ( o descrpon. Deermnaon of falure dsrbuon funcon Now wh a ceran ype of falure daa of N machne ool as an example, he falure dsrbuon funcon of he machne ool has been dscussed. The observed values of falures nerval me [.45, 69.53] have been dvded no 5 groups. The falure frequency and cumulave frequency were shown n able. Table.. Falure frequency and cumulave frequency Group On he Under he In he group frequency nerval range values number frequency Accumulave.45 46.47 78.46 5.357.357 46.47 8.48 4.47 3.43.574 3 8.48 48.5 35.49 3.43.7857 4 48.5 554.5 486.5.49.986 5 554.5 69.53 6.5.74. To he abscssa denoes he value n each me he group, each group of probably densy of he observed value as he ordnae. The calculaon f ˆ( was as follows: fˆ ( = k ( In he formula, k was he falure frequency number n each group falure nerval me; k was oal falure frequency, for 4 mes; Δ was class wdh for 36. h. Wh he abscssa denoes he value n each me he group, each group of cumulave frequency as he ordnae, hus make a F (k ( = / k scaer dagram. k 89
The resuls show ha he falures nerval me he ype of N machne ool was no subjec o normal dsrbuon or lognormal dsrbuon, was he exponenal dsrbuon or Webull dsrbuon. Suppose he dsrbuon was Webull dsrbuon, probably densy funcon and dsrbuon funcon of he Webull dsrbuon were: β γ β γ β γ ( exp[ ( ], γ f ( = ( = = f ( d exp[ ( β ], γ F ( (, < γ, < γ In he formula (, (, βwas he shape parameer, β> ; was he scale parameer,> ; γwas he poson parameers, γ>. hange γ affeced ranslaonal poson of he probably densy curve. Before =γ, he produc was no down; Afer =γ, he produc falure. In pracce, ofen assume ha when =, produc falure. So, ypes (, ( hen were smplfed as: β β β β f ( = ( exp[ ( ], (3 F( = exp[ ( ], (4 The dsrbuon of falures nerval me has been suded by Webull dsrbuon wh wo parameers. Fng Inspecon of Falure Dsrbuon Funcon Parameers Esmaon of he Webull Dsrbuon Suppose ha one-dmensonal lnear regresson equaon was: y = A + Bx (5 Accordng o he wo parameer Webull dsrbuon, o lnear ransformaon ype (4, avalable: y ln ln F( x = ln (7, A = β ln (8, B = β (9 If observed values of falure dsrbuon funcon were,, k, accordng o he ype (6, (7, he values were conversed no he x, y n ype (5. Inercep A and slope B of regresson sragh lne could be obaned by he leas squares. Thus wo parameers, βof Webull dsrbuon could be esmaed by ype (8, (9. Value x could be calculae by ype (7, namely x = ln ( Value y could be calculaed by ype (6. Before calculaon, he value F( should be esmaed frsly. General use medan esmaed: ˆ.3 F( k +.4 ( lxy In hs way, by he leas squares mehod, was avalable: B ˆ = lxx (, Aˆ = y Bˆ x (3. By ype (8 and (9, avalable: β ˆ = Bˆ (4, ˆ = exp( Aˆ / Bˆ (5 Accordng o he falure daa of N machne ool, wo parameers Webull dsrbuon have been esmaed. Falure daa were shown n able. 8
Table.. Falure es daa of a specal N machne ool No x F ( y x y.45.44.486 -.999 5.99 8.9945-7.38.96.56.8 -.744 6.56 4.333-5.345 3 5.37.73.875 -.57 7.47.47-4.95 4.66 4.6.569 -.4.7.474-5.599 5 43. 4.96.364 -.986 4.63.863-4.686 6 6.3 5.8.3958 -.6859 5.76.4697-3.4785 7 7.8 5.4.4653 -.4684 6.45.94 -.488 8 36.7 5.46.5347 -.67 9.87.77 -.463 9 349.49 5.86.64 -.76 34.3.58.4454 389.7 5.96.6736.3 35.57.8.674 48.4 6.4.743.367 36.44.94.85 45.5 6..85.55 37.33.654 3.478 3 56.95 6.5.889.759 39.4.5764 4.7435 4 689.53 6.54.954.65 4.7.44 7.34 By ype ( and (3, avalable: B ˆ =.779, A ˆ = 4.37678 By (4 and (5, avalable: β ˆ =.779, ˆ = 9. 4 For fng effec of Webull dsrbuon, he lnear correlaon es was used, so: k x y kxy ˆρ = = (6 k k x kx y ky = = When ρ > ρ, was seen as X and y lnear correlaon, whch conformed o he Webull dsrbuon. I was calculaed n relably of he nformaon managemen sysem [3], ρˆ =.957, When =.645/ k sgnfcan level =., correlaon coeffcen value ( ρ =.456. For ρˆ > ρ, we hough falure dsrbuon funcon of he N machne ool obeyed Webull dsrbuon. x y 8
Hypohess Tes of Webull Dsrbuon Hypohess es daa of a specal N machne ools were shown n able 3. Table.3. Hypohess es daa of a specal N machne ools No F ( (-/k /k d.45.79397.74.797.96.86988.74.49.5587 3 5.37.98575.49.43.57 4.66.35669.43.857.798 5 43..4396.857.357.887 6 6.3.4676.357.486.394 7 7.8.48578.486.5.48 8 36.7.5798.5.574.55 9 349.49.684358.574.649.46 389.7.7345.469.743.83 48.4.733.743.7857.55 45.5.7594.7857.857.4 3 56.95.7888.857.986.398 4 689.53.8546.986..4354 Take a sgnfcance level =., by emprcal formula:. D k, = =.36 n For D k <D k,a, receve he null hypohess, ha he average rouble-free workng me of he machne obeyed Webull dsrbuon. The falure probably densy funcon f( and dsrbuon funcon F( were : f ( =.77 ( 9.4 9.4 Falure Rae Funcon.3.77 exp[ ( ] (7,.77 F( = exp[ ( ] (8 9.4 9.4 By ype (9, falure rae funcon was:.3 f ( ( λ = =.6 (9 R( 9.4 In he falure analyss of he produc, βwas assocaed wh he faul mechansm of he produc, dfferenβvalues was wh dfferen faul mechansm. When he β<, showng he lfe dsrbuon of he early falure perod; When he β=, showng lfe dsrbuon of he random falure perod; When heβ>, showng lfe dsrbuon of he wear falure. was assocaed wh he load of workng condons, load was bg, he correspondng was small; And vce versa. From ype (9 analyss show haβ<,λ(was downward rend, he specal N machne ools was n he early falure perod when was no 8
approprae for he cusomer o buy he machne. The machne should be connued o early falure expermens, formulaed reasonable specfcaon of early falure of he expermen. On he premse of no damage o he machne ool, as far as possble elmnae early falure; mprove he relably of machne ool. If falure daa were daa samples of sngle falure, Bayesan esmaon mehod should be adoped. Selecng ncomplee βdsrbuon B ( θ, θ, a, b as pror dsrbuon was feasble; whena, b >, βmperfec dsrbuon densy funcon was p monoone decreasng funcon, accord ha he possbly was large ha falure probably p was small, he possbly was small ha falure probably p was large, and hen he parameers a,b, θ,θ, could be properly seleced. For example a=, parameer b obeyed unform dsrbuon of nerval [, ], θ =,θ p, he consan, accordng B θ,, a, b. o he ncomplee dsrbuonβdensy funcon ( θ a b ( ( p θ ( θ p f p ;, θ, a, b = a + b B( a, b( θ θ θ ( <θ < The pror dsrbuon formula has been deermned for [4] : h ( p p p <θ (3 b ( p b (, b( p db = (3 B For k, r =( m, m k,he es sample was, falure daa of acquson was, so he lkelhood funcon was: L, p = p (3 ( Accordng o he ype (7, he Bayesan esmaon of falure probably p of he pror dsrbuon of was followng as: p ( r = = p p B B b p ( p (, b( p b p ( p (, b( p b dpdb dpdb b = ( ( p ( + p ln p ( p + ( p + + ln 3 + ln If falure probably p was calculaed for each me, he relably parameers of specal N machne have been esmaed by weghed leas squares, and hen he falure probably funcon could be deermned. oncluson ( Based on he falure daa, n hs paper, he deermnaon mehod of falure rae funcon of he sngle sample specal N machne ool has been dscussed. ( Se up falure daa analyss model of a sngle sample specal N machne ool. (3 In he case of sngle falure daa and smooh daa, how o deermne he sngle sample falure rae funcon of specal N machne ools, he relably analyss of he machne ool has ye o be furher suded. References [] Yazhou J, Holm W, Zhxn J.Probably dsrbuon of machnng cener falures, Relably Engneerng and Sysem Safey, 5(995-5. [] Guo Yong j, Prncples of Relably Engneerng, Tsnghua Unversy Press, Bejng,. [3] Wang Y Q, Ja Y Z, Yu J Y, e al, Falure daabase of N Lahs, Inernanoal Journal of Qualy and Relably Managemen, 6(999 33-34 [4] Mng Han, Relably Analyss of No Falure Daa, hna Sascs Press, Bejng, 999. (33 83