Reliability Prediction for CNC Machine of Spindle System Z. Xiaocui1,a*, C. Fei1, b, W. Jili1, c, Y. Zhaojun1, d, L. Guofa1, e,z.

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1 Itertiol Symposium o Mechicl Egieerig d Mteril Sciece (ISMEMS 206) Reliility Predictio for CNC Mchie of Spidle System Z. iocui,*, C. Fei,, W. Jili, c, Y. Zhou, d, L. Guof, e,z. ige, f College of Mechicl Sciece d Egieerig, Jili Uiversity, Chgchu 30022, Chi c d zxc@lu.edu.c, chefeic@lu.edu.c, wgili00@26.com, ygz@26.com, e ligf@lu.edu.c,zhgxige@lu.edu.c Keywords: CNC mchie, Spidle system, Grey theory, Reliility predictio Astrct. Grey predictio of CNC mchie spidle system filure is modeled with GM(,) grey predictio s dt ccumultio weke rdomess d stregthe regulrity. The model result provides prevettive mitece of CNC mchie spidle system with theoreticl foudtio. The exmple lysis shows tht grey predictio model c exctly predict time of filure quickly. It is effective to e used i reliility predictio of CNC mchie spidle system. The grey predictio model lso c predict filure time of CNC mchie d the other susystem. Itroductio Spidle system is oe of the key CNC mchie susystems. It is relted to CNC mchies cpility d the qulity of mchied workpieces. Its reliility is very importt to CNC mchie. Spidle system of CNC mchie icludes spidle d spidle compoets (erig, spidle ecoder, spidle motor etc). I pst surveys, its dymic ehvior c e got y FEA, the estlishmet of the dymic model d trsiet respose of its prts. The dt structure of spidle prts provide the sis for desig d ssemle of spidle system. The reliility of spidle system c e lyzed y FTA [-3]. There is less report of spidle filure time predictio. The predictio time of spidle system is of gret vlue to theoreticl foudtio of spidle system prevettive mitece d CNC mchie reliility improvig. The pper hs certi type of CNC mchie spidle system fult iformtio s the kow iformtio. Susequet filure time of spidle system c e predicted y grey predictio theory. Dt Acquisitio d Processig of Spidle System Filure Dt Acquisitio of Spidle System Filure. The filure dt is from certi CNC mchie tht is trcked o the scee. It icludes receptio time, strtig dowtime, termitio of the dowtime d mitece time [4]. The fult messge of mesured CNC mchie c e got y filure dt record. The pper reserches the exmple tht is spidle filure dt of 5 CNC mchies s Tle. Tle. Filure time of spidle system. No Filure time/h No Filure time/h Geertig Process of Filure Dt. Susequet filure time c e got y geertig process of the collected dt. The method of geertig process icludes ccumultio geertio method, B- geerted method d me vlue geertig method [5-7]. Accumultio geertio method. Origil filure dt colum s follows: Copyright 206, the Authors. Pulished y Atltis Press. This is ope ccess rticle uder the CC BY-NC licese ( 365

2 = {x, x (2),..., x ()} Formul: superscript is origil filure dt colum, is the umer of vlues. Accumultio geertio dt colum s follows: = {x, x (2),..., x ()} (2) Formul: t x (t ) = x (i ) = x (t ) + x (t ), t =, 2,..., i = (3) Superscript expresses oce ccumultio geertio dt. B- geerted method. B- geerted method is tht frot d ck dt of origil filure dt sutrct. B- method is iverse opertio of ccumultio method. The ccumulted dt c e restored y B- method. The B- geerted formul s follows: = x (t ) x (t ), t =, 2,..., (4) I the formul: x = 0 Me vlue geertig method. The ew dt c e geerted y me vlue of dcet dt. The geerted formul s follows: Z (t ) = [ x (t ) + x (t )], t = 2,3,..., 2 (5) Predictio of Spidle System Reliility The predictio is to predict susequet filure time of spidle system while uildig model y grey theory. The model c predict the susequet filure. The grey model is used to process the dt which ccord with smooth discrete fuctio d the geerted dt possesses idex-to-e lw. Therefore, the dt should e checked whether ccordig with smooth discrete d idex-to-e lw efore uildig grey predictio model. While the dt ccords with the lw, the grey predictio model c e uilded. Checkig Smooth Discrete d Idex-to-e Lw of the Dt. Checkig smooth discrete. The formul of checkig smooth discrete s follows: ρ (t ) = x (t ) x (t ) (6) Checkig idex-to-e lw. The formul of checkig idex-to-e lw s follows: σ (t ) = x (t ) x (t ) (7) The filure time dt of tle is checked whether ccordig with smooth discrete. The result s follows: ρ( t ) =(2.87,0.77,0.75,0.52,0.42,0.3,0.28,0.3,0.27,0.22,0.9,0.8,0.5,0.4,0.3,0.4,0.3,0.2, 0.4,0.3), The ρ (4) = 0.52 ρ (5) = 0.42 < ρ (2) = 0.3 < 0.5, While t > 4, the dt ccords with smooth discrete lw. The filure time dt of Tle is checked whether ccordig with idex-to-e lw. The result s follows: σ (t ) = (3.87,.77,.75,.52,.42,.3,.28,.3,.27,.22,.9,.8,.5,.4,.3,.4,.3,. 2,.4,.3), σ (4) =.52 σ (5) =.42 [,.5],, σ (2) =.3 [,.5], While t > 4, the dt ccords with idex-to-e lw. Buildig Grey Predictio Model. The GM (,) is usully pplied to grey predictio model. Its 366

3 costructio is simple d coveiet to uild ut its result is exct. The model (,) presets differetil equtio of first order d vrile. Buildig the sic form of GM (,). The filure time colum hs oserved vlue, s = {x, x (2),..., x ()}. The ew ccumultio geertio dt colum s = {x, x (2),..., x ()}. is cotiuous fuctio of t. The uildig differetil equtio s follows: d x ( ) + x = dt (8) I the formul: is progress grey umer, is edogey commd grey umer. The sic form of grey predictio model c e got y the formul (8) x (t ) + z (t ) = (9) The prmeter estimtio of model. x ( 0 ) ( 2) z (2) ( 0) x (3) z (3) Y =... B = ( 0 ) ( ) () z, x To predict the progress grey umer y lest squre method s follows: = ( BT B ) BT Y (0) Followig Y = B () Fil result s follows: x (t ) z (t ) x (t ) z (t ) t=2 = 2 2 ( z ) ( z (t )) [ x (t ) + z (t )] = (2) Put the predicted, ito predictio model, d the grey predictio model GM (,) s follows: t (t + ) = ( )e + (3) So predictio model of the origil colum s follows: (t + ) = (t + ) (t ) = ( ) (e t e ( t )) (4) The fil predictio model of spidle system s follows: t t x (t ) = ( x ) e + = ( x ) e (5) c e restored y the formul (4) d (5). The check of predictio model. The uildig model should e checked to esure it is right or 367

4 wrog. The checkig method icludes residul exmitio, correltio test d the fter test rule. The uildig predictio model should e checked y the ove three methods to esure it resole. Residul exmitio. Residul exmitio is to clculte the solute error d reltive error etwee the origil dt colum (t ) d predictio, colum ^ (t ). The formuls re s follows: The solute error: (t ) = (t ) (t ), t =, 2,..., (6) The reltive error Φ (t ) = (t ) 00%, t =, 2,..., (t ) (7) The model precisio is higher while the solute error d reltive error is smller ccordig to experiece. Correltio test. The correltio test is to predict correltio etwee the predictio colum ^ (t ) d the origil colum (t ).The computtiol process s follows: ① Clcultig correltio coefficiet Supposig the predictio filure time colum s follows: (t ) = {x, x (2),...x (t )..., x ( )} The origil filure time colum s follows: = {x, x (2),...x (t )..., x ()} The defiitio of correltio coefficiet s follows: η ( ) = mi{ ( )} + ρ mx{ ( )} ( ) + ρ mx{ ( )} (8) I the formul: () mi { ( )} is miimum vlue of solute error, =, 2,...,, () mx { ( )} is mximum vlue of solute error, =, 2,...,, (c) ρ is resolutio rtio, 0 < ρ <,usully ρ = 0.5. ② Clcultig correltio degree The formul of correltio degree s follows: r= η ( ) = (9) The fter test rule. ①The fter test rule should firstly clculte the stdrd devitio of the origil dt colum. The formul s follows: 2 ( (t ) ) t = S = (20) I the formul: = (t ) t = (2) ② Clcultig the stdrd devitio of solute error colum. The formul s follows: S2 = 2 ( (t ) ) t = (22) I the formul: 368

5 = (t ) t = (23) ③ Clcultig the rtio of stdrd devitios. The formul s follows C = S 2 / S (24) ④ Estimtig the little proility of error: Estimtig the little proility of error ccordig to (t ) P = P{ (t ) < S s follows: < 0.645S } (25) Accurcy grde referrig to the Tle 2. The result c e got s Tle 3 while checkig the predictio model of spidle system y the ove tests. The reltive error is less th 0.6 ccordig to Tle 3. The degree of ssocitio: r = > The rtio of stdrd devitios: The model is verified y the fter test rule. Tle 2. Accurcy grde. P C ccurcy grde >0.95 >0.80 > <0.35 <0.50 < First grde: good Secod grde: qulified Third grde: Brely qulified Fourth grde: uqulified The predictio model of origil dt colum is got prelimirily s follows: t ( t ) ), t =, 2,..., x (t ) = ( x ) (e e (26) 0.034t, t =, 2,..., So x (t ) = e. Tle 3. The check result of predictio model. Origil dt Predictio dt Reltive error Correltio coefficiet Origil dt Predictio dt Reltive error Correltio coefficiet Buildig the fil predictio model. I order to mkig the predictio model more resole d more correct, the fil model c e got y weightig fctor, method sed o the origil model. The method steps s follows: ) The dt is divided ito three groups y the whole dt d deletig the lst oe d two dt. 2) The models of other two group dt c e got ccordig to the method of uildig predictio model of whole dt. 3) To predict the lst oe dt y the three predictio models. 4) To give the prmeters of three predictio models the empowermet ccordig to the differece etwee the predictio d ctul vlue. The priciple is tht the weight is ig while the differece is smll d the weight is smll while the differece is ig. Weight dditive is equl to. 5) The dditio of product tht prmeter multiply its weight is the prmeter of fil model. The fil models s Tle

6 Tle 4. Fil predictio model. Dt colum Complete dt First group dt (deletig the lst oe) Secod group dt (deletig the lst two) Predictio model 68.46*exp(0.03*t) 636*exp(0.068*t) 625*exp(0.067*t) The 2st filure time of spidle system is 5343h, 5384h, 5280h ccordig to the three models i Tle 4. Actully the 2st filure time of spidle system is 52.2h. So the differece etwee the secod predictio filure time d the rel filure time is the miimum. The differece etwee predictio of complete dt d the rel filure time is the secod miimum. The differece etwee predictio of first group dt d the rel filure time is the mximum. The weights of three model prmeters re 0.3, 0.2 d 0.5 ccordig to the ove lysis d weight priciple. The fil model s follows: y = 644 exp( t ) (27) The 22st filure time of spidle system is out h ccordig to the predictio model s there is 2 filure time dt i the ssessmet period. Coclusios The grey predictio model of spidle system c predict susequet filure time quickly d exctly. The model result provides prevettive mitece of CNC mchie spidle system with theoreticl foudtio. (2) The process proves tht the fesiility method c e pplied ito the complete CNC mchie d susystem. The method is uiversl. (3) The complete process does ot eed y ssumptio. The weight mkes the result of predictio model more correct. Ackowledgemet This work is supported y the Importt Ntiol Sciece d Techology Specific Proects of Chi (No. 206Z d JC ) Refereces [] Y. M. Zhg, C. S. Liu et l. FEA o the Spidle Assemly of High Speed Mchie Tool, J. Northester Uiversity (Nturl Sciece), 29(0) (2008) [2] L. H. Li,. A. Che et l. Dymic Alysis of The Spidle Compoets of Numericl Cotrolled Mchie Tool Bsed o Fiite Elemet Method, J. Mech. Str. 3(4) (2009) [3] R. H. Go, H. W. Liu et l. Reserch o the dt structure for spidle uit of mchie tool d the commuictio techology, J. Hefei Uiversity of Techology (Nturl Sciece), 24(3) (200) [4] G. F. He, H. B. u. Collectio d Alysis of Reliility Dt. Ntiol Defese Idustry Press, (995), pp. 2. [5] S. F. Liu, Y. G. Dg et l. Grey system theory d pplictio. Beiig: Sciece Press, 200. [6] Z. F. LI, W. S. Go. Mthemticl Sttistics d Stochstic Process. Jili Uiversity Press, (2000), pp. 0. [7] J. H. Co, K. Che. Reliility Mthemticl Itroductio. Higher Eductio Press, (2006). 370