Surface Roughness Simulation Using Fractal Interpolation to the Profilogram

Transcription

1 ISSN 39 3 MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol. 8, No.. Surfce Roughess Smulto Usg Frctl Iterpolto to the Proflogrm Loret MAČĖNAITĖ, Mts LANDAUSKAS, Vdmts Povls PEKARSKAS Fculty of Fudmetl Sceces, Kus Uversty of Techology, Studetu st.5, LT-5368 Kus, Lthu Receved Mrch ; ccepted Jue Ths pper presets the methodology d results of surfce roughess smulto of soft polymer mterls usg pproxmtos of frctl terpolto curves. The modelg lgorthm d ts relzto re mde so tht they c e ppled to y proflogrm. The comprso of creted models d rel proflogrms s doe usg utocorrelto fuctos d frctl dmesos. The three-dmesol model of the rougheed surfce s developed d equtos, reltg the re of the surfce to the chrcterstcs of rsve mterls,.e. to the sze of rsve gr, re oted. Those equtos re to e used forecstg the re of the surfce of utdee-styree ruer evdg the expermet tself. Keywords: rso, surfce roughess, three-dmesol model of rough surfce, utocorrelto fucto, frctl terpolto, frctl dmeso.. INTRODUCTION The rel cotct surfce re dhesve jot plys mportt role ts stregth d durlty. Ths prmeter depeds ot oly o the geometrcl dmesos of the surfce glued, ut lso o the roughess of the surfce [ 3]. It s lmost mpossle to fd the rel re of rough surfce y pplyg drect methods. Such re could e pproxmtely evluted from the umercl model of the surfce. I order to forecst the relto etwee the rel re of rough surfce d the regmes of surfce rso t s ecessry to fd the reltoshp etwee the rel cotct re d the dctors of surfce roughess. The sze of rsve gr of the rsve mterl, whch s used for surfce rougheg, hs fluece o the chrcterstcs of the surfce roughess. These chrcterstcs my e of two types: sttstcl d frctl, depedg o postos the profle of the rougheed surfce s treted. Sttstcl chrcterstcs of the profle re oted y tretg the profle of the surfce roughess s the relzto of stochstc process [4]. The ssumptos of sttory, ergodc d ormlly dstruted process re lso cosdered. However, those ssumptos re ot lwys stsfed d reserchers re ecourged to develop other models of surfce profle [5, 6]. Curretly, methods of frctl geometry d mportt qutttve property of frctls, frctl dmeso, re used to defe surfce roughess [7, 8]. It s mportt to crete surfce roughess model depedg o specfc prmeter chrcterstc for tht surfce oly. Such prmeter could e the frctl dmeso of the profle of the rough surfce [9 ]. The frctl dmeso dctes how desely oject ppers to fll the spce. The pprtus of frctl terpolto eles us to ot rough surfce model wth dfferet grdes of roughess, Correspodg uthor Tel.: ; fx: E-ml ddress: loret.mcete@ ktu.lt (L. Mčėtė) 38 sce prtculr frctl terpolto prmeters hve fluece o the frctl dmeso of the surfce profle. The three-dmesol model of surfce roughess descred ths pper employs qulty of self smlrty of two dmesol curve. The rel proflogrm my ot hold ths qulty, ut the model s formed wy tht t should correspod to the rel proflogrm ts shpe d frctl dmeso. Cosequetly, there re o extr codtos for proflogrm to stsfy ( order to e modeled).. FRACTAL INTERPOLATION CURVES AND THEIR ESTIMATES I ths pper, the preseted model performs frst two tertos of frctl terpolto for the set of pots {( x, y ) =, N}, chose from proflogrm. A smple wy to perform frctl terpolto s to pply sher trsformtos []. The umer of (two dmesol d ouded) sher trsformtos equls the umer of tervls etwee the terpolto pots. The -th ( = ; N ) sher trsformto s wrtte s x e ω, () here ( ) + x = c d y f, c, d, e d f re prmeters of the trsformto. The ltter prmeters re foud from the system ellow: c c x d y x d y It follows tht + e x = ; f y. () + e x =. f y

2 x x = ; x x y y y y c = d ; x x x x x x x x e = ; x x x y x y x y f = d x x x x y x. (3) Suppose, B s cotuous curve (tl set) coectg two terpolto pots, ( x, y ) d ( x N, y N ). Prmeters d d cotrol the level of stretch of the tl set log the scss d the ordte xes, respectvely. The prmeters d re rtrry, lthough d < must hold. The vlues c djust the level of sher of the tl set prtculr tervl. The prmeters e d f correspod to the dsplcemet of the tl set. The frst step performg frctl terpolto s to fd the uo of trsformtos ω ctg upo the tl set B: B = W B = ω B Υ ω B ΥΚ Υω B. (4) ( ) ( ) ( ) ( ) ( ) W N s treted s the tl set d s trsformed usg the sme trsformto W,. e. W ( B) = W ( W ( B) ). The frctl terpolto fucto (FIF), correspodg to the set B, s defed to e: FIF lmw B W B = W W B. (5) The resultg set of pots W ( B) = ( ), ( ) ( ) ( ) ( ) A llustrtve exmple s preseted Fg.. Here, ; ; 6 ;.5, the set of fve terpolto pots, ( ), ( 3 ), ( ) ( ;.5), ( ;), s cosdered Fg.. Frst terto of frctl terpolto of the dt: the tl set B s segmet, the tl set B s polygol le Fg. shows the resultg curve fter pplyg tertos to the dt depcted Fg.. It should e oted tht FIF does ot deped o the tl set B. If we restrct ourselves wth two (three, or 39 more) tertos, the frctl terpolto fucto (FIF) ecomes depedet o the tl set B (Fg. ) Fg.. Two tertos of the frctl terpolto: the tl set B s segmet, the tl set B s polygol le 3. RESULTS AND DISCUSSION 3.. The expermet The ojects of the expermet re the moolthc ruer smples (utdee-styree ruer wth the desty d hrdess ccordg Shore A, respectvely ρ =.5 g/cm 3 d H = 75.u.), whose surfce ws rded wth rsve pper of dfferet grdes. The pressure force of 8 N hs ee used the process of rso. Accordg to the FEPA s (Federto of Europe Producers of Arsves) stdrd stdg the Europe Uo, rsve mterls, wth respect to ther gress, re umered s follows: P4, P6, P, etc. The greter the grde of the rsve pper, the smller the sze of the rsve gr. I ths work, the followg grdes of rsve pper re used: P4, P4, P6 d P. The verge dmeter of rsve gr for ech grde of the rsve pper s show Tle. Tle. The relto etwee the grde of rsve pper d the verge dmeter of rsve gr The grde of the rsve pper ccordg to the FEPA The verge dmeter of rsve gr, (mm) P4.698 P4.38 P6.6 P.49 I ths pper, the roughess s treted s whole of mcro roughess locted reltvely close to ech other. From ths vewpot, surfce processed wth the grde P rsve pper s rougher th the oe processed wth the grde P4 rsve pper. All proflogrms were tested y Hommelwerke T5 surfce fsh tester proflogrph (Germy; mml lmt of mesurg s. µm) perpedculr to the drecto of rso. The proflogrms of the surfces rded wth the dfferet grde of the rsve pper re show Fg. 3.

3 c Fg. 3. The typcl proflogrms of rough surfce correspodg to the dfferet grde of roughess: P4; P4; c P6; d P 3.. The smulto of proflogrm The profle of surfce roughess the smple tervl of legth l = mm s expressed s set of pots M, = ; N, wth coordtes ( x, y ) defed. To smulte prtculr proflogrm, t frst cert mout of ts pots must e chose. These c e specfc pots (peks or mmums) wth respect to the lty to preserve the shpe of the proflogrm. But g, ths depeds o prtculr proflogrm d ts shpe. There re lots of specfc pots extremely rough proflogrm. They ofte resde esde ech other some res of the proflogrm d they c e fr from ech other other res. I such cse, tkg of the uo of sher trsformtos leds huge umer of ew pots res where the proflogrm s extremely rough,. e. the roughess of the model would e dequtely ueve. Becuse of ths property, t s ot dvsle to choose specfc pots from the proflogrm to perform frctl terpolto. Whe performg the reserch, the chose pots were dstruted uformly log the x xs, t the dstce u from ech other. Bellow, we demostrte the methodc of smulto of proflogrm y cosderg the excerpt from the proflogrm, whch represets the surfce rded wth the grde P4 rsve pper. The rel proflogrm d the selected pots for smulto re depcted Fg Fg. 4. The proflogrm of surfce rougheed usg rsve pper P4 d the set of selected pots for the smulto It s mportt to decde upo the umer of pots (from the proflogrm) to perform frctl terpolto. Ofte, we re forced to predct the resultg umer of pots the model. Otherwse, the clculto tme could d e wsted or model my hve too my or too few detls compred to the dt. By performg frctl terpolto d usg vrous tl sets ( segmet, polygol le), t ws foud out tht polygol le comprsg segmets fts est for smulto of the proflogrm. Such selecto of tl set s useful for esurg the model s dequcy to the dt. The totl umer of resultg pots the model s foud y pplyg smple clcultos. For exmple, f there re N terpolto pots d polygol le s chose s the tl set, the totl umer of pots fter the k-th terto equls ( N, k) = ( N ) + f. The smulto geertes the umer of resultg pots, whch s somewht equl to the umer of pots N the proflogrm ( f ( N, k) N ). Thus, for tertos we hve: ( ) N = +.5 N. (4) Accordg to the lyss completed, t ws otced tht there s o eed to perform my tertos, s t results huge umer of resultg pots the model. If eve 3 tertos re performed, the model gets extremely rough. O the other hd, the reducto of the umer of terpolto pots leds to the loss of the formto of the proflogrm s shpe, d does ot solve the ssue. The sher trsformtos, used to perform the frst terto of frctl terpolto, re depcted Fg Fg. 5. The result of the frst terto ppled to the chose pots from the proflogrm Prmeters d of sher trsformtos must e selected wy tht the model roughess were smlr to tht of the proflogrm. The rougher the proflogrm s, the k

4 greter d should e chose the cse of smooth proflogrms, vlues of d should e close to. Roughess of the surfce could e mesured y clcultg the stdrd devto of the profle s ordtes y k, k = ;, = N N. Thus, prmeters d re clculted ths wy: ( y, y, y ) c d = σ..., ; (5) here: s the dex of the tervl etwee the chose pots for terpolto; s the umer of the proflogrm s pots ths tervl d c s costt. I other words, prmeters d re rtos of the stdrd devto σ of the ordtes y k prtculr tervl d the rtrry costt. By performg expermets wth vrous vlues of c d evlutg the model smlrty to the proflogrm, the vlue c = 5 ws chose. The costt c s postve prtculr tervl f the me of proflogrm s ordtes o ths tervl s hgher th the rthmetcl me of the frst d the lst ordtes of the sme tervl d vce-vers. So, µ ( y, y,..., y ) > µ ( y, y ) mkes d >. After fte umer of tertos the resultg curve s treted s the model of the proflogrm. Two tertos trsform the tl set to the curve depcted Fg Fg. 6. Smulted profle of the rougheed surfce (usg rsve pper P4) fter tertos The foresd techque of proflogrm smulto d other clcultos hve ee progrmmed usg MATLAB 7.9. softwre. I order to evlute the model dequcy, the utocorrelto fuctos of the proflogrm d ts model hve ee used [3]. The utocorrelto fuctos of the proflogrm d ts model re clculted ccordg to the formul: ( τ ) E( ( μ)( μ) ) σ ACF, (6) = y + τ t y t where τ s cremet, y t, t = ; N τ ordtes of the proflogrm d μ, σ re the me d the vrce of the ordtes y t. The vlue of the utocorrelto fucto s dmesoless. By comprg vsully, the utocorrelto fuctos of the model d the proflogrm, t s oserved (Fg. 7) tht they re very smlr, provded τ s less th out 5. Thus, the orgl proflogrm d ts model correspod to ech other. It ws oted tht oth utocorrelto fuctos could e pproxmted y the sme lytcl expresso: ( ) ( τ ) = + ( α τ ) ACF, (7) where α s prmeter, oted y pplyg the lest squres method. ACF(τ),.u τ, mm Fg. 7. The comprso of the utocorrelto fuctos: ---- ACF of the rel proflogrm, ACF of the smulted proflogrm Tle shows tht the reltve error (RE) etwee the utocorrelto fuctos of the orgl d modelled proflogrms creses whe the grde of the rsve pper creses, provded the tl set for frctl terpolto s segmet. I the cse of polygol le, the shpe of the proflogrm s preserved etter f prmeters d re clculted the wy of eg proportol to stdrd devtos of the proflogrm s ordtes o ech tervl The model of the three-dmesol surfce Let us desgte the rry of ordtes of the surfce s { p, p, Κ, pn }. Let N represet the umer of pots comprsg the proflogrm. Due to the fct tht the proflogrm s cut of prtculr three-dmesol (3D) surfce, settg the wdth eles us to mke the 3D terpretto of surfce roughess (Model ). Whle mkg the three-dmesol model of proflogrm, we lso set the wdth to the estmte of ts FIF. If { f, f, Κ, f N } re the ordtes of the FIF s estmte of the proflogrm, the the three-dmesol terpretto of surfce roughess s wrtte mtrx Tle. The me squred (MSE) d the reltve (RE) errors etwee the utocorrelto fuctos of the rel proflogrm d ts model The grde of the rsve pper ccordg to the FEPA A segmet s the tl set for terpolto 4 A polygol le cosstg of segmets s the tl set for terpolto MSE RE MSE RE P P P P

5 Tle 3. The smulto of proflogrms hvg dfferet grde of roughess. The grde of the rsve pper ccordg to the FEPA Model Model P4 P form s follows f f F3 D = Μ Μ fn fn Κ Ο Κ f N Μ ; (8) f NN here N s rtrry d deotes the umer of zoes the model ( N > ) d f j = f j, =, N. The surfce s dvded to zoes log the drecto of the proflogrm, whe the model of surfce roughess (Model ) s lyzed. We set rdom wdth for ech of the zoes. The ech zoe s rdomly moved log the postve drecto of the x xs y the vlue whch s uformly dstruted o the tervl (.;.3 ). The result of ths process s depcted Fg. 8. Flly, the extr prts of the surfce zoes re dsplced to the frot of the model where free spce resdes ow. The three dmesol model of surfce roughess s frctl-stochstc. It s due to the rdomess of ts coordtes d the self-smlrty of the model s profle. Creto of t volves the rdom selecto of the pots of the proflogrm; evertheless, the zoes whch we dvde the model re rdomly offsetted spce The comprso of models wth dfferet surfce roughess Ech type of the proflogrm hs ee vestgted order to compre the models of surfce roughess. Tle 3 shows models formed y performg two tertos of frctl terpolto. A polygol le comprsg of segmets hs ee chose s the tl set for frctl terpolto. The surfce model covers the rel re of mm o the ple xz. Frctl dmeso hs ee used to compre the model of proflogrm to the rel proflogrm. Such comprso s gve Tle 4. A segmet s the tl set Fg. 8. The 3D models of surfces: Model, Model of frctl terpolto hs ee used, tertos hve ee performed d the re of the surfce modelled s mm mm. 4

6 Tle 4. Comprso of models of the surfces The grde of the rsve pper ccordg to the FEPA The surfce re of Model S p, mm The surfce re of Model S m, mm The estmte of frctl dmeso of the rel proflogrm, D p The estmte of frctl dmeso of model of the proflogrm, D m P P P P The three-dmesol models cosdered cover the sme re ( mm ) o the ple xz. By multplyg the legth of the proflogrm s model y ts wdth, the surfce re s clculted. The res of surfces Model d Model re deoted y S p d S m, respectvely. The re S p s greter for the surfce modeled ccordg to the rougher proflogrm. Tle 4 shows tht usg segmet (s the tl set) leds to the estmtes of frctl dmesos D p d D m eg comprle etwee the proflogrms of grde 4 d d ther models. Tle 5 shows the comprso of the models of surfce roughess whe polygol le comprsg segmets s chose. Usg the polygol le (s the tl set) results hgher precso of the models (D p d D m dffer less). The proflogrm of grde 4 s excepto. Ths occurs due to the use of the polygol le whch mkes the model of the proflogrm rougher The relto etwee the re of the rough surfce d the grde of the rsve pper The stregth of the dhesve jots depeds o the re of cotct surfces. Ths dctor depeds ot oly o the geometrcl dmesos of the sustrte, ut lso o the surfce roughess. If the pproprte recept s cosdered, the glue c fll up ll the roughess orgted the process of surfce grdg or polshg. The resultg roughess of the surfce depeds o ts er structure, s well s o the roughess of the rsve pper. The the ssumpto out the relto etwee the grde of the rsve pper d the surfce could e drw. We hve lyzed the relto etwee the re of the rough surfce d the grde of the rsve pper r, referrg to the ssumpto ove. We showed tht the greter the sze of rsve gr (. e. the smller the grde of the rsve pper), the smller the re of the rougheed surfce (Fg. 9). The reltoshp etwee the rel re of the cotct S p d the re of the orgl Tle 5. Comprso of models of the surfces surfce S, c e wrtte s follows:. 5 S p = S.998+ ; (9) R here: S = mm r, R =, r the verge dmeter of r s rsve gr (see Tle ), r s s the dmeter of rsve gr correspodg to the smllest grde of the rsve pper. Accordg to the clculted coeffcet of the determto, ths reltoshp predcts the dt wth ccurcy of %. S p /S,.u R,.u. Fg. 9. The reltoshp etwee the re S p d the vlue R Whle forecstg the re S m, the ler relto fts etter: S m = S (.86. 5R). () Accordg to the clculted coeffcet of the determto, ove reltoshp (Fg. ) predcts the dt wth ccurcy of %. Due to the fct tht the grde of the rsve pper s lwys kow, the equtos oted eles the resercher to predct the re of surfce roughess, ffected y the rsve pper of y grde wthout performg the expermet tself. The grde of the rsve pper ccordg to the FEPA The surfce re of Model S p, mm The surfce re of Model S m, mm The estmte of frctl dmeso of the rel proflogrm, D p The estmte of frctl dmeso of model of the proflogrm, D m P P P P

7 S m /S,.u R,.u. Fg.. The relto etwee the re S m d vlue R 4. CONCLUSIONS The use of frctl terpolto for modellg surfce roughess hs ee vestgted the pper. It hs ee determed tht usg polygol les s tl sets for frctl terpolto s preferle whle costructg the model of proflogrm. The model oted correspods to the rel proflogrm terms of oth the shpe of profle d the degree of fllg the spce. It ws foud y clcultg the utocorrelto fucto d the estmte of frctl dmeso for the rel proflogrm d ts model. The reltoshp etwee the rel re of the rough surfce of utdee-styree ruer d the grde of the rsve pper hs ee determed. So, the resercher s le to predct the re of surfce roughess, ffected y the rsve pper of y grde, wthout performg the expermet tself, ecuse the grde of the rsve pper s lwys kow. REFERENCES. Nem, P., Lu, T., Kw, K. The Effect of Surfce Chrcterstcs of Polymerc Mterls o the Stregth of Boded Jots Jourl of Adheso Scece d Techology (4) 996: pp Pckhm, D. E. Surfce Eergy, Surfce Topogrphy d Adheso Itertol Jourl of Adheso & Adhesves 3 3: pp Prologo, S. G., Rosro, G., Ure, A. Study of the Effect of Sustrte Roughess o Adhesve Jots y SEM Imge Alyss Jourl of Adheso Scece d Techology (5) 6: pp Petrtee, S., Brzdzus, R., Pekrsks, V. The Sttstcl Chrcterstcs d Ther Iterrelto of Arsves d Surfce Corse fter the Process wth Them Mterls Scece (Medžgotyr) (4) 997: pp Ptrkr, R. M. Modelg d Smulto of Surfce Roughess Appled Surfce Scece 8 4: pp To, Q., Lee, H. P., Lm, S.P. Cotct Mechcs of Surfces wth Vrous Models of Roughess Descrptos Wer 49 : pp Yu, Q., L, J., Y, X. P., Peg, Z. The Use of the Frctl Descrpto to Chrcterze Egeerg Surfces d Wer Prtcles Wer 55 3: pp Mjumdr, A., Te, C. L. Frctl Chrcterzto d Smulto of Rough Surfces Wer 36 99: pp Mčėtė, L., Pekrsks, V. P. Ivestgto of Relto etwee Durlty of Adhesve Jots of Soft Polymerc Mterls d Roughess Chrcterstcs of Glued Surfces Mechk 4 6: pp. 6.. Mhovc Poljcek, P., Rsovc, D., Furc, K, Gojo, M. Comprso of Frctl d Proflometrc Methods for Surfce Topogrphy Chrcterzto Appled Surfce Scece 54 8: pp Jh, R., Truckerodt, H. A Smple Frctl Alyss Method of the Surfce Roughess Jourl of Mterls Processg Techology 45 4: pp Brsley, M. F. Frctls Everywhere. d ed. Cmrdge: Acdemc Press Professol, 993: 5 p. 3. Mčėtė, L., Msloeė, K., Pekrsks, V. P. The Modelg of Soft Polymer Mterls Surfce Profles Mterls Scece (Medžgotyr) (4) 4: pp