ANALYSIS OF STEADY WEAR PROCESSES FOR PERIODIC SLIDING

Transcription

1 Joural of Computatioal ad Applied Mechaics, Vol. 1., No. 2., (215), pp ANALYSIS OF STEADY WEAR PROCESSES FOR PERIODIC SLIDING Istvá Páczelt Istitute of Applied Mechaics, Uiversity of Miskolc H-3515 Miskolc, Miskolc Egyetemváros, Hugary Zeo Mróz Istitute of Fudametal Techological Research, A. Pawińskiego 5B, 2-16 Warsaw, Polad Attila Baksa Istitute of Applied Mechaics, Uiversity of Miskolc H-3515 Miskolc, Miskolc Egyetemváros, Hugary [Received: Jue 14, 215, Accepted: September 2, 215] Dedicated to Professor Bara Szabó o the occasio of his eightieth birthday ad to Professor Imre Kozák o the occasio of his eighty-fifth birthday Abstract. The relative slidig motio of two elastic bodies i cotact iduces the wear process ad cotact shape evolutio. The trasiet process at the costat relative velocity betwee the bodies teds to a steady state occurrig at fixed cotact stress ad strai distributio. This state correspods to a miimum of the wear dissipatio power. The optimality coditios of the fuctioal provide a cotact stress distributio ad a wear rate compatible with the rigid body puch motio. The preset paper is devoted to the aalysis of wear processes occurrig for periodic slidig of cotactig bodies, assumig cyclic steady state coditios for mechaical fields. From the coditio of the rigid body wear velocity a formula for summarized cotact pressure i the periodic steady state is derived. The optimizatio problem is formulated for calculatio of the cotact surface shape iduced by wear i the steady periodic state. Mathematical Subject Classificatio: Keywords: steady wear process, periodic slidig, uilateral cotact, p-versio of fiite elemet method, shape optimizatio 1. Itroductio The wear process o the frictioal iterface of two bodies i a relative slidig motio iduces shape evolutio. I may practical idustrial applicatios it is very importat to predict the form of wear shape ad cotact stresses. Usually the simulatio of the c 215 Miskolc Uiversity Press

2 232 I. Páczelt, Z. Mróz, A. Baksa cotact shape evolutio is performed by umerical itegratio of the modified Archard wear rule expressed i terms of the relative slip velocity ad cotact pressure. For cases of mootoic slidig motio the miimizatio of the wear dissipatio power provides the cotact pressure distributio ad rigid body wear velocities directly without time itegratio of the wear rule util the steady state is reached, cf. [1, 2, 3, 4]. (The steady state is reached whe the cotact stress is fixed with respect to the movig cotact domai ad the rigid body wear velocity is costat i time.) The quasi-steady wear state is reached for the stress distributio depedet o a slowly varyig cotact domai S c (t). It is importat that i geeral cotact coditios the vector of wear rate is ot ormal to the cotact surface ad has a tagetial compoet [1]. A fudametal assumptio was itroduced, amely, i the steady state the wear rate vector is colliear with the rigid body wear velocity of a slidig body allowed by boudary costraits. I [1] a ew idea of the wear rate vector ad ew form of the wear dissipatio power was preseted. This ew priciple was applied i the aalysis of the steady wear states i disk ad drum brakes. Next, this approach was exteded i [2] by the authors of previous aalysis to specificatio of steady-state cotact shapes with coupled wear ad thermal distortio effects take ito accout. The wear rule was assumed as a o-liear relatio of wear rate to shear stress ad relative slidig velocity. The aalysis of wear of disks ad drum brakes was preseted with the thermal distortio effect. I [3] a improved umerical aalysis of the thermo-elastic cotact coupled with wear process was developed. The coupled thermo-mechaical problem was umerically treated by applyig the operator split techique. For larger values of relative slidig velocities ad movig frictioal heat fluxes the thermal aalysis requires applicatio of the upwid techique. Neglectig temperature depedece of material parameters, it was cocluded that the cotact pressure distributio i the steadystate is ot affected by temperature field, but the cotact surface shape reached i the wear process strogly depeds o the thermal distortio. A brake system with differet shoes support was ivestigated, derivig the cotact pressure distributio also for the steady wear state. I [4] the umerical aalysis of coupled thermo-elastic steady wear regimes was preseted: wear aalysis of a puch traslatig o a elastic strip ad wear iduced by a rotatig puch o a toroidal surface. The wear ad frictio parameters were assumed as fixed or temperature depedet. The icremetal procedure for temperature depedet parameters was established. Three trasverse frictio models were discussed accoutig for the effect of wear debris motio. It was demostrated, that the cotact pressure distributio depeds oly o the trasformed wear velocities, frictio coefficiet ad wear parameter b, ad is ot depedet o relative velocity ad wear parameter β i (see (1)). The cotact coformity coditio was defied. I the cases of wear of puch ad wear of two bodies the cotact pressure distributio i the steady state is govered by the relative rigid body motio iduced by wear.

3 Aalysis of steady wear processes for periodic slidig 233 O the other had, whe oly wear of the substrate takes place, the cotact pressure distributio is specified from the cotact coformity coditio ad depeds o the elastic moduli of cotactig bodies. I the literature there are umerous works dealig with frettig problems, whe i the cotact domai both adhesio ad slip sub-regios ca develop, [5, 6, 7, 8, 9]. Periodic cotact slidig was treated i some papers, cf. [1, 11]. The extesio of variatioal method is preseted for the case of multi-zoe cotact problems for steady wear states i [12] which both trasiet ad steady states have bee aalyzed. Paper [13] was aimed at extedig the results of previous aalyses [1, 2, 3, 4] of steady state coditios to cases of periodic slidig of cotactig bodies, assumig the existece of cyclic steady state coditios. I the time itegratio it was assumed that cotact pressure distributio is fixed durig the semi-cycle ad varies discotiuously durig slidig reversal i cosecutive semi-cycles. The p-versio of the fiite elemet method is well suited for solvig the cotact problems with high accuracy, usig the bledig techique for approximatio of the shape. Wear predictio was made i the brake system by usig the averagig techique of results from mootoic motios. The cotact pressure distributio has bee derived i the discretized form for 3 cases usig the Gree fuctios. Case 1 : wear of both puch ad substrate, Case 2 : wear of substrate oly, Case 3 : wear of puch oly. I particular, the body B 1 ca be regarded as a puch traslatig ad rotatig relative to the substrate. Several classes of wear problems ca be distiguished ad discussed for specified loadig ad support coditios for two bodies i the relative slidig motio. Class 1 : The cotact zoe S c is fixed o oe of the slidig bodies (like a puch) ad traslates o the surface of the other body (substrate). The rigid body wear velocity is compatible with the specified boudary coditios. The steady state coditio is reached whe the cotact pressure distributio correspods to the wear rate proportioal to the rigid body velocity [2, 3]. The relative velocity betwee the bodies is costat i time. Class 2 : Similarly to Class 1, the cotact zoe S c is fixed but the wear process occurs for periodic slidig motio. Class 3 : Similarly to Class 1, the relative velocity is costat, but the load is periodic i time. Class 4: Similarly to Class 2, the cotact zoe is fixed, but the wear process reaches the steady state for periodic load ad periodic slidig motio (for istace i the brakig process). I the case of Class 1, from miimizatio of the wear dissipatio power it is easy to derive the formulae for cotact pressure distributio [2, 3]. Paper [6] presets the aalysis of wear for the case of periodic slidig of cotactig bodies, assumig cyclic steady state coditios ad takig ito the heat geeratio at the cotact surface. I particular, the body B 1 ca be regarded as a puch traslatig ad rotatig relative to the substrate B 2. It is assumed that strais are small ad the materials of the cotactig bodies are liearly elastic. I discretizatio of the cotactig bodies for the displacemet ad temperature determiatio, the p-versio of fiite elemets was used [13, 14], assurig fast covergece of the umerical process ad providig a high level accuracy of geometry for shape optimizatio.

4 234 I. Páczelt, Z. Mróz, A. Baksa The specific examples are related to the aalysis of puch wear iduced by reciprocal slidig alog a rectiliear path o a elastic strip. The exteral loads actig o the puch are ot symmetric. Specifyig the steady state cotact pressure distributios for a arbitrarily costraied puch, it is oted that the pressure at oe cotact edge vaishes, ad the maximal pressure is reached at the other edge. It was show that by summarizig pressure values for cosecutive semi-cycles, a resultig distributio is obtaied that correspods to the rigid body displacemet of the puch. The aalysis of the same example takig heat geeratio ito accout demostrates that the thermal distortio affects essetially the cotact shape ad the trasiet cotact pressure distributio [15]. However, it was show that i the steady wear state for reciprocal slidig, the cotact pressure reaches the same distributio as that obtaied for the case of eglect of heat geeratio, but the steady state cotact shapes are differet. I the case of periodic slidig motio, the steady state cyclic solutio should be specified ad the averaged pressure i oe cycle ad the averaged wear velocity ca be determied from the averaged wear dissipatio i oe cycle. I our ivestigatio betwee the bodies it was assumed that the stick zoe o loger exists ad the whole cotact zoe udergoes slidig. The tagetial stress ca the be directly calculated from the cotact pressure ad the coefficiet of frictio. 2. Wear rule ad wear rate vector The modified Archard wear rule [1] specifies the wear rate ẇ i, of the i-th body i the ormal cotact directio. Followig previous work [1, 2] it is assumed that ẇ i, = β i (τ ) bi u τ ai = β i (µ p ) bi u τ ai = β i (µ p ) bi v ai r = β i p bi v ai r, i = 1, 2 (1) where µ is the frictio coefficiet, β i, a i, b i are the wear parameters, βi = β i µ bi, v r = u τ is the relative tagetial velocity betwee the bodies, costraied by the boudary coditios. Figure 1. Referece frame ad wear rate vectors o the cotact surface S c. Coaxiality rule of ẇ R ad e R. The shear stress at the cotact surface is expressed i terms of the cotact pressure p by the Coulomb frictio law τ = µp. I geeral cotact coditios the wear rate vector ẇ i is ot ormal to the cotact surface ad results from the costraits imposed o the rigid body motio of puch B 1. Itroducig the local referece triad

5 Aalysis of steady wear processes for periodic slidig 235 e τ1, e τ2, c o the cotact surface S c (see Figure 1), where c is the uit ormal vector, directed ito body B 2, i is the uit surface ormal of the i-th body, e τ1 is the uit taget vector coaxial with the slidig velocity ad e τ2 is the trasverse uit vector, the wear rate vectors of bodies B 1 ad B 2 are ẇ 1 = ẇ 1, c + ẇ 1,τ1 e τ1 + ẇ 1,τ2 e τ2, ẇ 2 = ẇ 2, c ẇ 2,τ1 e τ1 ẇ 2,τ2 e τ2. (2) The cotact tractio o S c ca be expressed as follows [3] t c = t c 1 = t c 2 = p ± ρ ± c, ρ ± c = c ± µ e τ1 + µ d e τ2 (3) where ρ ± c specifies the orietatio ad magitude of tractio t c with referece to the cotact pressure p ad µ d is the trasverse frictio coefficiet. The sig + i (2) correspods to the case whe the relative tagetial velocity is u τ = u (2) τ u (1) τ = u τ e τ1 = v r e τ1 with the correspodig shear stress actig o the body B 1 alog e τ1. The fudametal coaxiality rule was stated by Páczelt ad Mróz [1, 2, 3, 4], amely: i the steady state the wear rate vector ẇ R is colliear with the rigid body wear velocity vector λ R, thus ẇ R = ẇ R e R, e R = λ R λ λ F + = λ M r R λ F + λ, (4) M r where r is the positio vector. The coaxiality rule is illustrated i Figure 1. The ormal ad tagetial wear rate compoets ow are ẇ = ẇ R cos χ, ẇ τ = ẇ R si χ = ẇ ta χ (5) where χ is the agle betwee c ad e R. The wear rate compoets i the tagetial directios are ẇ τ1 = w R si χ cos χ 1, ẇ τ2 = w R si χ si χ 1 (6) where the agle χ 1 is formed betwee the projectio of ẇ R o S c ad e τ,1 as show i Figure 1. Let us ote that the slidig velocity v r = u τ is specified from the boudary coditios ad the wear velocity vectors λ F ad λ M should be determied from the solutio of a specific problem. I the aalysis of slidig wear problems the elastic term of relative slidig velocity is usually eglected. 3. Steady state coditios for mootoic motio It has bee show i [1, 2] that the steady state coditios for mootoic motio ca be obtaied from miimizatio of the wear dissipatio power subject to equilibrium costraits for body B 1. The wear dissipatio power for the case of wear of two bodies equals 2 2 D w = (t c i ẇ i ) ds = C i. (7) i=1 S c i=1

6 236 I. Páczelt, Z. Mróz, A. Baksa The global equilibrium coditios for the body B1 ca be expressed as follows Z ± f = ρ± c p ds + f = Sc Z m= (8) ± r ρ± c p ds + m = Sc where f ad m deote the resultat force ad momet actig o the body B1. The formula for cotact pressure at steady wear state ca be foud i papers [1, 2, 4] ad for multi-cotact zoe cases the cotact pressure s formula ca be foud i [12]. 4. Wear dissipatio i periodic motio ad summed pressure i periodic steady wear state I this sectio we shall aalyze the wear process iduced by the reciprocal strip traslatio. It is assumed that oly the puch udergoes wear (see Figure 2), that is i our case β 1 6=, β 2 =. a) b) Figure 2. Periodic slidig o the cotact iterface betwee puch ad strip. The umber of fiite elemets i body 1 alog the x directio is 8, ad i vertical z directio is 7. The lies are drow through the Lobatto itegral coordiates. I the aalysis the cotact pressure distributio is assumed as fixed durig semicycle ad varies discotiuously durig slidig reversal i cosecutive semi-cycles. The temperature distributio varies cotiuously durig each cycle period [15]. The coupled thermo-mechaical problem was solved by the operator split techique [16]. The wear effect is calculated icremetally by applyig the Archard type wear rule (1). The wear is accumulated at the ed of the half period of motio, so the cotact pressure is fixed (at the iteratio level), ad the trasiet heat coductio problem is ext solved for the give temperature field at the begiig of the half period.

7 Aalysis of steady wear processes for periodic slidig 237 The steady state cotact pressure distributio i the wear process iduced by periodic slidig does ot deped o the value of wear factor β 1 or geerated temperature field, but the wear iduced cotact surface shape is strogly affected. Durig the steady periodic respose the wear icremet accumulated durig oe cycle should be compatible at each poit x S c with the rigid body puch motio. The wear dissipatio work for periodic motio is E w = 1 T /2 2 (t c+ i ẇ + i 2 ) ds dτ i=1 S (i) c T i=1 T /2 S (i) c (t c i ẇ i ) ds dτ (9) where t c+ i, t c i is the cotact tractio vector ad ẇ + i, ẇ i is the wear velocity of the i-th body i the progressive ad reciprocal motio directio, T is the period of slidig motio, T = 2π/ω. Figure 3. The wear process occurrig o the cotact iterface betwee puch ad strip traslatig with the relative velocity v r = u ω si ωτ, u τ = v r e τ1. The segmet MÑ of the substrate takes part i the wear process. I our case the tagetial velocity of body 2 is (see Figure 3): u τ = u (2) τ u (1) τ = u ω si ωτ e x = u ω si ωτ e τ1 = v r e τ1 (1) with the correspodig shear stress actig o the body B 1 alog e τ1. The itegral of the relative velocity betwee the bodies is T /2 v r dτ = T T /2 v r dτ = 2u. (11) I view of the wear rule (1) the wear dissipatio for the puch of Figure 2 is E w = 1 T /2 1 p + ẇ 1, + 2 ds dτ + 1 T 1 p ẇ1, 2 ds dτ (12) i=1 S (1) c i=1 T /2 S (1) c

8 238 I. Páczelt, Z. Mróz, A. Baksa ad for β 1, β 2 =, a 1 = b 1 = 1 there is E w = 2u β1 S (1) c { ( p + ) 2 ( ) + p 2 E + )} ds = w + 2u β1 E w 2u β1. (13) I the steady wear state E w reaches a miimum value. Let us ote that p + ad p are ot uiformly distributed at the cotact iterface. Takig the coordiate x = 113 x it ca be stated that p(x) = p( x) durig the cosecutive semi-cycles of reciprocal slidig. It is very importat that durig the steady wear periodic state the wear icremet accumulated durig oe cycle should be compatible at each poit p(x) = p( x) with the rigid body puch motio. The mai idea for derivatio of the wear icremet ad summed pressure for 2D system with cylidrical cotact surface is collected i the Appedix. Assume the rigid body wear velocities for left ( ) ad right (+) directios of the substrate i the followig λ F = λ F e z, λ M = λ M e y, λ+ F = λ + F e z, λ+ M = λ + M e y. (14) Thus the velocities at a arbitrary poit at the puch, Figure 2b, are ( λ + F + λ + M x)e z, or ( λ F λ M x)e z. The displacemets resultig from this velocities are where ( λ + F + λ+ M x)e z ad ( λ F λ M x)e z (15) λ + F,M T /2 = λ + F,M dt, λ F,M T T /2 Thus, the total wear accumulated durig oe slidig cycle is λ F,M dt. w = w + + w = ( λ F + λ+ F ) ( λ M λ+ M ) x. (16) This value of wear ca be calculated from the wear law supposig β 1, β 2 =, a 1 = b 1 = 1, thus accordig to (A.12) where w = w + + w = Q (p + + p ) = 2 Q p m = Q p Σ (17) p m = (p + + p )/2 = p Σ /2 ad Q = β 1 T /2 u τ dt. Comparig (16) ad (17), it is see that the distributio of the sum of cotact pressure values of cosecutive semi-cycles must be a liear fuctio of positio, thus p m = p C m + p L m x. (18)

9 that is Aalysis of steady wear processes for periodic slidig 239 w = w + + w = ( λ F + λ+ F ) ( λ M λ+ M ) x = β 1 T u τ dt 2 (p C m+p L m x), where λ ± is the icremet of rigid body wear velocities i the half period time. F,M Usig the equilibrium equatios for summed loads, the summed pressure for the steady wear state is determied as p C m = F S c 3F ( L + 2 x F ) L S c, p L m = 6F ( L + 2 x F ) L 2 S c, p Σ = 2p m = p + + p = 2(p C m + p L m x) where x F is the coordiate of the resultat load F = F (p ). For o-egativity of p m there should be L/2 x F 2L/3. At x F = L/2 the results of [5, 6] are obtaied. Here S c is the area of cotact zoe. The wear icremet i oe period (ote that the cotact pressure is fixed i half period) equals w 1, = β 1 [ p + + p ] (u ω) which usig (11) provides the simple relatio T /2 (19) si ωτ dτ, (2) w 1, = β 1 [ p + + p ] 2u = Q p Σ, (21) where Q = β 1 2u ad the averaged wear rate i oe period equals ẇ 1, = w 1, T = β 1 [p + + p ] T 2u. (22) If the rigid body wear velocity λ + M = λ M =, (at the supports see Figure 2a), the i the steady periodic wear regime the uiform wear icremet is accumulated durig full cycle at each poit of the cotact zoe ad the followig coditio should be satisfied: p + + p = 2p m = cost. (23a) Remark: If a 1 = 1 b = b 1 1 the i a periodic steady state there must be ( p + ) b + ( p ) b = 2(pm ) b = cost2 (23b) where p m is the cotact pressure at the ceter of the puch cotact zoe, at x = 11. Because at x = 17, p = the cotact pressure is p + (x = 17) = 2 1/b p m At the other perimeter at x = 113 it holds that p + = ad p (x = 113) = 2 1/b p m. (23c) (23d)

10 24 I. Páczelt, Z. Mróz, A. Baksa Performig time itegratio of the wear rate rule for a 1 = b 1 = 1 durig the oe half period, the wear icremet is calculated i the followig way: tp+t /2 w (j) 1, = t p β1 p (j) (τ) u ω si ωτ dτ = β 1 p (j) (t p + T /2) where t p is the time of start of the half period, p (j) T /2 = p (j) (t p + T /2). u ω si ωτ dτ The accumulated wear at the ed of half period at the iteratioal step j equals or i other otatio (24) w (j) 1, (t p + T /2) = w 1, (t p ) + w (j) 1, = w 1,(t p ) + β 1 p (j) 2 u (25) w (j) 1, (t p + T /2)= tp+t /2 w (j) 1, = tp w 1, + w (j) 1,. (26) This j type iteratioal process is repeated util j = J whe the followig covergece criterio for cotact shape is satisfied, thus ( ) ( ) e w = 1 tp g+ w (j) 1, ds tp g+ w (j 1) ( ) 1, ds / tp g + w (j 1) 1, ds S c Here tp g is the iitial gap at the begiig of the half period. S c S c.1. (27) Remark: If a 1 = 1, b = b 1 1 the the wear icremet durig the oe half period is w (j) 1, = β ( ) b 1 p (j) 2 u. (28) I practical calculatios the iterative scheme of cotact pressure ad wear shape correctio ca be modified after k half cycles, so we ca write t p+kt /2 w 1, = tp w 1, + k w (J) 1,. (29) I our case we chose the extrapolatio factor k i the followig way: for the umerical steps 5, k = 1; for 5 < 1, k = 5 ad whe > 1, the k = 1. The umber of the half periods i the iterval the is 5 1 hp = 5 + ( 5) 5, 1 hp = 3 + ( 1) Examples 5.1. Example 1: wear of puch iduced by periodic slidig of the substrate. Let us aalyze the wear of the puch (Body 1) show i Figure 2. We would like to examie two types of costraits, oe whe the puch ca move oly i the vertical directio (see Figure 2a), ad secod whe the puch has additioal rotatio aroud a pi (see Figure 2b). The poit M i the puch has coordiates: x = 17, z = 1. (3)

11 Aalysis of steady wear processes for periodic slidig 241 The followig geometric parameters are assumed: the puch width is L = 6 mm, its height is h = 1 mm, the thickess of puch ad strip is t th = 1 mm. The wear parameters are: β 1 = 1.25π 1 8, β 2 =, a 1 = 1, b 1 = 1, the coefficiet of frictio is µ =.25. The horizotal displacemet of the substrate is u τ = u cos ωτ, where u = 1.5 mm, ω = 1 rad/s, τ is the time. The material parameters are preseted i Table 1. Table 1. Mechaical parameters of two materials Youg modulus Material desity Poisso ratio MPa kg/m 3 Material 1 (steel) Material 2 (composite) The upper parts of the puch ad strip are assumed to be made of the same material, (Material 1, see Table 1). The lower puch portio of height 2 mm is characterized by the parameters of Material 2 (see Table 1) Symmetric load. The puch is loaded o the upper boudary z = 2 mm by the uiform pressure p = MPa correspodig to the resultat vertical force F = 1. kn. The wear parameter is b = 1. This problem was eglectig the heat geeratio i [13], ad accoutig for heat geeratio i [15]. The umerical results of paper [13] are collected i Table 2 ad i Figure 4 for l z = 4mm. Let us deote the cotact pressure for the puch of Figure 2a by p ( λ F ), for the puch of Figure 2b for l z = 2 ad l z = 4 by p ( λ F, λ M, l z = 2) ad p ( λ F, λ M, l z = 4), respectively. After time itegratio of the Archard wear rule the cotact pressures at poit M are collected i Table 2 versus the umerical time steps for differet puch costraits. It is clear that covergece to the pressure MPa proceeds for all cases of costraits. I the case l z = 4mm the evolutio of the shape ad cotact pressure is demostrated i Figure 4. Because the loadig distributio is symmetric, the distributio of the pressure ad shape is also symmetric. The optimal solutios (marked by... ) correspod to the mootoic relative motio. Also it is observed that after 15 the pressure distributio does ot chage ad the cotact profile is preserved, movig alog the puch axis like a rigid lie. I this case λ F = λ+ F, λ M = λ+ M, that is i the wear process the accumulated puch wear is the same durig each period, the pressure distributio is p m = p C m = p, ad the summed pressure p Σ = p + + p = 2p m = 2p.

12 242 I. Páczelt, Z. Mróz, A. Baksa Table 2. Cotact pressure at the poit M versus umerical time steps o. of half period hp p ( λ F ) p ( λ F, λ M, l z = 2) p ( λ F, λ M, l z = 4) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E F =1. kn, Cotact pressure Optimal solutios: (..) =15 (. ) =2 ( ) =25 ( ) =3 (.) Rightward slidig directio F =1. kn, Shape evolutio [mm] =3 p Leftward slidig directio Shape [mm] =25 = x~=113.3 = x~=113 a) b) Figure 4. a) Cotact pressure at differet time steps ad slidig directios, b) evolutio of shape of puch for reciprocal motio, l z = 4 mm, the load p = MPa ad resultat force s coordiate is x F = L/2. The wear parameter is b 1. Let us ivestigate the periodic wear process at β 1 = 1.25 π µ.2 1 8, β2 =, ad a 1 = 1, b = b 1 = 1.2, µ =.25. The displacemet of body 2 is: u = u cos ωτ, where u = 1.5 mm, ω = 1 rad/s. Performig time itegratio of (1) we see that after the umber of half periods ( 11) hp 13 the wear process reaches its steady state. I this case the value 2 (p m ) b = cost2 = 61.48, where p m = MPa, p + = p = MPa.

13 Aalysis of steady wear processes for periodic slidig 243 Comparig the cotact pressure ad shape of puch i the steady state, we see that the pressure for b = 1.2 at the border of cotact zoe is lower tha that for b = 1, ad at the ceter of the puch the pressure is higher. The cotact shape for b = 1.2 is show by the curve placed above that predicted for b = 1 (cf. Figure 5). O the other had, for the case b =.8, the cotact pressure is higher tha that for b = 1 at the perimeter poits, ad the cotact shape curve is lower tha that for b = 1. It also is oted that for the wear parameter value β 1 = 1.25 π µ.2 1 8, which is the smallest, the steady state is reached at = 25. The the pressure i the ceter is p m = MPa, the value 2 (p m ) b = cost2 = , the pressure at the perimeter poits are p + = p = [ MPa ad calculated value is (p + ) b = (p ) b = The calculatio error 1 (p + ) b 2 (p m ) b] /2 (p m ) b =.55 is very small. Also p + (x = 17) = 2 1/b p m, p m = MPa. p F =1. kn, Cotact pressure by differet wear parameter b =3, b=.8 (.) =15, b=1 ( ) =15, b=1.2 ( ) Rightward slidig directio Shape [mm] 6 x F =1. kn, Shapes by differet wear parameter b [mm] =15, b=1.2 ( ) =15, b=1 ( ) =3, b=.8 (.) 1 5 Leftward slidig directio x~= x~=113 a) b) Figure 5. The effect of the wear parameter b o the periodic steady wear state, a) distributio of the cotact pressure, b) cotact shape of puch (maximal shape fuctio ordiate is 6 µm) No-symmetric load. Let us ow aalyze the case of eccetric load whe the resultat vertical force equals F = 1 kn ad its positio coordiate is i the iterval L/2 = 3 x F 2L/3 = 4. The first case. The pressure p = 2 MPa is applied i the iterval 1 x 6. The resultat positio coordiate is x F 35 mm. This load case represets the variat 2. The results are preseted i Figure 6. Figures 6a,c demostrate the pressure at differet umbers of half cycles ad Figures 6b,d preset the summed pressure p Σ = p + + p = 2p m. It is see that after 1 the summed pressure practically does ot chage. Its distributio is preseted by a liear fuctio. A small oscillatio is observed because i the solutio of the cotact problem the positioal techique has ot bee used [17]. I our calculatio it is required that the gap i poit x = 113 mm of the cotact zoe be fixed durig cosecutive iteratios. The cotact shapes are show i Figures 7a,b. If the pi height is l z = 2mm, the obtaied pressure

14 244 I. Páczelt, Z. Mróz, A. Baksa F =1 kn variat 2, l z =4 mm, p evolutio =1:.., =5:, =1: =15:., =25: o, =35: F =1 kn variat 2, l z =4 mm, 2p m evolutio =1:.., =5:, =1: =15:., =25: o, =35: + Pressure p Rightward slidig directio Leftward slidig directio Pressure 2p m x[mm] 6 5 a) b) F =1 kn variat 2, l z =4 mm, p evolutio =5:, =55:, =6:. =65: o, =7: x[mm] 6 5 F =1 kn variat 2, l z =4 mm, 2p m evolutio =5:, =55:, =6:. =65: o, =7: + Pressure p Rightward slidig directio Leftward slidig directio Pressure 2p m x[mm] 6 5 c) d) F =1 kn variat 2, l z =2 mm, p evolutio =5:, =55:, =6:. =65: o, =7: x[mm] 6 5 F =1 kn variat 2, l z =2 mm, 2p m evolutio =5:, =55:, =6:. =65: o, =7: + Pressure p Pressure 2p m x[mm] x[mm] e) f) Figure 6. Periodic wear process for the load variat 2: a), c), e) evolutio of pressures, b), d), f) evolutio of the summed pressure p Σ = 2p m.

15 Aalysis of steady wear processes for periodic slidig F =1 kn variat 2, l z =4 mm, Shape evolutio [mm] =5:, =1:, =15:. =25: o, =35: F =1 kn variat 2, l z =4 mm, Shape evolutio [mm] =5:, =55:, =6:. =65: o, =7: Shape [mm].8.6 Shape [mm] a) b) F =1 kn variat2, l z =2 mm, Shape evolutio [mm] =1:, =1:, =15:. =2: o, =35: F =1 kn variat 2, l z =2 mm, Shape evolutio [mm] =5:, =55:, =6:. =65: o, =7: Shape [mm].8.6 Shape [mm] c) d) Figure 7. Evolutio of cotact shapes for periodic wear process at the load variat 2: a), c), = 1 35, b), d) = 5 7. distributio is show i Figure 6e, ad the summed pressure is show i Figure 6f. The cotact shapes are preseted i Figures 7c,d. Because i the equatio for summed pressure (19) the height l z is abset, the summed pressures for l z = 2 mm, ad l z = 4 mm must be the same. This fact is also demostrated by the umerical time itegratio results (compare Figures 6d ad 6f). The secod case. The pressures p 1 = 25 MPa, p 2 = 12.5 MPa act i the itervals: p 1 : 3 x 6, p 2 : 1 x 3. The resultat vertical force is F = 1 kn, the resultat positio coordiate is x F 37.5 mm. This load case correspods to variat 3. The pressure distributio ca be see i Figures 8a,c ad the summed pressure i Figures 8b,d. The maximum of the pressure is higher tha before, because the resultat coordiate x F is larger by 2.5 mm. I this case the high pressure at the border of cotact domai very quickly decreases. For the periodic steady wear state the maximum of the pressure ca be calculated from the summed pressure, which is predicted without time itegratio! For each half period the cotact gap has bee

16 246 I. Páczelt, Z. Mróz, A. Baksa specified. For the rightward slidig directio the maximum of the cotact pressure is o the left border of cotact zoe, but for the leftward slidig directio the maximum is i the iterior of the cotact zoe. Pressure p F =1 kn variat 3, l =4 mm, p evolutio z =1:.., =5:, =1: =15:., =25: o, =35: + Rightward slidig directio Leftward slidig directio Pressure 2p m F =1 kn variat 3, l z =4 mm, 2p m evolutio =1:.., =5:, =1: =15:., =25: o, =35: Pressure p x[mm] a) b) F =1 kn variat 3, l z =4 mm, p evolutio =25:, =35:, =45:. =6: o, =65: + Rightward slidig directio Leftward slidig directio Pressure 2p m x[mm] F =1 kn variat 3, l z =4 mm, 2p m evolutio =25:, =35:, =45:. =6: o, =65: x[mm] x[mm] c) d) Figure 8. Periodic wear process for the load variat 3: a), c) evolutio of the pressures, b), d) evolutio of the summed pressure p Σ = 2p m. The wear is larger for loadig variat 3 tha for the variat 2. However, the character of wear process is the same. The shape evolutio is preseted i Figures 9a,b Example 2: Periodic steady wear state for a disk brake system. Cosider the periodic tagetial relative displacemet of body B 2 (the disk) with respect to body B 1 i the directio e τ u τ = u cos ωτ e τ (31) where u ad ω are the amplitude ad agular velocity of the motio.

17 Aalysis of steady wear processes for periodic slidig F =1 kn variat 3, l z =4 mm,shape evolutio [mm] =5:, =1: =15:., =25: o, =35: F =1 kn variat 3, l z =4 mm, Shape evolutio [mm] =25:, =35:, =45:. =6: o, =75: Shape [mm].15.1 Shape [mm] a) b) Figure 9. Evolutio of the cotact shapes for periodic wear process at the load variat 3: a) = 5 35, b) = The relative slidig velocity ad the cycle period are u τ = u ω si ωτ e τ = v r e τ (32a) v r = u τ = ω u si ωτ = v si ωτ, T = 2π ω, v = ω u (32b) z [mm] a) b) Figure 1. Brake system, a) α = 3, the resultat force F = 1 kn, thickess of bodies t th = 1 mm; b) fiite elemet mesh of the half part of the system, umber of cotact elemets is 11, umber of elemets i radial directio is 4, the p-versio of the fiite elemets has p = 8 polyomial degree. The liers are draw through the Lobatto itegral coordiates. The shoe (body B 1 ) is loaded by the force F = F e z. I our case F = 1 kn. The Lagragia multiplier λ F represets the vertical wear velocity.

18 248 I. Páczelt, Z. Mróz, A. Baksa p =1 F =1. kn, Cotact pressure Pressure (p (+))+(p ( )): * Rightward slidig directio: Leftward slidig directio: Pressure p σ theory: o p F =1. kn, Cotact pressure 25 =32 =4:, * Pressure =1: (p, * * (+))+(p ( )): * =18:., *. * 2 Rightward slidig directio: Presssure (p (+))+(p ( )): * Leftward Rightward slidig slidig directio: Leftward slidig directio 15 Pressure p σ theory: o Presssure p σ theory: o p F =1. kn, Cotact pressure 25 =42 Pressure (p (+))+(p ( )): * 2 Rightward slidig directio: Leftward slidig directio: 15 Pressure p σ theory: o 1 a) b) p F =1. kn, Cotact pressure 25 =65 Pressure (p (+))+(p ( )): * 2 Rightward slidig directio: Leftward slidig directio: 15 Pressure p σ theory: o p F =1. kn, Cotact pressure 25 =92 =92 Presssure (p (+))+(p ( )): * 2 Pressure (p (+))+(p ( )): * Rightward slidig directio: 15 Leftward slidig directio: Pressure p σ theory: p σ theory: o o 1 c) d) p F =1. kn, Cotact pressure 25 =121 Pressure (p (+))+(p ( )): * 2 Rightward slidig directio: Leftward slidig directio: 15 Pressure p σ theory: o e) f) Figure 11. Cotact pressure ad summed pressure (p Σ = p σ theory) distributio at differet time steps.

19 Aalysis of steady wear processes for periodic slidig 249 It is easy to calculate the average ormal wear rate for body 1. (The ormal wear vector is ẇ 1, = ẇ 1, c ) ẇ 1, = 1 T T The vertical average wear rate is β 1 p b v a1 r dτ = β 1 p b T T v a1 r dτ = β 1 p b v a1 r (33) ẇ R = ẇ 1, / cos α (34) I view of (32)-(34), the average vertical wear rate i oe period equals ẇ R = 1 β 1 [p +b + p b ] (u ω) a1 cos α T which for a 1 = b = 1 provides the relatio T /2 si ωτ a1 dτ (35a) ẇ R = 1 β 1 [p + + p ] 2u, w R = ẇ R T = 1 cos α T cos α (p+ + p ) β 1 2u where w R is the vertical wear icremet for oe motio cycle. (35b) Let us ote that p + ad p are ot uiformly distributed o the cotact iterface. To assure the uiform wear icremet w R accumulated durig full cycle at each poit of the cotact zoe, the followig coditio should be satisfied accordig to results of (A.2) i the Appedix: where w R = w 1, cos α = Qp Σ cos α = Q2pC m = cost (36) p Σ = p + + p = 2p m = 2p C m cos α, Q = β 1 2u (37) The wear parameters are β 1 =.5π 1 8, β2 =, a 1 = b 1 = 1. The slidig parameters are u = 1.5 mm, ω = 1 rad/s. Usig time itegratio of the wear rule i the usual way, the obtaied cotact pressure evolutio is demostrated i Figure 11 at the begiig of umerical steps = 1. The umber of the half periods is calculated by (1). With icreasig umber of cycles coditio (37) is progressively better satisfied, see Figure 11. Here p Σ = p Σ (α) = (p + + p ) = 2p C m cos α. At = 42, p Σ () = MPa, at = 65 p Σ () = MPa, at = 72 p Σ () = 1.93 MPa, at = 87 p Σ () = 1.71 MPa, at = 92 p Σ () = 1.69 MPa, at = 1 p Σ () = 1.66 MPa ad at = 121 p Σ = 1.62 MPa, that is p Σ () = 2p C m. The value of p Σ () as the fuctio of is demostrated i Figure 12. At the begiig of the wear process the drop i pressure p Σ () is very high, ext it expoetially decreases to the value p Σ () = 2p m = 1.57 MPa. This value is calculated from (A.15). The evolutio of the cotact shape is also iterestig. I the iitial phase the wear is high i the middle cotact portio, ad ext the shape teds to its steady form, which traslates vertically as a rigid lie, see Figure 13.

20 25 I. Páczelt, Z. Mróz, A. Baksa 7 F =1. kn, Summa of cotact pressures at the x= Presssure (p (+))+(p ( )) Figure 12. Satisfyig the costrait of uiform vertical wear icremet. 3.5 x 1 3 F =1. kn, Vertical shape evolutio [mm] =1.7 F =1. kn, Vertical shape evolutio [mm] = =13 Shape [mm] Shape [mm] =11 =9 =7 =5 Shape [mm] 1 =4 =5.5 =3 =2 = a) b) F =1. kn, Vertical shape evolutio [mm] =2 =15 =25 Shape [mm] =3 = F =1. kn, Vertical shape evolutio [mm] =55 =5 =45 =4 =35 = c) d) Figure 13. Evolutio of cotact shape i the wear process.

21 Aalysis of steady wear processes for periodic slidig 251 F =1 kn, Optimal vertical shape for shoe with oscillatio of the disk.1 Shape from average momoto slidig motio: ( ).9 Shape from time itegratio, =65: ( ).8.7 Shape [mm] Figure 14. Predictio of cotact shape from the averaged mootoic slidig motio betwee the shoe ad disk. The averagig techique for predictio of the shape form [13] provides a overestimated wear form, see Figure 14. I [13] it was demostrated that the shape does ot deped o the coefficiet of frictio. At the ceter poit the averaged shape fuctio has the value.753 mm, ad the shape fuctio obtaied by time itegratio of the wear rule has the value.7456 mm. The error is less tha 1%. A small asymmetry has bee foud from time itegratio. 6. Optimizatio problem 6.1. Specificatio of the iitial wear form. Let us aalyze the wear of the puch (Body 1) show i Figure 2a. We would like to fid the steady cotact shape for periodic motio usig the results of mootoic strip slidig i the leftward or rightward directio ad develop a ew optimizatio techique. The puch ow is allowed to execute a rigid body wear velocity λ F [13, 15] which is ormal to the cotact iterface. The optimal pressure for steady wear state is uiform, p + = p = p. The calculatio of the iitial gap that is the wear shape is performed by loadig separately each body by the optimal cotact pressure ad frictio stress. I this case the bodies are ot allowed rigid body motio i the vertical directio. For mootoic slidig the equatio requirig the total cotact gap to vaish specifies the wear gap g, thus d = u (2) u (1) λ F + g = (38) where the rigid body wear velocity of the puch is kow from the statioary coditio, so that λ F = λ F t s, where t s is the selected time istat specifyig iitiatio of the steady state. The steady state shapes ca be foud i Figure 15, where at leftward slidig it is set at g(x = 17) =, ad at rightward slidig set at g(x = 113) =.

22 252 I. Páczelt, Z. Mróz, A. Baksa.35.3 F =1. kn, "Average shape" at steady state Average shape : Optimal solutio at leftward slidig: Shape [mm] Optimal solutio at rightward slidig: o x~=113 Figure 15. Shapes for the steady wear states iduced by the strip mootoically traslatig i leftward or rightward directios. The average shape is Shape (a) Deotig by shape (l), shape (r) the resultig wear shape curves durig the leftward ad rightward mootoic slidig (see Figure 15), assume the shape curve for reciprocal Wear 5 x shapes for average mootoic ad for periodic motio at steady state 1 3 Shape [mm] Average shape from leftward ad rightward mootoic slidig motio : + Shape for periodic motio at =1: Shape for periodic motio at =15: x~=113 Figure 16. Predictio of the wear shape for periodic wear process from the results of mootoic slidig. The average shape is Shape (a) slidig to be approximated by the sum of mootoic shape curves, thus Shape (a) = shape (l) + shape (r) 2 shape (l) (x = 11), (39) where the last term specifies the traslatio of the curve alog the z-axis i order to obtai the zero value at the mid cotact poit (see Figure 16). It is see that the predictio is ot close to the actual wear form at the cotact edges. It is also oted that shapes at = 1 ad = 15 are practically the same, so the wear process has reached its steady state at = 1.

23 Aalysis of steady wear processes for periodic slidig x 14 F =1. kn, E w E w ( ) x 1 4 Figure 17. Evolutio of the wear dissipatio E w /2u β1 = S c (p ) 2 ds The evolutio of the wear dissipatio eergy for oe cycle is plotted i Figure 17. The cotiuous lie correspods to the leftward ad the dotted lie to the rightward slidig directio of the substrate. The wear dissipatio eergy very quickly decreases ad teds to its miimum level. a b Figure 18. δw = w 1, mi w 1, = ( w 1, + + w 1, ) mi( w 1, + + w 1, ) at differet time periods, a) at the begiig of the wear process, b) i the steady state.

24 254 I. Páczelt, Z. Mróz, A. Baksa Theoretically calculatig the value δw = w 1, mi w 1, (4) it is expected that i the steady state it must vaish. Here δw is the wear differece after oe slidig period. I Figure 18 the evolutio of δw is show at the begiig of the wear process (a), ad at the periodical steady state (b). I the iitial period of the wear process, δw is 1 times greater tha that at the steady state. I the steady state it reaches a stabilized small value. It seems to be impossible to reduce δw to zero i the umerical calculatio process Solutio of the optimizatio problem usig splies. I view of the precedig aalysis, the followig optimizatio problem ca be stated for calculatio of the wear shape mi g { max δw = w 1, max w 1, mi p ±, d ±, p ± d ± =, τ + = µp +, τ = µp, f =, m = } (41) where w 1, max, w 1, mi are the maximum ad miimum values of wear attaied i oe cycle, f =, m = are the puch equilibrium equatios. The global equilibrium coditios for the body B 1 ca be expressed as (8). Accordig to Sigorii cotact coditios i the ormal directio the cotact pressure must be positive i the cotact zoe ad distace after deformatio betwee the bodies must also be positive, thus d ± = u (2)± u (1)± + g (42) where u (i) = u (i) c is the ormal displacemet of the i-th body, g is the iitial gap (shape of body 1 i the periodic steady state which is ot give, but must be foud i the optimizatio process). The Sigorii coditios for the whole period the are p ± d ± =, p ±, d ±. (43) The objective fuctio ca be stated as I δw = S c δw ds, or I δw max δw. The steady state coditio the is I δw =, also max δw =. Numerically this extremum caot be reached. I our examples it is foud that I δw max δw First step i the solutio of the optimizatio problem (41). The optimizatio problem is solved i two steps. A. First we take the average shape for mootoic motios [5], see Figure 16(-+) lie, to build a cubic splie for the followig poits:

25 Aalysis of steady wear processes for periodic slidig 255 x mm Shape mm 1.17E E E E E E E E E E E E E E E E E+4.E E E E E E E E E E E E E E E E E-2 Shape [mm] 5 x F =1. kn, Shape modificatios [mm] Average shape for mootoic motios, m=: * m=1 : m=2:. m=3: m=4: m=5:. m=6: m=7: From time itegratio of the wear law: o x~=113 Figure 19. Shape modificatio specified with 9th polyomial order. B. The trasform the data i the followig way: m (x 11) q / (L/2) q for x 11, m ( x + 11) q / (L/2) q for x < 11. We take q = 9, =.4 mm, m = 1, 2,..., 7. This procedure is amed polyomial iteratio. The shapes are demostrated i Figure 19. I this figure the curve obtaied from time itegratio of the wear rule (- - o) is preseted. It is see that the approximatio curve at m = 7 is lower tha the curve (- - o). That is, the splie approximatio must be modified Secod step i the solutio of the optimizatio problem (41). I the secod step we suppose that the ew wear fuctio ca be approximated by the Taylor series 16 δw = δw () + j=1 (44) (δw) a j a j, (45)

26 256 I. Páczelt, Z. Mróz, A. Baksa that is we ca calculate the splie parameters a j. The derivative (δw) / a j is determied i the umerical way, so that (δw) a j I our case s =.2. δw(a () 1,..., a() j + s,..., a () 16 ) δw() s. (46) Shape [mm] 5 x F =1. kn, Shape modificatios [mm] Average shape for mootoic motios: * After poliomial iteratio: Splie modificatio ms=1:. Splie modificatio ms=2: Splie modificatio ms=3: + From time itegratio of the wear law: o x~=113 Figure 2. Shape modificatio specified by splie modificatio. For each j the cotact problem must be solved ad the wear is calculated for oe slidig period. For the cotrol of the coditio δw =, 16 poit zoes i the cotact domai are take, ad usig the Raphso iteratio techique, ew splie poit coordiates ca easily be foud: a ew j = a () j + a j (47) F =1. kn, Cotact Pressure at the differet polyomial shape modificatios p Average shape for mootoic motios, m=: * m=1 : m=2:. m=3: m=4: m=5:. m=6 m=7 + Rightward slidig directio F =1. kn, Cotact Pressure at the differet polyomial shape modificatios p Average shape for mootoic motios, m=: * m=1 : m=2:. m=3: m=4: m=5:. m=6 m=7 + Leftward slidig directio x~= x~=113 a) b) Figure 21. Cotact pressure evolutio due to differet polyomial shape modificatios, a) for rightward slidig motio, b) for leftward slidig motio.

27 Aalysis of steady wear processes for periodic slidig F =1. kn, Cotact Pressure at the splie modificatios Average shape for mootoic motios: * After polyomial iteratio: Splie modificatio m s =1:. Splie modificatio m s =2: Splie modificatio m s =3: + Rightward slidig directio F =1. kn, Cotact Pressure at the splie modificatios Average shape for mootoic motios: * After polyomial iteratio: Splie modificatio m s =1:. Splie modificatio m s =2: Splie modificatio m s =3: + Leftward slidig directio p 25 2 p x~= x~=113 a) b) Figure 22. Cotact pressure evolutio due to splie modificatios, a) for rightward slidig motio, b) for leftward slidig motio. where the algebraic system is [ ] (δw)i a j i,j=1,16 a 1.. a j. a 16 = δw 1. δw j. δw 16 Usig this techique, after m s = 1, 2, 3 splie modificatios we obtai a ice result, see Figure 2. The calculated shape is practically the same as that obtaied from time itegratio. The cotact pressure distributios are preseted i Figures 21 ad 22. Whe the averaged mootoic shape is applied, the cotact pressure has a high value at the perimeter poits chagig with the slidig directio, see Figure 21. After polyomial iteratio the pressure value is lower tha that i the steady state, see lies ( +) i Figure 21. After the ed of the secod step, the pressure exhibits a very small oscillatio, so the optimal solutio is very close to the umerically specified result (see Figure 22). It ca be cocluded that the recommeded optimizatio process provides correct results Solutio of the optimizatio problem by applyig the pealty techique. The objective fuctio ca be preseted i a differet form, usig the pressure costrait i the periodic steady wear state. Defie the pressure differece for the rigid body wear velocity λ ± F, λ ± M = (see (23a)) () (48) p = p + + p 2p m. (49) If p = at each poit of cotact zoe, the the correspodig cotact shape is correct. If ot, the the shape must be modified. Usig the idea of pealty techique [18], we ca write p = p + + p 2p m = c (u + + g + + u + g ) 2p m, (5)

28 258 I. Páczelt, Z. Mróz, A. Baksa where g + ad g are the shapes (gaps) at the ed of the + or slidig directio, c is the pealty parameter used for the ormal cotact problem. a) b) Figure 23. Cotact pressure distributio ad shape evolutio. a) p at steady state, b) shape modificatio i the i-th (+ directio slidig) ad (i + 1)-th (- directio slidig) iteratioal step. If p the shape must be chaged, that is istead of (5) it ca be writte p ± = p + + p 2p m = c (u + + g + u + g ± ) + c g ± 2p m (51) The optimizatio problem ca be writte i the followig form mi { g S c 1 2 (p+ + p 2p m ) 2 ds p ±, d ±, p ± d ± =, τ + = µ p +, τ = µ p, Equilibrium equatios for puch} (52) where the miimum of (52) provides the cotact pressure distributio satisfyig (49) if p =. The shear stress τ ± acts o the cotact surface of Body 1 i the directio of x-axis. For solutio of the miimizatio problem (52) a special iteratioal process is recommeded. I each step the Sigorii cotact coditios p ± d ± =, p ±, d ± ad the Coulomb dry frictio law τ + = µp +, τ = µp must be satisfied i the solutio of the cotact problem ad ext the modified cotact shape should be determied. The shape modificatio is take from equatio (51).

29 Aalysis of steady wear processes for periodic slidig 259 Cosider the half cycle i of the + slidig directio, ext i + 1 of the slidig directio ad similarly the cosecutive half cycle i+2 for + slidig directio ad i+3 for directio of slidig ad so o. Accordig to Figure 23 i the iterval x l x x cl the cotact pressures are p +, p = i the iterval x cl x x cr the pressures are p +, p ad i the iterval x cr x x r the pressures are p + =, p. Let us begi the i-th half cycle. The p ± = p + + p 2p m = c (u + + g + u + g ± ) + c g ± 2p m (53) i the iterval x l x x cl because g +(i) = g (i 1) i the right directio of slidig at the ed of half cycle the ew shape is modified by g +(i). The modificatio of the shape is p + c (u +(i) I the iterval x cl x x cr i right directio p + = p +(i) p + c + p (i 1) 2p m = = c (u +(i) (u +(i) + g +(i). It is supposed that + g (i 1) ) = g +(i) 2p m = g +(i). (54) c + g (i 1) + u (i 1) + u (i 1) + g (i 1) ) + c g +(i) 2p m, (55) + 2g (i 1) ) = g +(i) 2p m c = g +(i). (56) For umerical calculatio it is supposed that at the poit x cr of the cotact domai the modificatio of the gap is equal to zero, that is g +(i) g +(i) +(i) um (x cr ) = g ad the ew shape at the ed of + directio motio is g +(i) = g (i 1) I the iterval x cr x x r i the left directio we have that is p = p (i+1) p c (u (i+1) 2p m = c (u (i+1) +(i) um + g. (57) + g +(i) ) = g (i+1) 2p m ad i the iterval x cl x x cr for the left directio there is p = p +(i) where g +(i) p = p +(i) ad + p (i+1) 2p m = = c (u +(i) p c = g (i+1) + p (i+1) (u +(i) + (g (i 1) + g +(i), thus 2p m = c (u +(i) + u (i+1) + g +(i) ) + u (i+1) + u (i+1) + g +(i) ) + c g (i+1) 2p m (58) c + 2g +(i) ) = g (i+1) 2p m Sice i poit x cl the modificatio of the gap is equal to zero g (i+1) g (i+1) (x cl ) = g = g (i+1) (59) + g +(i) ) + c g (i+1) 2p m + 2g +(i) ) + c c g (i+1) 2p m (6) c (i+1) um = g (i+1). (61)