4 : 355 F ig 1 Koch curves w ith the scale of 1g3 1 r = 1g3, L (r) = (4g3) n M (r) = 4 n,,, L (r) = M (r) r = r 1- D (1) : D (1),

Size: px

Start display at page:

Download "4 : 355 F ig 1 Koch curves w ith the scale of 1g3 1 r = 1g3, L (r) = (4g3) n M (r) = 4 n,,, L (r) = M (r) r = r 1- D (1) : D (1),"

奚狂皮
5 years ago
Views:

1 17 4 V o l 17, N o TR IBOLO GY D ec, 1997 ( ) 3 ( ), 4,, W eierstrass2m andelbro t,, TH , R a,, ( ) 80, [ 1 5 ], 1,, r Κr, 1 (Koch) r, r= 1g3, L (r) = 4g3, ( ) M (r) = 4 1 ; r= (1g3) 2, L (r) = (4g3) 2 M (r) = 4 2 ; ; r = (1g3) n 3 g g , ,, 1995, 1995, Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

2 4 : 355 F ig 1 Koch curves w ith the scale of 1g3 1 r = 1g3, L (r) = (4g3) n M (r) = 4 n,,, L (r) = M (r) r = r 1- D (1) : D (1), D = 1 - log L (r) log (r) = 1 - n log 4 3 n log 1 3 = log (4) log (3) = , (1) : : C L (r) = M (r) r = C r 1- D (2) (2) log M (r) = log C - D log r (3) (3) log M 2log r 2 : a,, ; b n (Yard stick m ethod), Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

3 , 2 (a) n ri (i= 1, 2,, n), L i, [ r1,l 1 ], [ r2,l 2 ],, [ rn, L n ], 2 (b) : Α 2 2 (Box d im en sion m ethod) D = 1- Α (4) 1, 2 n F ig 2 T he p rincip le of the yard stick m ethod fo r the fractal dim ension of rough p rofiles 2, 2 - n, 3, M (n), 2 3 (Var ia tion m ethod) D = logm (n) lim n n log2 (5) 4, r, i H i, r, H i, V (r) : Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

4 4 : 357 V (r) = 6 rh 6 i r 2 = H i (6) r V ( r) r D = 2- Α (7) F ig 3 T he box dim ension m ethod fo r the fractal dim ension of rough p rofiles 3 F ig 4 T he variation m ethod fo r the fractal dim ension of rough p rofiles (Structure function m ethod) z (x ) S (Σ) = z (x + Σ) - z (x ) 2 = CΣ 4-2D (8) : z (x + Σ) - z (x ) 2, Σ Σ S (Σ), log S 2log Σ Α, D Α D = 2- Α 2 (9) 4, : a, Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

5 ; b ; c, 3 (Covar ia tion we ighted m ethod) [ 6 ] Z (T ) Z (T ) - Z (T 0) = ΦgT - T 0g 2- D (10) T 0= 0, Z (0) = 0 V (T ) = E Z (T ) - Z (T ) 2 T 4-2D (11) Ρ(T ) = V (T ) 1g2 T 2- D (12) (10), T T,, n T i (i= 1, 2,, n) Ρ(T ), log Ρ2log T, Α D = 2- Α (13),, (12) Ρ(T ) = C T 2- D (14) : C, C, C = 1, ΡC = E (C - C θ ) 2 (15) ΡC ;, ΡC,, C u = C θ + K ΡC (16) : K, K 0 995, K = 2 99 C u (14) C (16) (14),, 7 Γ Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

6 4 : 359 Φ= 1-7 Γ (17), 4 [ 7 ] W 2M, D, g (t) = 6-2) t= - Κ(D t [1- co s (Κ i t) ] (18) : Κ> 1, 1< D < ( 18) Κ= Expected fractal dim ension 1 W -M Table 1 The com par ison on com puted fractal d im en sion of the W -M curves Computation m ethods Computed fractal dim ension D eviation g% P recision g% 1 2 Yard stick V ariation Box dim ension Structure function Covariance w eighted Yard stick V ariation Box dim ension Structure function Covariance w eighted Yard stick V ariation Box dim ension Structure function Covariance w eighted W 2M ( D 1 2, ), 10% 1 02%, 99% D = 1 5 W 2M, 2 1 1%, 5 4%, 4 1% 4 4% Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

8 4 : 361 M inim um scale g% 2 D = 1 5 W -M Table 2 The com par ison on the stability for the fractal d im en sion (D = 1 5) with d ifferen t m ethods Scale range g% Yard stick m ethod V ariation m ethod Structure funtion m ethod Covariance W eigh t m ethod F luctuationg% , Κ W 2M, D 1 2, Κ= 1 3, D 1 185, , Κ= 1 8, D 1 212, , 5,,, 1 M ajudar A, T ien C L F ractal characterization and sim ulation of rough surfaces W ear, 1990, 136: Stupark P R F ractal characteristics of rubber w ear surfaces as a functrion of load and velocity W ear, 1990, 141: B row n C A, Savary G D escribing ground surfaces texture using contact p rofilom etry and fractal analysis W ear, 1991, 141: Ganti S, Bhushan B Generalized fractal analysis and its app lication to engineering surfaces W ear, 1995, 180: T rico t C, Ferland P, Baran G F ractal analysis of wo rn surfaces W ear, 1994, 172: Feders J F ractals N ew Yo rk: P lenum P ress, Falconer K J :, Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

9 The Com putation M ethods for the Fractal D im en sion of Surface Prof iles Ge Sh irong Suo Shuangfu (M echanical E ng ineering D ep artm ent Ch ina U niversity of M ining and T echnology X uz hou Ch ina) Abstract T he basic concep t abou t fractals and the fractal cu rves w ere in troduced Fou r k inds of the m ethods cu rren tly u sed fo r com puting the fractal dim en sion of rough p rofiles w ere illu strated It is show ed that a h igh deviation ex isted fo r the resu lts of fractal dim en2 sion by these cu rren tly u sed m ethods and a com paratively large fluctuation in fractal di2 m en sion w as ob served w hen u sing differen t ru ler scaleṡ T herefo re, it w as difficu lt to ac2 cu rately reflect the true fractal behavio r of p rofiles by the fractal dim en sion ob tained from these m ethodṡ O n the basis of the exponen tial relation of the roo t m ean square of heigh ts of a rough p rofile to the scales, the covariance w eigh ted m ethod w as p ropo sed to evaluate the fractal dim en sion Com pared w ith W eierstrass2m andelb ro t function cu rves, the co2 variance w eigh ted m ethod p resen ted h igher accu racy and less fluctuation of the com puted fractal dim en sion T he availab le com putation m ethod fo r the fractal dim en sion w ou ld sim 2 p lify the characterization of su rface roughness in o rder to describe the com p lex characteris2 tics of rough su rfaces in tribo logy Key words su rface p rofile fractal dim en sion tribo logical com putation roo t m ean square of heigh ts w eigh ted m ethod Cla ssify ing num ber TH Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved

7 53 1 [5, 7 ],,,,,,,,,,, 111, (,,, ) : (a,m, b ) [A,M B ]; (a,m, b ), [A,M, B ], (a, m, b, (a, m, b ), [A,M, B ], A, A + ( ) ( ), A - ( ) ( ) ;M,M +,

7 53 1 [5, 7 ],,,,,,,,,,, 111, (,,, ) : (a,m, b ) [A,M B ]; (a,m, b ), [A,M, B ], (a, m, b, (a, m, b ), [A,M, B ], A, A + ( ) ( ), A - ( ) ( ) ;M,M +, 1999 7 7 α (, 200331), ; ; ; A nalysis on the Co re System M odel P rincip le of the Basic T heo ry of Ch inese M edicine GU AN X iao si (D ep ṫ of Basic Science, Shanghai T iedao U n iversity, Shanghai