22 5 V o l 22, N o 5 2002 9 TR IBOLO GY Sep, 2002,, (, 100084) :, (FFT )., 2 (Green ),,, 3,. ; ; FFT; ; TH 117. 2 A 100420595 (2002) 0520390205,.,,,. L ub rech t [1 ]. (FFT ) [2 5 ]., [4, 5 ].,L iu [6 ] FFT., FFT, FFT, ;,. 1 1 p (x ) p (x, y ) 1, U (x ) U (x, y ) H ertz [7 ] : U (x ) = - 4 3 lngx - sgp (s) ds. (1) U (x, y ) = 2 ΠE κ 3 p (Ν, Γ) (x - Ν) 2 + (y - Γ) 2dΝdΓ. (2),, U (x i) = - U (x i, y j) = 4 3 K (x i- x k) p (x k). (3) k= 0 M - 1 2 3 k= 0 K (x i- x k, y j- y l) p (x k, y l). l= 0 : K (x i- x k) K (x i- x k, y j- y l), (4) k (k, l) i (i, j ) ( 1 ), K i k ( K i, j k, l),. (3 4), K 2,,. 2 2. 1., (19992150). 2001211214; 2002204221g, E2m ail: w angw z@po sṫ p im. tsinghua. edu. cn.,, 1971,,.
5 391 K i, j k, l= κ 8 h (x i- Ν, y j- Γ) n (Ν, Γ) dνdγ. (5) : 8 (x k, y l) ( 1 F ig 1 Influence coefficient 1 ), n (Ν, Γ),, h (x, y ), Green,, h (x ) = ln (x ) ;, h (x, y ) = 1g x 2 + y 2., K i, j k, l 2,, K i, j k, l,. n (Ν, Γ),,. n (Ν, Γ),, (6) K k, l= X k + x X k - x g2 + y h (Ν, Γ) dνdγ. - y h (Ν, Γ), K k, l= h (x k, y l) x y, Green,,,,. Green,. 2. 2 (3, 4), 2,,.,. 2. 2. 1 2. 2 N,M a h H A, [8 ] : K k, l, (5) K k, l= X k + x X k - x g2 + y ck= h k- na n, k= 0, 1,,M + N - 2. (7) h (Ν, Γ) n (Ν, Γ) dνdγ. (6) n= 0 - y 2 N M, : X k N + M - 1, 2 N ( 1 ). a h h a, (6), n (Ν, Γ) :, n (Ν, Γ) h a= h a= c (0) c (1) c (2) c (3) c (4) c (5) c (6) c (0) c (1) c (2) c (3) 2 2,, 2, 2, = = ck= h < k- n> N an, k= 0, 1,,. (8) n= 0, 2 N N, < n> N n N.,, N = M = 4, h (0) 0 0 0 h (1) h (0) 0 0 h (2) h (1) h (0) 0 h (3) h (2) h (1) h (0) 0 h (3) h (2) h (1) 0 0 h (3) h (2) 0 0 0 h (3) h (0) h (3) h (2) h (1) h (1) h (0) h (3) h (2) h (2) h (1) h (0) h (3) h (3) h (2) h (1) h (0) a (0) a (1) a (2) a (3) a (0) a (1) a (2) a (3),,.. ; 2, h, h
392 22., D F T {c (n) }= D F T {h (n) } D F T {a (n) }. (9) 2 2.,,,. h, a (n) h (n), N + M - 1 a (n) h (n), 2,. FFT. 2. 2. 2 FFT [ 3 5 9 ], FFT, ; (3, 4)., (3 4), P N,, K 2,.,, 3N - 2,. 2 ( ), 2, F ig 2 P retreatm ent of influence coefficient (L 2computation dom ain,l 2extended dom ain) [ 6 ]. (1) L p (x ) N {P i}n, 2N - {PN i}2. 2 (L 2, L 2 ) 1 (2) 2, {K i}2,. : {K i}2 N - 1 K 1 K {KN i}2 KN N + 1 KN 2, { K i }2 N K N K 2 {KN i}2 N KN 1 KN N. (3) {PN i}2 {KN i}2, {P δ N i}2 {K δ N i}2. (4) {P δ N i}2 {K δ N i}2 {T δ i}2. {T δ i}2 {T i}2. (5) U i= T i, i= 1, 2,, N, {U i}n.,,., O (N 2 ) O (N lnn ). 3 FFT, 1,, (3 6). P m, P h H ertz (GPa). :,,,. FFT, H ertzian, M LM I FFT, 2.,. Pen tium g 400, V isual Fo rtran 6. 6, FFT IM SL. 2 2., FFT, 1g3
5 393 1 Table 1 The computation cases Case Load Dom aingmm Loading fo rm Case 1 U nifo rm - a x a L ine loading Case 2 T riangle - a x a L ine loading Case 3 U nifo rm 2a 2b Po int loading Case 4 H ertzian p ressure C ircle w ith R a Po int loading F ig 3 D efo rm ations due to a unifo rm p ressure ( line loading) 3 F ig 4 D efo rm ations due to a triangle p ressure ( line loading) 4 F ig 5 D efo rm ations due to a unifo rm p ressure on a rectangle area 2a 2b 5 2a 2b F ig 6 D efo rm ations due to a H ertzian p ressure app lied to a circular region 6 H ertz 2 Table 2 Time needed for FFT-based method and MLM I tgs FFT2based m ethod M LM I 513 513 5. 490 18. 29 257 257 1. 200 4. 55 129 129 0. 280 1. 10 65 65 0. 060 0. 27 33 33 < 0. 001-3 Table 3 Relative errors to closed form solution Relative erro r g% FFT2based m ethod M LM I D irect summ ation 513 513 0. 021721 0. 021720 0. 021720 257 257 0. 043488 0. 043488 0. 043488 129 129 0. 088167 0. 088168 0. 088167 65 65 0. 177570 0. 177570 0. 177560 33 33 0. 338140-0. 338140
394 22. 3 2,. 2 Green, 3, FFT M LM I, ( )., FFT. 4 a. 2 2,. b., FFT M LM I, M LM I 3. c.,. [ 1 ] L ubrech t A A, Ioannides E. A Fast So lution of the D ry con2 tact P roblem and A ssociated Surface Stress F ield, U sing M ul2 tilevel Techniques [ J ]. A SM E Journal of T ribo logy, 1991, 113: 1282133. [ 2 ] H u Y Z, Barber Gary C. N um erical A nalysis fo r the E lastic Contact of Real Rough Surfaces[J ]. STL E T ribo logy T rans, 1999, 42: 4432452. [ 3 ] N ogi T, Kato T. Influence of a hard surface layer on the lim it of elastic contacṫ Part 1: A nalysis using a real surface model [J ]. A SM E Journal of T ribo logy, 1997, 119: 4932500. [ 4 ] Ju Y, Farris T N. Spectral A nalysis of Two2D im ensional Con2 tact P roblem s [ J ]. T ransactions of the A SM E, 1996, 118: 3202328. [ 5 ] Po lonsky IA, Keer L M. Fast m ethods fo r so lving rough con2 tact p roblem s: a comparative study[j ]. A SM E Journal of T ri2 bo logy, 2000, 122: 36241. [ 6 ] L iu S, W ang Q, L iu G. A V ersatile M ethod of D iscrete Con2 vo lution and FFT (DC2FFT ) fo r Contact A nalyses[j ]. W ear, 2000, 243: 1012111. [ 7 ] Johnson K L. Contact m echanics[m ]. L ondon: Cam bridge U 2 niversity P ress, 1985. [ 8 ] Burrus C S, Park s T W. D FT gfft and Convo lution A lgo2 rithm s theo ry and imp lem entation [M ]. N ew Yo rk: R ice U ni2 versity, 1984, 1985. [ 9 ] Co lin F, L ubrech t A A. Comparison of FFT 2M LM I fo r E lastic D efo rm ation Calculations [ J ]. A SM E Journal of T ribo logy, 2001, 123: 8842887. Fast Com putation of Surface D eformation in L ubr icated Con tact W AN G W en2zhong, W AN G H u i, HU Yuan2zhong (S tate K ey L aboratory of T ribology, T sing hua U niversity, B eij ing 100084, Ch ina) Abstract: Based on the exam ination of calcu lation p rocess of su rface defo rm ation and discrete convo lu tion, an FFT 2based m ethod fo r calcu lation of su rface defo rm ation w as derived and described. T he m ethod requ ires a p retreatm en t fo r the discrete data of influence coefficien ts and p ressu re distribu tion so as to get rid of the pe2 riodic erro rs. N um erical examp les are given to validate the m ethod and to compare the resu lts w ith tho se from o ther fast app roacheṡ T he resu lts show that the FFT 2based m ethod is very accu rate and efficien ṫ It has a great po ten tial in app lication to the num erical analysis of m echan ical param eters, such as su rface defo r2 m ation s and temperatu re riseṡ Key words: influence coefficien t; su rface defo rm ation; FFT; m u lti2level m u lti2in tegration; discrete convo lu2 tion Author: W AN G W en2zhong, m a le, bo rn in 1971, Ph. D. studen t, E2m a il: w angw z@po sṫ p im. t singhua. edu. cn.