Методика обработки профилограмм с использованием вейвлет-фрактального анализа..pdf

!" 620.179:621.81:531.19
.!. "#$%&'(, ).*. +,$,-(., /. . 01'23, *.4. 35$66.
!"#$-%&$&'()'*+#,- *.+)/!'+$0&""1- )",0&'+,$&$ ,"2.'3!4,.""15
$&5".6.*,-, 3&5!",#, , .7$,#,, !"#$-%&$&'()'*, 8.++,9
)7+"8/9
":; :"+9/ <;"=/>"?; ))
4 /4<">@A"! B/7) !7C!>7+-=; 9+ >@B"?"
B >/A
:++6&/)&$+9 #.''&694,9 «;0.6<4,9 /,"!3,=&+#.- +,+$&31 — ;0.6<4,9 #!=&+$0! $')>,5+9
7.0&'5".+$&-».
7.3.>?< $',(.3&$',=&+#.- )+$!".0#, «@8:ABC-2» (16, 7'.0&/&"1 .71$1 7.
7&'&/!=& 0.D0'!$".-7.+$)7!$&6?".*. /0,E&",9 $'&",&3 "! .('!D4!5 6!$)",. F")$'&""99 /,"!3,#!
$'&",9 (16! ,++6&/.0!"! 0 '&D)6?$!$& .('!(.$#, 7'.2,6.*'!33 + 7.3.>?< 3".*.)'.0"&0.*. 0&-06&$-'!D6.E&",9, ! $!#E& '!+=&$.0 2'!#$!6?"15 '!D3&'".+$&- 7.6)=&""15 7'.2,6.*'!33. F '&D)6?$!$& !"!6,D! /!""15 (16, 7.6)=&"1 *'!2,#,, 5!'!#$&',D)<>,& 0")$'&""<< /,"!3,#) 7'.4&++!:
;0.6<4,9 7!'!3&$'.0 G&'.5.0!$.+$, Ra, 7'&/&6?"1& D"!=&",9 #)3)69$ , #.;22,4,&"$1 H&'+$!
=!+$.$"15 #.37."&"$.0 +,*"!6.0 7'.2,6.*'!33. A16. )+$!".06&"., =$. 7.#!D!$&6? H&'+$! ,3&&$
#.6&(!$&6?"1- 5!'!#$&', #.$.'1- )#!D10!&$ "! #.6&(!$&6?".+$? )+$.-=,0.+$, , "&)+$.-=,0.+$, +.+$.9",9 7.0&'5".+$".*. +6.9 0D!,3./&-+$0)<>,5 #."$'7!'.
9DEF6.%6 -D(.,: $'&",&, 0&-06&$, 2'!#$!6, #',$&',- H&'+$!, G&'.5.0!$.+$?, +7&#$'!6?"!9 76.$".+$? ;"&'*,,, «@8:ABC».
A.V. Opryshko, M.U. Tarasov, I.A. Utkin, U.S. Andreev
St. Petersburg State University of Information Technologies,
Mechanics and Optics, St. Petersburg, Russia
PROFILOGRAMMS PROCESSING TECHNIQUE
USING WAVELET-FRACTAL ANALYSIS
Investigate the correlation between «evolutions of a dynamical system – the evolutions of the
quality of the rubbing surfaces». An experiment was carried on friction machine «Tribal-2» for transfer
the reciprocating motion with friction on the sample of brass. The internal dynamics of friction was investigated using multilevel wavelet decomposition and the calculation of fractal dimensions obtained profilogramms. After analysis of the data were obtained graphics describing the internal dynamics of the
process: the evolution of the roughness parameters Ra, cumulates and Hurst coefficient for frequency
component signals profilogramms. It was found that the Hurst exponent has an oscillatory character,
which indicates to the oscillation stability and instability of the state of the surface layer of interacting
counter-pairs
Key words: friction, wavelet, fractal, Hurst coefficient, surface roughness, the spectral energy
density, Tribal.
!"#"$%"
# $%&'()*+, &'%'-+ .&&/+01+'&) 2(03(0 4 1&'%$(5/+$.6 4(77+/)8.. «95(/68.) 0.$%:.;+&4(, &.&'+:< – 95(/68.) 4%;+&'5% '71*.3&) 2(5+73$(&'+,». &'%$(5/+$.+ 0%$$<3 =%5.&.:(&'+, )5/)+'&)
118
>%=(, %5'(:%'.=%8.. 4($'7(/) 4%;+&'5% 2(5+73$(&'+, 5 27(8+&&%3 '7+$.). ?% 4%@+07+ :+3%'7($.4. AB>C DEFG & 2(:(*-6 &.&'+:<
«EHDIJK-2» ></. 27(5+0+$< (2<'< 2( '7+$.6. #$1'7+$$)) 0.$%:.4% '7+$.) ></% .&&/+0(5%$% 5 7+=1/-'%'+ (>7%>('4. & 2(:(*-6 5+,5/+'-%$%/.=%, % '%4L+ 7%&;+'%:. @7%4'%/-$<3 7%=:+7$(&'+, & 2(:(*-6
27(M7%::< Fractan 2(/1;+$$<3 27(@./(M7%:: '7.>(2%7. D&&/+0(5%$.+ 5$+N$+, 0.$%:.4. &(5+7N+$( 2( :+'(0.4+, (2.&%$$(, 5 [1]. # 7+=1/-'%'+ 7%>('< $% EHDIJK+ ></. 2(/1;+$< 53(0$<+ . 5<3(0$<+
0%$$<+ 27(8+&&% '7+$.), 4('(7<+ 5 2(&/+016*+: ></. (>7%>('%$< &
2(:(*-6 2%4+'% System Identification Toolbox &.&'+:< Matlab. # .'(M+
27+0&'%5/+$< 2(/1;+$$<+ 7+=1/-'%'<.
&$'(%) *"+,-,!'.,/.% 0,!"+-$,/.%
F.47(M+(:+'7.) 2(5+73$(&'. 27+0&'%5/)+' &(>(, M+(:+'7.;+&4., (>O+4', ('$(&)*.,&) 4 4/%&&1 @7%4'%/(5. # (&$(5+ (8+$(4 N+7(3(5%'(&'. /+L%' 2(03(0< 4 5<;.&/+$.6 &'(3%&'.;+&4.3 3%7%4'+7.&'.4 @7%4'%/-$<3 M+(:+'7.;+&4.3 (>O+4'(5, 4('(7<+ .$'+727+'.716'&) 4%4 &.M$%/< 5 $+/.$+,$<3 &.&'+:%3 [1].
1,2')'."(3 4"+/.'
D&&/+0(5%$.+ 57+:+$$(M( 7)0%, 5 7(/. 4('(7(M( 5<&'12%+' N+7(3(5%'(&'-, (&1*+&'5/)+'&) & 2(:(*-6 :+'(0% $(7:.7(5%$$(M( 7%=:%3%. !%$$<, :+'(0 27.:+$)+'&) 27. .&&/+0(5%$.. 2(&/+0(5%'+/-$(&'.
.=:+7+$., '+:2+7%'17< %':(&@+7$(M( 5(=013%, 4(/.;+&'5% (&%04(5,
2%7%:+'7(5 &'(4% 7+4 . '.2. [3].
?(7:.7(5%$$<, 7%=:%3 (27+0+/)+'&) &/+016*.: (>7%=(:. B1&'x(t) – $+4('(7%) &/1;%,$%) 5+/.;.$%, 7%&&:%'7.5%+:%) 5 $+4('(7<+
0.&47+'$<+ 27(:+L1'4. 57+:+$. ti 5 '+;+$.+ 2+7.(0% $%>/60+$., P,
1 $
x P ! " # x ti ! – ++ &7+0$++ =$%;+$.+, %
P i "1
S P! "
2
1 P
%( x ti ! ' x P ! &)
#
P i "1
(1)
&'%$0%7'$(+ ('4/($+$.+ x.
t
X t , P ! " # %( x u ! ' x P ! &).
(2)
u "1
119
B7.5+0+$$(+ 5<7%L+$.+ )5/)+'&) $%4(2.5N.:&) ('4/($+$.+:
=$%;+$., &/1;%,$(, 5+/.;.$< x(t) (' ++ &7+0$+M( =$%;+$.) <x> =% 57+:) t. H%=$(&'- :+L01 :.$.:%/-$<: . :%4&.:%/-$<: =$%;+$.):. X(t,P)
$%=<5%+'&) 7%=:%3(: R(P).
R P ! " max X t , P ! ' min X t , P ! ,
(3)
M0+ 1<t<P. H%&&:%'7.5%+:<, 7%=:%3 R(P) )5$( =%5.&.' (' 2+7.(0% P .
7%&'+' 5:+&'+ & $.:. I+=7%=:+7$(+ ('$(N+$.+ R/S 2(=5(/)+' &7%5$.5%'- 7%=:%3 0/) 7%=$<3 )5/+$.,. I</( 5<)5/+$(, ;'( $(7:.7(5%$$<,
7%=:%3 R/S 3(7(N( (2.&<5%+'&) 9:2.7.;+&4.: &(('$(N+$.+:
H
R *P+
", - .
S .2/
(4)
!%$$(+ &(('$(N+$.+ 2(/1;./( $%=5%$.+ =%4($% Q+7&'%, % 2(4%=%'+/- H – 4([email protected]@.8.+$' Q+7&'%.
"5!(".-0+",6+'),!'$%"
!/) 0.&47+'$(M( &.M$%/% 4($+;$(, 0/.$< N 5+,5/+'-27+(>7%=(5%$.+ 5<;.&/)+'&) 0/) 0.&47+'$<3 =$%;+$., 2%7%:+'7(5 :%&N'%>%
a = 2j . &05.M% d = k2j, M0+ k, j – 8+/<+ ;.&/%. #+,5/+'- . :%&N'%>.716*.+ @1$48.. .:+6' 5.0
0 j ,k t ! "
1 * t
* t
+
+
0 , j ' k - , 1 j ,k t ! "
1, j ' k -.
j
.2
/
/
2
2 .2
1
j
(5)
!.&47+'$(+ 27+(>7%=(5%$.+ &.M$%/% $% j-: 17(5$+ 7%=/(L+$.)
:(L$( 5<7%=.'- &/+016*.: (>7%=(::
kmax
kmax
k "0
k "0
W j " # a j , k 1 j ,k t ! 2 # d j ,k 0 j ,k t !.
(6)
#+,5/+'-4([email protected]@.8.+$'< aj,k . dj,k =%0%6'&) .$'+M7%/%:.
a j ,k " 3 x t ! 1*j ,k t ! dt , d j ,k " 3 x t ! 0*j ,k t ! dt.
(7)
?% 2(&/+0$+: 17(5$+ 7%=/(L+$.) m @(7:.716'&) $%>(7< 4([email protected]@.8.+$'(5 %227(4&.:%8.. 2(&/+0$+M( 17(5$) . 0+'%/.=.716*.3 4([email protected]@.8.+$'(5 5&+3 17(5$+,. !/) 5(&&'%$(5/+$.) &.M$%/(5 2( .=5+&'$(:1 $%>(71 4([email protected]@.8.+$'(5 .&2(/-=1+'&) 4%&4%0$<, %/M(7.': (>7%'$(120
M( 5+,5/+'-27+(>7%=(5%$.). J$%/.=.71+:<, &.M$%/ 7%5+$ &1::+ &M/%L+$$(M( 4(:2($+$'% 2(&/+0$+M( 17(5$) (Am) . 0+'%/+, 5&+3 17(5$+,
7%=/(L+$.) (Dm,…,D1):
kmax
m kmax
k "0
j "1 k " 0
x ti ! " Am ti ! 2 Dm ti ! 2 ...D1 ti ! " # am ,k 1m ,k t ! 2 ## d j , k 0 j , k t !. (8)
1'+'7".+8 #"5/.!%9 /0"2.+'(3$,5 0(,.$,/.% :$"+;%%
A2+4'7%/-$%) 2/('$(&'- 9$+7M.. 7%5$% 45%07%'1 @17-+-27+(>7%=(5%$.) &.M$%/%:
E f!" X f ! "
2
3 x t!e
'2 4ift
2
dt ,
(9)
M0+ X f ! – &2+4'7 &.M$%/%.
R%&'('$(+ $%4(2/+$.+ &2+4'7%/-$(, 2/('$(&'. 9$+7M.. (41:1/)f2
'%) 5 27+0+/%3 2(/(&< ;%&'(' f1 , f 2 ! (27+0+/)+'&) 4%4 S " 3 E f ! df .
f1
H%=:+7$(&'- 41:1/)'< (27+0+/)+'&) 5 !L/& = #'.
# 4%;+&'5+ 3%7%4'+7.&'.4 &2+4'7%/-$(, 2/('$(&'. 9$+7M.. .&2(/-=16'&) $(:.$%/-$<, . .$'+75%/-$<, 2%7%:+'7< 0+,&'5.):
h1 "
E
5*
, h2 " max ,
f 2 ' f1 ! f max
f max
(10)
M0+ Emax – :%4&.:%/-$(+ =$%;+$.+ &2+4'7%/-$(, 2/('$(&'. 9$+7M..,
f max – :%4&.:%/-$%) ;%&'('%, &(('5+'&'516*%) Emax , S* – 27+0+/-$(+
=$%;+$.+ &2+4'7%/-$(, 2/('$(&'. 9$+7M... ?%.>(/++ 8+/+&((>7%=$(
.&2(/-=(5%$.+ 2%7%:+'7% h1.
&!.,2,++"(9<%,$$'9 % !)'%7$'9 2,++"(9<%,$$'9 =>$2<%9
B( =%2.&): 53(0$(M( u(t) . 5<3(0$(M( &.M$%/(5 y(t), 2(/1;+$$<3
27. .&2<'%$.)3 (>7%=8(5, (8+$.5%+'&) %5'(4(77+/)8.($$%) . 5=%.:$%) 4(77+/)8.($$%) @1$48... J5'(4(77+/)8.($$%) @1$48.) &/1;%,$(M( 27(8+&&% 3%7%4'+7.=1+' (>*16 =%5.&.:(&'- =$%;+$., 27(8+&&% 5
$+4('(7<, 0%$$<, :(:+$' 57+:+$. (' =$%;+$., 5 071M(, :(:+$'. #=%.:$%) 4(77+/)8.($$%) @1$48.) 0513 &.M$%/(5 3%7%4'+7.=1+' (>*16
=%5.&.:(&'- =$%;+$., (0$(M( &.M$%/% y(t) (' =$%;+$., 071M(M( u(t) [1].
121
?//("#,!'$%9
D&&/+0(5%$.+ 27(8+&&% '7+$.) 27(5(0./(&- $% '7.>(:+'7.;+&4(,
1&'%$(54+ «EHDIJK» [2]. #(=57%'$(-2(&'12%'+/-$(+ 05.L+$.+ .&&/+01+:<3 (>7%=8(5 (&1*+&'5/)/(&- & 2(&'()$$(, &4(7(&'-6, 27. @.4&.7(5%$$(, 5+/.;.$+ $(7:%/-$(M( $%M71L+$.), 5 7+L.:+ &05.M% @%= 47.5<3
2+7+:+*+$.) (5 0%$$(: 7+L.:+ (&1*+&'5/)+'&) 05.L+$.+ & '7+$.+:
&4(/-L+$.)). # 4%;+&'5+ .&&/+01+:(M( :%'+7.%/% ></ .&2(/-=(5%$ &2/%5
K56, 27+05%7.'+/-$( (>7%>('%$$<, 2( 9-:1 4/%&&1 '(;$(&'. – N+7(3(5%'(&'- .&&/+01+:(, 2(5+73$(&'. Ra &(&'%5./% 0,22 :4:. !/) 27(5+0+$.)
.&2<'%$., ></. .=M('(5/+$< (>7%=8< 0/) '7+3'(;+;$(M( 4($'%4'%.
G2<'< '7+$.) 27(5(0./.&- &/+016*.: (>7%=(:: &$%;%/% 1&'%$%5/.5%/&) 2+75<, $%>(7 .&&/+01+:<3 (>7%=8(5, 5 '+;+$.+ (27+0+/+$$(M( 57+:+$. 0/./(&- .&2<'%$.+, 2(&/+ 27(5+0+$.) (2<'% 2+75<, $%>(7 &$.:%/&) 0/) 0%/-$+,N.3 .&&/+0(5%$., N+7(3(5%'(&'. 2(&/+ '7+$.), % 5 EHDIJK 1&'%$%5/.5%/&) &/+016*., $%>(7 2%7 '7+$.) 0/)
27(5+0+$.) .&2<'%$.) 27. .$(, 0/.'+/-$(&'. (2<'%. #&+M( ></( 27(5+0+$( 5 .&2<'%$., $% '7+$.+ & 7%=/.;$<:. 57+:+$$<:. 27(:+L1'4%:.: 30, 40, 60, 90, 120 :.$. # 4%L0(: .&2<'%$.. &.&'+:(,
«EHDIJK» '%4L+ @.4&.7(5%/.&- 53(0$<+ u(t) . 5<3(0$<+ y(t) 0%$$<+
0/) 2(&/+016*+, .0+$'[email protected]%8.. 27(8+&&% '7+$.).
B(&/+ (2<'(5 '7+$.) ></( 27(5+0+$( .&&/+0(5%$.+ N+7(3(5%'(&'. 2(5+73$(&'.. A 2(:(*-6 27(@./(M7%@% =%2.&<5%/&) 27(@./- 2(5+73$(&'+, '7+$.) 0/) 4%L0(M( (>7%=8%. A$)'.+ 27(@./(M7%:: 0/)
4%L0(M( .= $.3 (&1*+&'5/)/(&- 5 '7+3 $%27%5/+$.)3: 50(/-, 2(2+7+4 .
2(0 1M/(: 45° ('$(&.'+/-$( 5(=57%'$(-2(&'12%'+/-$(M( 05.L+$.), &(5+7N%+:(M( (>7%=8%:..
@6+'6,.2' 0,(>A"$$8- #'$$8-
B7. (>7%>('4+ 2(/1;+$$<3 N+7(3(5%'(&'+, ></. 27.:+$+$< :+'(0< 5+,5/+'[email protected]%4'%/-$(M( %$%/.=% & .&2(/-=(5%$.+: 2%4+'% Matlab.
B7(@./(M7%::% 27+0&'%5/)+' &(>(, 0.&47+'$<, 7)0
6 x t !7
i
N
i "1
, :4:,
=$%;+$., 2.4(5 . 52%0.$ 7+/[email protected]% 2(5+73$(&'. '7.>(2%7<. !/) (>7%>('4. .&2(/-=(5%/&) :$(M(17(5$+5<, 5+,5/+'-%$%/.= [4]. A$%;%/% ></( 27(.=5+0+$( 7%=/(L+$.+ &.M$%/% 27(@./(M7%:: 0( 17(5$) N = 3, 5
7+=1/-'%'+ ;+M( 2(/1;+$< 0+'%/.=.716*.+ 4([email protected]@.8.+$'<. !%/++, 0/)
$%3(L0+$.) ;%&'('$<3 4(:2($+$' &.M$%/% 27(5+0+$( 5(&&'%$(5/+$.+
('0+/-$( 2( 4%L0(:1 $%>(71 0+'%/.=.716*.3 4([email protected]@.8.+$'(5. !/)
4%L0(M( 4(:2($+$'% &.M$%/% ></ 27(5+0+$ &2+4'7%/-$<, %$%/.= . 2(122
/1;+$< M7%@.4. &2+4'7%/-$(, 2/('$(&'. 9$+7M.. . $%4(2/+$.) 9$+7M.. – 41:1/)'% (7.&. 1).
H.&. 1. T$+7M+'.;+&4.+ &2+4'7< . 41:1/)'< (>7%=8%.
A'(3%&'.;+&4.+ 3%7%4'+7.&'.4.. R%&'('$<+ 4(:2($+$'< &.M$%/%
27(@./(M7%:: .:+6' @7%4'%/-$16 7%=:+7$(&'-. !/) (8+$4. 3%('.;$(&'. 4(:2($+$' (5+,5/+'-4([email protected]@.8.+$'(5) .&2(/-=(5%/&) 2(4%=%'+/- Q+7&'%. B(4%=%'+/- Q+7&'% H (2.&<5%+' 5+7()'$(&'- '(M(, ;'( 05% &(&+0$.3
(';+'% :(M1' ><'- (0.$%4(5<:.. D:++' :+&'( @(7:1/% H = 2–DF, M0+
DF – @7%4'%/-$%) 7%=:+7$(&'-, 4('(7%) (27+0+/)+'&) 4%4 27+0+/ [4]:
DF " lim 580
ln M 5 !
.
ln 1/ 5 !
(11)
!/) 5<;.&/+$.) 4([email protected]@.8.+$'% Q+7&'% .&2(/-=(5%/%&- 27(M7%::% Fractan.
# 7+=1/-'%'+ %$%/.=% 0%$$<3 ></. 2(/1;+$< M7%@.4. 95(/68..
2%7%:+'7(5 N+7(3(5%'(&'. Ra, 27+0+/-$<3 =$%;+$., 41:1/)' . 4([email protected]@.8.+$'(5 Q+7&'% ;%&'('$<3 4(:2($+$' &.M$%/(5 27(@./(M7%::.
?% 7.&. 2, 3, 4, 5 &(('5+'&'5+$$( 27+0&'%5/+$< M7%@.4. 95(/68.,
0/) $%27%5/+$., 50(/- (5+73$.+ . $.L$.+ (>7%=8<) . 2(2+7+4. U%:+'+$ 4(/+>%'+/-$<, 3%7%4'+7 .=:+$+$.) N+7(3(5%'(&'., ;'( :(L+'
><'- (>O)&$+$( 27.7%>('4(, 2(5+73$(&'+,. ?%>/60%+'&) 4(/+>%'+/-$<, 3%7%4'+7 .=:+$+$.) 2(4%=%'+/) Q+7&'% $% 5&+3 ;%&'('%3 &.M$%/%.
E%4L+ 5.0$(, ;'( 95(/68.) N+7(3(5%'(&'. 4(77+/.71+' & 2%7%:+'7%:.
0+,&'5.) $% 5<&(4.3 ;%&'('%3.
123
H.&. 2. T5(/68.) N+7(3(5%'(&'. 5+73$.3 (>7%=8(5 (50(/-)
H.&. 3. T5(/68.) N+7(3(5%'(&'. $.L$.3 (>7%=8(5 (50(/-)
124
H.&. 4. T5(/68.) N+7(3(5%'(&'. 5+73$.3 (>7%=8(5 (2(2+7+4)
H.&. 5. T5(/68.) N+7(3(5%'(&'. $.L$.3 (>7%=8(5 (2(2+7+4)
125
B'2(CA"$%"
# 7+=1/-'%'+ %$%/.=% 0%$$<3 ></. 2(/1;+$< M7%@.4., 3%7%4'+7.=16*.+ 5$1'7+$$66 0.$%:.41 27(8+&&%: 95(/68.) 2%7%:+'7(5 N+7(3(5%'(&'. Ra, 27+0+/-$<+ =$%;+$.) 41:1/)' . 4([email protected]@.8.+$'< Q+7&'% ;%&'('$<3 4(:2($+$'(5 &.M$%/(5 27(@./(M7%::. I</( 1&'%$(5/+$(, ;'( 2(4%=%'+/- Q+7&'% .:++' 4(/+>%'+/-$<, 3%7%4'+7, 4('(7<,
14%=<5%+', $% 4(/+>%'+/-$(&'- 1&'(,;.5(&'. . $+1&'(,;.5(&'. &(&'()$.) 2(5+73$(&'$(M( &/() 5=%.:(0+,&'516*.3 4($'72%7.
D%6(%,;+'=%A"/2%5 /0%/,2
1. F1&%/.:(5 #.F., #%/+'(5 #.J. !.$%:.4% @7.48.($$(M( 5=%.:(0+,&'5.). – AB>., 2006. – 191 &.
2. G&$(5< '7.>($.4. / #.F. F1&%/.:(5, J.J. A.=(5%, V.". D5%$(5%, ?.J. "7</(5, J.K. E4%;+5. – AB>., 2009. – 72 &.
3. "%/1N W.J., K(M.$(5 #.F. B(4%=%'+/- Q+7&'% . +M( &47<'<+
&5(,&'5% // A.>.7&4., L17$%/ .$01&'7.%/-$(, :%'+:%'.4.. – 2002. –
E. V, X 4 (12). – A. 29–37.
4. A:(/+$8+5 ?.". G&$(5< '+(7.. 5+,5/+'(5. #+,5/+'< 5
MATLAB. – "+:+7(5(: "+:+7. M(&. 1$-', 2003. – 200 &.
References
1. Musalimov V. M, V.A.Dinamika's Jacks frictional nteractions [Dinamika frikcionnogo vzaimodejstvija], SPb., 2006. – 191 p.
2. Musalimov V.M., Sizova A.A., Ivanova E.K., Krylov N.A.,
Tkachev A.L. Osnovy triboniki, SPb., 2009. – 72 p.
3. Kalush J.A., Loginov V.M. Pokazatel Hurst and its hidden Properties [Pokazatel' Hersta i ego skrytye svojstva], the Siberian magazine of industrial mathematics. – 2002. – Vol, X 4 (12). – P. 29–37.
4. Smolencev N.K. Osnovy teorii vejvletov. Vejvlety v MATLAB. –
Kemerovo: Kemer. gos. un-t, 2003. – 200 p.
@6 '!.,+'@0+8*2, &("2/"5 %2.,+,!%A (A%$4'-B+'+7>17M, H(&&.)) –
&'10+$' @%41/-'+'% '(;$(, :+3%$.4. . '+3$(/(M., A%$4'-B+'+7>17M&4(M( M(&10%7&'5+$$(M( 1$.5+7&.'+'% [email protected](7:%8.($$<3 '+3$(/(M.,, :+3%$.4. . (2'.4. (197101, A%$4'-B+'+7>17M, 27. "7($5+74&4.,,
0. 49, e-mail: [email protected]).
126
E'+'/,! F%-'%( G+3"!%A (A%$4'-B+'+7>17M, H(&&.)) – :%M.&'7
@%41/-'+'% '(;$(, :+3%$.4. . '+3$(/(M., A%$4'-B+'+7>17M&4(M( M(&10%7&'5+$$(M( 1$.5+7&.'+'% [email protected](7:%8.($$<3 '+3$(/(M.,, :+3%$.4.
. (2'.4.(197101, A%$4'-B+'+7>17M, 27. "7($5+74&4.,, 0. 49, e-mail:
[email protected]).
H.2%$ ?!'$ &$'.,(3"!%A (A%$4'-B+'+7>17M, H(&&.)) – :%M.&'7
@%41/-'+'% '(;$(, :+3%$.4. . '+3$(/(M., A%$4'-B+'+7>17M&4(M( M(&10%7&'5+$$(M( 1$.5+7&.'+'% [email protected](7:%8.($$<3 '+3$(/(M.,, :+3%$.4.
. (2'.4.(197101, A%$4'-B+'+7>17M, 27. "7($5+74&4.,, 0. 49, e-mail:
[email protected]).
&$#+""! G+%5 I"+;""!%A (A%$4'-B+'+7>17M, H(&&.)) – %&2.7%$' @%41/-'+'% '(;$(, :+3%$.4. . '+3$(/(M., A%$4'-B+'+7>17M&4(M(
M(&10%7&'5+$$(M( 1$.5+7&.'+'% [email protected](7:%8.($$<3 '+3$(/(M.,, :+3%$.4. . (2'.4.(197101, A%$4'-B+'+7>17M, 27. "7($5+74&4.,, 0. 49, e-mail:
[email protected]).
About the authors
Opryshko Alexey Viktorovich (St.-Petersburg, Russia) – student of
faculty of Exact mechanics and technologies of the St.-Petersburg state university of information technology, mechanics and optics (197101, St.-Petersburg, avenue Kronverksky, pr.49, e-mail: [email protected]).
Tarasov Michael Yurevich (St.-Petersburg, Russia) – master of faculty of Exact mechanics and technologies of the St.-Petersburg state university of information technology, mechanics and optics (197101, St.-Petersburg, avenue Kronverksky, pr.49, e-mail: [email protected]).
Utkin Ivan Anatolevich (St.-Petersburg, Russia) – master of faculty
of Exact mechanics and technologies of the St.-Petersburg state university
of information technology, mechanics and optics (197101, St.-Petersburg,
avenue Kronverksky, pr.49, e-mail: [email protected]).
Andreev Yury Sergeevich (St.-Petersburg, Russia) – postgraduate
student of faculty of Exact mechanics and technologies of the St.-Petersburg
state university of information technology, mechanics and optics (197101,
St.-Petersburg, avenue Kronverksky, pr.49, e-mail: [email protected]).
B(/1;+$( 15.05.2011
127