Modelagem e Simulação da Dinâmica de Alto

Modelagem e Simulação da Dinâmica de Alto

!"#$%&"'('$ ') #(*$ ') %") ')& &"!)& + !"&"!)&
,"-!,"(& $.(/(& $ /$,!)*01",(&
2%)1%(3( "!/$%'"&,"2*"!(% '$ 20&+1%(' (45) $3 ,)32 /(45)
(2*",('( + 2"2,(
!6#$* 3$&/%(')
7$(! &,83"/8
!"#$%&#' # ()'*$%+,! "% -)./')0% "# 1$2!34%$%.2#5
#' 6%)7%5 10852)0%5 (#$%"%5
&5) *$)2)*')
9:;;
!"# $%&'()&
!"#$%&#' # ()'*$%+,! "% -)./')0% "# 1$2!34%$%.2#5
#' 6%)7%5 10852)0%5 (#$%"%5
!""#$%&'() "*+,#%!-& . &/&0!&'()
1),) $#2*!"!%) 3&$1!&0 3&$& & )+%#4'()
-) 5$&* -# 6#"%$# #, 7),3*%&'()
830!1&-&
9$!#4%&-)$: ;$)< $= >*!? ;&*0) >*4&
-# 90!/#!$&
$*+ ,!+-+,.+
/011
!" # $%$&'($)*' +$ ",-&! $)*'
!""#
$%#&'%( )*+,
-./*0+1*# * &#20+34. /+ 5&,6#&$+ /* 70'.89+0+,'*:
*# ;+&<+: 7$=:'&$+: *0+/+: > ?.@ )*+, $%#&'%A B 4.
C*.?.0/.( DEFFA
G! HAI &0A $.0.@J !E$#A
K@.?.:'+ L#*:'@+/.M B N,&O*@:&/+/* /. P+0* /. Q&. /.:
&,.: 8 NRS SRT A ;&U,$&+: V<+'+: * W*$,.0X1&$+: [email protected]@+#+
S,'*@/&:$&?0&,+@ /* KX:8Y@+/2+34. *# ;.#?2'+34. 7?0&$+/+
8 KSK;7( 4. C*.?.0/.( ZQ[Q ( DEFFA T@&*,'+/.@I /*
T0&O*&@+( C2&\ K+20. C2,+A
FA 70'.89+0+,'*:A
DA &:'*#+: 5&,6#&$.:A
!A ;.#?.@'+#*,'. $+X'&$. ,.: :&:'*#+:A SA /* T0&O*&@+( C2&\
K+20. C2,+A SSA W]'20.A
;5N G^FA^_AE^_
"F`Aa!^
;+'+0.1+34. ,+ ?2b0&$+34.I
Z&b0&.'*$c@&+ ;+@0+ -+@&+ Y.20+@' /* -.@+*: 8 ;QZ FE>FD"D
!" #$%&'$()*+
!" #$% &'() #(*+' #,$ $ #(*+' (-#, &"- ."/" '&"(" $ (*0$*.(1" /(2&$*2'/"2 *$22$2
'*"2 /$ $2.%/"3
!" #$% '#"- 4'*/-' 56"-$2 7"/-(8%$2 &$6" (*0$*.(1" $ 0"#&-$$*2," 9%$ 2$#&-$
#$ /$%3
!" :-";3 <-3 =%(> :'%6" =%*' /$ ?6(1$(-') &$6' '#(>'/$ $ "-($*.'@," /$2.$ .-'A'6+"3
B C!:D4 &$6" '&"(" E*'*0$(-"3
B 2$0-$.'-(' /" :F:C!3
!" '#(8" G(2+'# G'2+$# H%+'##'/ &$6" '&"(" *' $/(@," /" .$I."3
! ."/"2 "2 #$%2 '#(8"2) 9%$ /$ '68%#' ;"-#' 0"*.-(A%J-'# &'-' ' -$'6(>'@," /$2.$
.-'A'6+"3
!"#$%& "'(" ()*+*,-& *& ."/ 0*$1
2 .$3-* .4"1 2 .$3-* $).4 "
*& ."/ *.&) 5*3#)*6
! "#$%# &'# (# )*+# ) '") $,-) ./#.) 0)").( -,1%) ), (#' %)")$2, ,+.3.$)145
!"#$% &'()%#'( !
6#"*+#7(#
&'#
9,$2#9."#$%,45
!"#$% &'()%#'( !
)(
8#((,)(
8,/#"
%.+)+
%'/,
/#
-,9:;
"#$,(
,
(#'
!"#$%
! "#$%&'"#"($) * % +,-(.-+"# -$)/ )/ 01"#01), 2-2$)/" 3) 2%/ ) * $"/4*/ %
+,-(.-+"# ,)2+%(256)# +%, 3-2$%,78)2 9),"3"2 ()2$)2 2-2$)/"2: ;2$% 2) 3)6) +,-(.-+"#/)($)
<2 (=%&#-()",-3"3)2 /).>(-."2 3% "#$%&'"#"($): ?2$13%2 ,).)($)2 219),)/ '%,$)/)($) 01)
)2$"2 (=%&#-()",-3"3)2 +%3)/ +,%6%.", /1#$-+),-%3-.-3"3) ) .%/+%,$"/)($%2 ."@$-.%2
)2+A,-%2 (% /%6-/)($% 3% .%() 3%2 "#$%&'"#"($)2: ;2$% -/+#-." (" "3-7=% 3) ',)01B(.-"2
)2+A,-"2 "% 2%/ %,-9-("# ) (=% C3)#-3"3): ?22)2 )2$13%2 +%,*/D (=% #)6"/ )/ .%(2-3),"7=%
" "+#-."7=% 3%2 "#$%&'"#"($)2 )/ ."-E"2 ".A2$-."2D 01) * " 21" /"-%, 1$-#-F"7=%:
!
%4G)$-6% +,-(.-+"# 3)2$) $,"4"#H% * )2$)(3), "01)#)2 )2$13%2D #)6"(3% )/ .%(2-3),"7=%
"2 ."-E"2 ".A2$-."2D ) 6),-C.", 01"# " -(I1B(.-" 3%2 6%#1/)2 C(-$%2 (" 3-(>/-." 3) 1/
"#$%&'"#"($): !2 ,)21#$"3%2 3)2$) $,"4"#H% /%2$,"/ 01) % 6%#1/) C(-$% 3) 1/" ."-E"
".A2$-." $)/ '%,$) -(I1B(.-" (" C3)#-3"3) 3) 1/ "#$%&'"#"($) +%, ,)31F-, % "+",).-/)($%
3) /1#$-+),-%3-.-3"3) ) ."%2 )2+A,-%2: J%/% ,)9," 9),"#D $)/&2) 01) /)(%,)2 6%#1/)2
-(-4)/ $"-2 .%/+%,$"/)($%2 )2+A,-%2D /"2D "% /)2/% $)/+%D ,)31F)/ " 3-(>/-." 3)
,)2+%2$" 3% "#$%&'"#"($): ?($=% 1/" )2.%#H" .,-$),-%2" 3% 6%#1/) 3" ."-E" ".A2$-." 3)6)
$%/", % /)(%, 6%#1/) +%22K6)#D 2)/ 2".,-C.", " 3-(>/-." 3% "#$%&'"#"($) .%(2-3),"3%:
!"!#$!%&'(!#)*
2-2$)/"2:
L#$%&M"#"($)2D N-2$)/"2 O-(>/-.%2D J%/+%,$"/)($% ."@$-.% (%2
! "#$ !"#$%&'() *($ +',-&*.'#( #/ .0% 1#-$23%*4%52 67(*,'82 '( 9&#2%$ :#;%2<
!"#$ %#
1#-$23%*4%52 *5% .0% ,*'( '.%, '( *(7 0')0 =$%&'.7 2#-($ 272.%,> *($ *5% .0%
,*'( 5%23#(2'?&% /#5 $'2.#5.'#(2 '( 2#-($ 5%35#$-8.'#(@ A0% ,*'( 5%*2#( /#5 .0*. '2
.0% 35%2%(8% #/ (#(&'(%*5'.'%2 '( .0%'5 /-(8.'#('()@ B%8%(. 2.-$'%2 2.5#()&7 2-))%2. .0*.
.0%2% (#(&'(%*5 %C%8.2 8*( &%*$ .0% 23%*4%5 .# ,-&.'3%5'#$'8 *($ 80*#.'8 ?%0*D'#52@ A0'2
',3&'%2 '( $'2.#5.'#(2 2'(8% *$$'.'#(*& 23-5'#-2 /5%E-%(8'%2 *5% *$$%$ .# .0% #5')'(*& 2#-($@
F#G%D%5> .0#2% 2.-$'%2 $# (#. 8#(2'$%5 *( ',3#5.*(. '22-% '( 5%*& 23%*4%5 -2*)%> G0'80
'2 .0% *8#-2.'8 %(8&#2-5% '( G0'80 '. '2 -2-*&&7 '(2.*&&%$ '(@ A0% ,*'( )#*& #/ .0'2 G#54
'2 .# %;.%($ .0#2% ,%(.'#(%$ 2.-$'%2 .*4'() .0'2 *$$'.'#(*& /*8.#5 '(.# 8#(2'$%5*.'#(> *($
.# D%5'/7 G0*. '(H-%(8% * &','.%$ %(8&#2-5% 8*( 8*-2% #( 23%*4%5 $7(*,'82@ A0% 5%2-&.2
2-))%2. .0*. .0% =('.% D#&-,% #/ .0% *8#-2.'8 %(8&#2-5% 8*( 2.5#()&7 *C%8. .0% =$%&'.7
#/ * &#-$23%*4%5> 5%$-8'() .0% 23-5'#-2 ,-&.'3%5'#$'8'.7 *($ 80*#2@ I2 * )%(%5*& 5-&%>
2,*&&%5 %(8&#2-5%2 '(0'?'.2 2-80 23-5'#-2 ?%0*D'#-52 ?-.> *. .0% 2*,% .',%> $','('20%2
.0% $7(*,'8*& 5%23#(2% #/ .0% &#-$23%*4%5@ A0%5%/#5%> * 85'.%5'#-2 D#&-,% 80#'8% 20#-&$
&%*$ .# .0% 2,*&&%5 3#22'?&% D#&-,% 0#G%D%5 G'.0#-. 2*85'/7'() .0% &#-$23%*4%5 ,-2'8*&
$7(*,'82@
%&'()*+,$ 1#-$23%*4%52> 67(*,'8*& +72.%,2> 90*#2@
!"#$%&
*
!"#$%&'()% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
% +,#%-.+,+"#/ /,/#$%&!"01!2% ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
"!#
$ %$& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! #'
"!"
()*+,$*-&.*/$ 0$ -1/$2(-1-*/. .1./3$0,*4&,+$ ! ! ! ! ! #5
"!'
&$0.1$% &-/.&6/,+$% 7-3- -1/$2(-1-*/.% ! ! ! ! ! ! ! ! ! ! #
3
4!4#/1+4 &!"01!2%4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! **
'!#
,*/3$0)89$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ""
'!"
%,%/.&-% 0,%+3./$% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ""
'!'
%,%/.&-% +$*/:*)$%
'!5
&-7-% 0. 7$,*+-3< . 0,-=3-&-% 0. >,()3+-8?.% ! ! ! ! ! ! "@
'!;
%,%/.&-% 0,*4&,+$% +-A/,+$% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 'B
5
7
9
&!"01!2+ 2+6#!2+ /1 +,#%-.+,+"#/4 ! ! ! ! ! ! ! ! ! ! ! 3*
.%$1',+()% &% 1%&/,% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 38
$/4',#+&%4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 53
!#
3
! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ";
,*(1)C*+,- 0$ D$1)&. 0- +-,E- -+F%/,+- *- 0,*4&,+0$% -1/$2(-1-*/.% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 5'
!"
,*(1)C*+,- 0$ D$1)&. 0- +-,E- -+F%/,+- *- 3.%7$%/.& (3.G)C*+,- 0$% -1/$2(-1-*/.% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;'
:
2%"2,'4)% / #$+;+,<%4 .'#'$%4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! 78
;!;,!%=$+.!+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 9*
!"#$ %& '!()*$"
!"#$%& '()
*"&#%&+& ,- $+ ."./-+& ,- .0+ 12."30( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( )4
!"#$%& '('
*"&#%&+& ,- $+ &5/067&5&8/- 9
!"#$%& '(>
?-@%-.-8/&AB0 ,- -C$"D&5E83"& -8/%- 0. -5-+-8/0. -5F/%"30.: &3G./"30.
!"#$%&': );;)<(
( ( ( ( ( ( ( ( ( ( ( )=
- +-3H8"30.: 08,- rE I%-."./E83"& -5F/%"3&:
rA I%-."./E83"& &3G./"3&:
rM I%-."./E83"& +-3H8"3& %-/"5J8-&: rR I%-."./E83"& +-3H8"3& %0/&3"08&5:
LI"8,$/H83"&: M I"8-%/H83"&: mI+&..&: I I+0+-8/0 ,- "8F%3"&:
CE I3&@&3"/H83"& -5F/%"3&: CA I3&@&3"/H83"& &3G./"3&: CM I30+@5"H83"&:
CR I30+@5"H83"& %0/&3"08&5( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( )
!"#$%& '(4 K0,-50 -C$"D&5-8/- ,- $+ &5/067&5&8/- -5-/%0,"8H+"30( K&". L
-.C$-%,&: & %-@%-.-8/&AB0 ,& @&%/- -5F/%"3&: 80 3-8/%0 0. 30+@08-8/-.
-C$"D&5-8/-. ,& @&%/- +-3H8"3& - M8&5+-8/- L ,"%-"/& & @&%/- &3G./"3& ,0
+0,-50 9%!()$: );;N<( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( )
!"#$%& '(= ?-@%-.-8/&AB0 ,- $+ ."./-+& +&..&6+05& 70%A&,0 - 30+
&+0%/-3"+-8/0( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( );
!"#$%& >()
O%2M30 ,0 +&@& 50#J./"30 30+ 1 < µ < 3( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( '4
!"#$%& >('
P&3"& ,- &/%&AB0 @&%& 0 ."./-+& ,- Q-808 @&%& !"#$ - %!$#&( ( ( '=
!"#$%& >(>
R+& S%1"/& @-%"S,"3& 80 -.@&A0 ,- 7&.- ,- $+ @E8,$50 T&%+U8"30
70%A&,0 - &+0%/-3",0( V -"W0 x F 0 ,-.503&+-8/0 ,& +&..& 30+ %-5&AB0 &
@0."AB0 -+ C$- 0 M0 @%-.0 & -5& -8308/%&6.- 8& @0."AB0 D-%/"3&5 - 0 -"W0 y
%-@%-.-8/& & D-503",&,- ,- ,-.503&+-8/0 ,& +&..&( ( ( ( ( ( ( ( ( ( ( ( ( ( 'N
!"#$%& >(4
X%1"/& 3&S/"3& 80 -.@&A0 ,- 7&.- ,- $+ @E8,$50 T&%+U8"30 70%A&,0
- &+0%/-3",0( V -"W0 x F 0 ,-.503&+-8/0 ,& +&..& 30+ %-5&AB0 & @0."AB0
-+ C$- 0 M0 @%-.0 & -5& -8308/%&6.- 8& @0."AB0 D-%/"3&5 - 0 -"W0 y %-@%-.-8/&
& D-503",&,- ,- ,-.503&+-8/0 ,& +&..&( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 'N
!"#$%& >(=
YW-+@50. ,- &@5"3&AZ-. ,- [0"83&%F @&%& $+ ."./-+& 1","+-8."08&5
9-.C$-%,&< - @&%& $+ ."./-+& /%","+-8."08&5 9,"%-"/&<( ( ( ( ( ( ( ( ( ( ( ( ( '
!"#$%& >(N
K&@&. ,- [0"83&%F ,0 ."./-+& ,0 @E8,$50 T&%+U8"30 70%A&,0
&+0%/-3",0( \& -.C$-%,& $+& S%1"/& @-%"S,"3&: 8& ,"%-"/& $+ %-#"+- 3&S/"30( ']
!"#$%& >(
*"&#%&+& ,- 1"7$%3&AB0 @&%& 0 +&@& 50#J./"30( ( ( ( ( ( ( ( ( ( ( ( ( ';
!"#$%& 4()
^$%D&. ,- %-..08H83"& @&%& ,"7-%-8/-. 30%%-8/-. 9_ D&5-8,0 ): ' -
4`< +-,",&. 80 &%( ` 5"8T& 308/J8$& %-@%-.-8/& & 3$%D& ,& "+@-,H83"& & 5"8T& /%&3-a&,& %-@%-.-8/& & 3$%D& ,- 7&.- ,0 &5/067&5&8/-( ` 7%-C$E83"&
,- %-..08H83"& F 0 @08/0 08,- & "+@-,H83"& @0..$" +&"0% D&50% 9
./ 012: 'bb
&<(
*+,-#
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( >>
!"#$%& '()
6
*+,-./%0 1- 2%-3$45."&+ ,&%& 1"2-%-5/-+ .0%%-5/-+ 678 9-1"1&+ 50 &%
!"#$% &' ()*: );;<&8(
!"#$%& '(=
>"&#%&9& 1- ?"2$%.&@A0 1- $9 &B/0C2&B&5/- 9-1"10 50 &% 6
&' ()*: );;<?8(
!"#$%& '('
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ='
!"#$%
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( =D
>$&+ +-@E-+ 1- F0"5.&%G ,&%& 0 &B/0C2&B&5/-( H& ,&%/- 6&8 & +-@A0
20" 2-"/& 50+ I&B0%-+ 9JK"90+ 10 +"+/-9& - 5& ,&%/- 6?8 & +-@A0 20" 2-"/& 50+
I&B0%-+ 9L5"90+ 10 +"+/-9& 6 #$+,, &' ()*: );; 8( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( =M
!"#$%& '(D
HA0CB"5-&%"1&1- &++$9"1& ,&%& kef f ,%0,0+/0 -9 6-$./,+0
&' ()*:
);; 8( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( =<
!"#$%& '(M
N%JO.0 1& I-B0."1&1- 6 v[m/s]8 - 1-+B0.&9-5/0 6x[m]8 -9 %-B&@A0 &
!"#$%& '(<
>"&#%&9& 1- ?"2$%.&@A0 ,&%& 0 901-B0 ,%0,0+/0 -9 6 -$./,+0 &' ()*:
.0%%-5/- 6I0 [A]8 ,&%& 0 901-B0 ,%0,0+/0 -9 6 -$./,+0 &' ()*: );; 8( ( ( ( ( ( =
);; 8( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( =
!"#$%& '(
P-@A0 1- F0"5.&%G ,&%& 2%-3$45."& 1- %-++05Q5."& f = 53Hz -
.0%%-5/- 1- -K."/&@A0 I0 = 2.7A ,&%& 0 901-B0 ,%0,0+/0 -9 6 -$./,+0
&' ()*: );;
8( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( =
!"#$%& D(R
H& -+3$-%1& & %-,%-+-5/&@A0 10 901-B0 1- 6 -$./,+0 &' ()*: );; 8 -
!"#$%& M(R
H& -+3$-%1& +-#$- & O#$%& .09 0+ %-/%&/0+ 1- 2&+- 10+ &/%&/0%-+ 10
10+ -K,-%"9-5/0+ 1- 6 #$+,, &' ()*: );; 8( H& 1"%-"/& $9& .&"K& &.S+/".& $+$&B( =T
&B/0C2&B&5/- "5+/&B&10 5& .&"K& &.S+/".&( U -"K0 y %-,%-+-5/& & I-B0."1&11- 1-+B0.&9-5/0 10 .05- -9
m
:
s
0 -"K0 x %-,%-+-5/& 0 1-+B0.&9-5/0 10 .05-
-9 m - 0 -"K0 V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 ( H& 1"%-"/&
0 1"&#%&9& 1- ?"2$%.&@A0 0?/"10 & ,&%/"% 1& O#$%& 1& -+3$-%1&( U -"K0
V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 - 0 -"K0 x %-,%-+-5/& 0
1-+B0.&9-5/0 10 .05- -9 m( V .0%%-5/- 1- -K."/&@A0 $/"B"W&1& 20" 1- MV( (
!"#$%& M() >"&#%&9& 1- ?"2$%.&@A0 ,&%& $9& .0%%-5/- 1- -K."/&@A0 1- ;:DV( U
-"K0 V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 ( U -"K0 x %-,%-+-5/&
0 1-+B0.&9-5/0 10 .05- -9 m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& M(= >"&#%&9& 1- ?"2$%.&@A0 ,&%& $9& .0%%-5/- 1- -K."/&@A0 1- RV( U
-"K0 V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 ( U -"K0 x %-,%-+-5/&
0 1-+B0.&9-5/0 10 .05- -9 m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& M(' >"&#%&9& 1- ?"2$%.&@A0 ,&%& $9& .0%%-5/- -K."/&@A0 1- )V( U -"K0
V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 ( U -"K0 x %-,%-+-5/& 0
1-+B0.&9-5/0 10 .05- -9 m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& M(D >"&#%&9& 1- ?"2$%.&@A0 ,&%& $9& .0%%-5/- 1- -K."/&@A0 1- =V( U
-"K0 V0 %-,%-+-5/& 0 I0B$9- 1& .&"K& &.S+/".& -9 m3 ( U -"K0 x %-,%-+-5/&
0 1-+B0.&9-5/0 10 .05- -9 m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
'=
'D
'D
'M
'M
!"#$%& '('
)"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, ,6/"5&012 +, 78( 9 ,"62
V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45& 2
+,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(> )"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, ?8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(@ )"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, '8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(
)"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, >8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(AB )"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, @8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(AA )"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, 8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(AC )"&#%&*& +, -".$%/&012 3&%& $*& /2%%,45, +, ,6/"5&012 +, AB8( 9
,"62 V0 %,3%,:,45& 2 ;2<$*, +& /&"6& &/=:5"/& ,* m3 ( 9 ,"62 x %,3%,:,45&
2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(AD )"&#%&*& +, -".$%/&012 3&%& 2 ;2<$*, +, BEBBA m3 ( 9 ,"62
I0 %,3%,:,45& & /2%%,45, +, ,6/"5&012 ,* A( 9 ,"62 x %,3%,:,45& 2
+,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(A7 )"&#%&*& +, -".$%/&012 3&%& 2 &<52F.&<&45, &2 &% <";%,( 9 ,"62 I0
%,3%,:,45& & /2%%,45, +, ,6/"5&012 &3<"/&+& ,* A( 9 ,"62 x %,3%,:,45& 2
+,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(A? )"&#%&*& +, -".$%/&012 3&%& 2 ;2<$*, +, /&"6& &/=:5"/& +, BEBA
m3 ( 9 ,"62 I0 %,3%,:,45& & /2%%,45, +, ,6/"5&012 &3<"/&+& ,* A( 9 ,"62 x
%,3%,:,45& 2 +,:<2/&*,452 +2 /24, ,* m( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
!"#$%& '(A' G&: H#$%&: +& ,:I$,%+& 5J*F:, 2: %,5%&52: +, .&:, 3&%& +".,%,45,:
;&<2%,: +, ;2<$*,( 9 ,"62 y %,3%,:,45& & ;,<2/"+&+, +, +,:<2/&*,452 +2
/24, ,* ms ( 9 ,"62 x %,3%,:,45& 2 +,:<2/&*,452 +2 /24, ,* m( G&
+"%,"5& 5J*F:, &: H#$%&: /2%%,:324+,45,: &2: %,5%&52: +, .&:, 3&%& +".,%,45,:
;2<$*,:( 9 ,"62 f %,3%,:,45& & .%,I$J4/"& ,* Hz ( 9 ,"62 8*3<"5$+,
%,3%,:,45& & &*3<"5$+, +&: .%,I$J4/"&: ,* dB ( K2+&: &: H#$%&: .2%&*
#,%&+&: 3&%& $*& /2%%,45, +, ,6/"5&012 /24:5&45, +, AB8( ( ( ( ( ( ( ( ( (
7>
7>
7@
7@
7
7
?B
?A
?A
?C
?7
!
"#$%&' () * +', -$%&', .' /,0%/&.' 1/23,/ 4, &/1&'.4, ./ 5',/ 6'&' .#5/&/71/,
8'94&/, ./ :4&&/71/ ./ /;:#1'<=4) > /#;4 y &/6&/,/71' ' 8/94:#.'./ ./
./,94:'2/714 .4 :47/ /2
m
)
s
> /#;4 x &/6&/,/71' 4 ./,94:'2/714 .4 :47/
/2 m) +' .#&/#1' 1/23,/ ', -$%&', :4&&/,647./71/, '4, &/1&'14, ./ 5',/
6'&' .#5/&/71/, :4&&/71/, ./ /;:#1'<=4) > /#;4 f &/6&/,/71' ' 5&/0%?7:#'
/2 Hz ) > /#;4 @269#1%./ &/6&/,/71' ' '269#1%./ .', 5&/0%?7:#', /2 dB )
A4.', ', -$%&', 54&'2 $/&'.', 6'&' %2 849%2/ :47,1'71/ ./ !B!* m3 )
+/,1/ :',4 4 ,#,1/2' '6&/,/71' %2 ./,/26/7C4 .#7D2#:4 7=4 ,'1#,5'1E&#4) ) F(
"#$%&' () G +', -$%&', .' /,0%/&.' 1?23,/ 4, &/1&'14, ./ 5',/ 6'&' .#5/&/71/,
8'94&/, ./ :4&&/71/ ./ /;:#1'<=4) > /#;4 y &/6&/,/71' ' 8/94:#.'./ ./
./,94:'2/714 .4 :47/ /2
m
)
s
> /#;4 x &/6&/,/71' 4 ./,94:'2/714 .4 :47/
/2 m) +' .#&/#1' 1?23,/ ', -$%&', :4&&/,647./71/, '4, &/1&'14, ./ 5',/
6'&' .#5/&/71/, :4&&/71/, ./ /;:#1'<=4) > /#;4 f &/6&/,/71' ' 5&/0%?7:#'
/2 Hz ) > /#;4 @269#1%./ &/6&/,/71' ' '269#1%./ .', 5&/0%?7:#', /2 dB )
A4.', ', -$%&', 54&'2 $/&'.', 6'&' %2 849%2/ :47,1'71/ ./ !B! m3 ) ) ) ) F*
"#$%&' *)
H#'$&'2' ./ I#5%&:'<=4 6'&' 4 '91435'9'71/ '4 '& 9#8&/ J849%2/
#7-7#14K) > /#;4 I0 &/6&/,/71' ' :4&&/71/ ./ /;:#1'<=4 '69#:'.' 74 ,#,1/2'
/2 A) > /#;4 x &/6&/,/71' 4 ./,94:'2/714 .4 :47/ /2 m) @ 5&/0%?7:#'
%1#9#L'.' 54# ./ (! Hz ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (!
"#$%&' *)M
H#'$&'2' ./ I#5%&:'<=4 6'&' 4 849%2/ ./ !B!
m3 ) > /#;4 I0
&/6&/,/71' ' :4&&/71/ ./ /;:#1'<=4 '69#:'.' 74 ,#,1/2' /2 A) > /#;4 x
&/6&/,/71' 4 ./,94:'2/714 .4 :47/ /2 m) @ 5&/0%?7:#' %1#9#L'.' 54# ./ (!
Hz ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (!
!"#$%&'()%
! "#$%&" '#()*" +"&, +,-.,/,- #() 0)() (#%1" 0-)*&, &, %*1,*2%&)&,2 2"*"-)23
%21" 43 +"&, "#$%- &,2&, "2.%5)67,2 ."( (#%1" +"#.) $)-%)68" &, +-,228" &" )- )14 "2.%5)67,2
&, +-,228" (#%1" 9"-1,2: ; &%9,-,*6) ,*1-, ) +-,228" 2"*"-) ()%2 9-).) <#, +"&,("2
+,-.,/,- =" 5%(%)- &) )#&%68" >&? ,( @ABC , ) +-,228" ()%2 9"-1, <#, +"&,("2 2#+"-1)=" 5%(%)- &) &"- D>&? ,( @ABC 4 &, ()%2 &, #( (%5'8" &, $,B,2: ! "#$%&" *8" -,)0,
&, 9"-() 5%*,)- ) ,22)2 $)-%)67,2 &, *E$,5 2"*"-": F"- ,G,(+5"3 +)-) 1,-("2 ) 2,*2)68"
&, <#, ,22, *E$,5 &"/-"# 4 *,.,22H-%" )#(,*1)- ) +"1I*.%) 2"*"-) ,( &,B $,B,2 =
!""#$3
D>>JC:
;54( &%22"3 " "#$%&" '#()*" 4 .)+)B &, &,1,.1)- 9-,<#I*.%)2 &,*1-" &, #() 9)%G)
<#, $)% &, .,-.) &, D> AB ) D>>>> AB , 4 )51)(,*1, 2,5,1%$"3 +"&,*&" &%21%*0#%- .,-.)
&,
K>> 9-,<#I*.%)2 %*&%$%&#)%2: ; -,2+"21) &" 2%21,() )#&%1%$" 1)(/4( 4 /)21)*1,
-H+%&)3 -,<#,-,*&" )+,*)2 +"#."2 .%.5"2 &) "*&) +)-) -,."*',.,- 2#) 9-,<#I*.%) = %!&&#'3
D>>LC: M +"- .)#2) &,22) +-,.%28" *" )+)-,5'" )#&%1%$" '#()*" <#, "2 ,<#%+)(,*1"2 &,
H#&%" +-"N22%"*)%2 &,$,( 2,- &, )51) <#)5%&)&,3 +)-) <#, +"22)( -,+-"&#B%- "2 2"*2 ."(
N&,5%&)&, ."(+)1E$,5 O .)+).%&)&, &" 2%21,() )#&%1%$":
P( #( ,<#%+)(,*1" &, H#&%" &, )51) N&,5%&)&,3 "2 )51"Q9)5)*1,2 28" "2 %1,*2 ()%2
.-E1%."2:
R, 9)1"3 "2 )51"Q9)5)*1,2 28" "2 ,5,(,*1"2 <#, +"22#,( ()%"- $)-%)68" ,(
<#)5<#,- 2%21,() &, 2"(3 , 28" -,2+"*2H$,%2 +,5)2 &%9,-,*6)2 )#&%1%$)2 <#)*&" " (,2("
2"( 4 1".)&" ,( 2%21,()2 &%9,-,*1,2: ; .)+).%&)&, &, #( )51"Q9)5)*1, -,+-"&#B%- #( 2%*)5
2,( )&%.%"*)- &%21"-67,2 4 2%0*%N.)1%$)(,*1, (,*"- &" <#, <#)5<#,- "#1-" ,<#%+)(,*1" &,
H#&%": F"- ,G,(+5"3 ) &%21"-68" ')-(S*%.) ,( #( )51"Q9)5)*1, 1E+%." +"&, 2,- &, >> )14
>>> $,B,2 ()%"- <#, ) &, #( )(+5%N.)&"- = (")*+,-3 TLUC: !2 9)/-%.)*1,2 ."*2%&,-)(
5%*,)- #() -,2+"21) ,( 9-,<#I*.%)2 &, #( )51"Q9)5)*1, <#)*&" ,22, +"22#% #() $)-%)68"
,*1-, ± V &? =V &? ,<#%$)5, )" &"/-" &, %*1,*2%&)&,C: W" .)2" &, #( )(+5%N.)&"- ,21)
9)%G) 4 /,( ()%2 ,21-,%1)3 2,*&" +,-(%1%&" )+,*)2 $)-%)67,2 (,*"-,2 <#, >3 &?:
!"#$%& '()"*# $"+*# ,% -.$*/0-.-1$%#2 3*&* *# $4"$$%)#2 )"55*1 % *# ,%
&*'"&%1$$-67* ,- 5*5"1- &8'%.9 #$% :.$"&* #%)( * *5;%$* ,% %#$:,* 1%##%
$)-5-.<*9 & :& -.$*/0-.-1$% ,% 5*5"1- &8'%. $=+"3* 3 " 2"( 4 +-"&#B%&" )
+)-1%- &" ("$%(,*1" &, #() (,(/-)*) =."*,C3 0,-)*&" "2.%5)67,2 &, +-,228": P21)
(,(/-)*) 4 +"21) ,( ("$%(,*1" ) +)-1%- &, #() 9"*1, 2"*"-) ."*,.1)&) )" )51"Q9)5)*1,:
; (,(/-)*) *8" 4 -E0%&)3 , +"22#% &%$,-2)2 &,9"-()67,2 ,5H21%.)2: P21)2 &,9"-()67,2
+"&,( 2,- "/2,-$)&)2 1)(/4( ,( "#1-"2 ."(+"*,*1,2 &" )51"Q9)5)*1,3 ."(" ) )-)*') , )
2#2+,*28": P21)2 &,9"-()67,2 )22".%)&)2 )" ("$%(,*1" &" 2%21,() &" )51"Q9)5)*1, ."("
#( 1"&"3 +"22#,( *8"Q5%*,)-%&)&,2 <#, +"- 2#) $,B 0,-)( &%$,-2)2 &%21"-67,2 *" 2"(:
;2 *8"Q5%*,)-%&)&,2 )22".%)&)2 )"2 )51"Q9)5)*1,2 9"-)( -,.,*1,(,*1, ,21#&)&)2 +"=%.)++ /0 1233 D>>JC , =*.&#+), /0 1233 D>>JC: W"2 1-)/)5'"2 "2 )#1"-,2 2#0,-,( 9"-1,(,*1,
!
" #$%&%'(" )% *"+& %, "-.+/0"-"'.%& )%12)+ " '3+/-2'%"$2)")%& %, &%4 &2&.%,"5 6%&&%&
.$"7"-8+&9 +& "4.+$%& %.4)"$", + "-.+/0"-"'.% '+ 1:*4+ % "+ "$ -21$% % '3+ -%1"$", %,
*+'&2)%$"(3+ "& )20%$%'("& )% #$%&&3+ ;%$")"& <4"')+ +& "-.+/0"-"'.%& &3+ 4.2-2=")+& %,
*"2>"& "*?&.2*"&9 <4% $%#$%&%'." " ,"2+$2" )% &4"& "#-2*"(@%&5
A +7B%.21+ #$2'*2#"- )%&.% .$"7"-8+ C %&.%')%$ "<4%-%& %&.4)+&9 -%1"')+ %,
*+'&2)%$"(3+ "& *"2>"& "*?&.2*"&9 % 1%$2D*"$ <4"- " 2'E4F'*2" <4% 4, 1+-4,% D'2.+ %,
4," *"2>" "*?&.2*" #+)% *"4&"$ '" )2'G,2*" )+ "-.+/0"-"'.%5
H ,%.+)+-+;2" " &%$
%,#$%;")" '%&.% .$"7"-8+ C " ,%&," &4;%$2)" #+$ I !"#$%& '( )*+9 !JJKL5 M&." ,%.+)+-+;2"
C ",#-",%'.% 4.2-2=")" '" -2.%$".4$" #"$" "':-2&% )% &2&.%,"& )2'G,2*+&5 N%&4-.")+&
&4;%$%, 4," 0+$.% 2'E4F'*2" )+ 1+-4,% )" *"2>" "*?&.2*" '+ *+,#+$.",%'.+ )2'G,2*+
)+ "-.+/0"-"'.%5
O+,+ &%$: 12&.+ '%&.% .$"7"-8+9 + 1+-4,% )" *"2>" "*?&.2*" .%, 4, 2,#+$."'.%
#"#%- '" D)%-2)")% )+ "-.+/0"-"'.%5 P% 4, ,+)+ ;%$"-9 <4"'.+ ,%'+$ " *"2>" "*?&.2*"9
,"2+$ " D)%-2)")%5 Q"& .",7C, &%$: "#$%&%'.")+ <4% '3+ 7"&." %&*+-8%$ + ,%'+$ 1+-4,%
)% *"2>" "*?&.2*" #+&&R1%-9 % <4% %&." %&*+-8" )%1% &%$ .+,")" *+, 7"&."'.% *42)")+5 P%
0".+9 " ,%.+-+;2" 4.2-2=")" '%&.% .$"7"-8+ 2')2*" 1+-4,%& )% *"2>" "*?&.2*" <4% ,%-8+$", +
)%&%,#%'8+ )+ "-.+/0"-"'.% %, .%$,+& )% D)%-2)")%5 H ,%.+)+-+;2" )% %&*+-8" )+ 1+-4,%
)" *"2>" "*?&.2*" " &%$ "#$%&%'.")" '%&.% .$"7"-8+ #+)% &%$ -%1")" " <4"-<4%$ "-.+/0"-"'.%
*+'.$2742')+ #"$" ,%-8+$%& #$+B%.+& )% *"2>"& "*?&.2*"& &%-")"&5
6+ *"#R.4-+ ! &%$: "#$%&%'.")+ 7$%1%,%'.% + 04'*2+'",%'.+ )% 4, "-.+/0"-"'.% % +&
&%4& ,+)%-+& ,".%,:.2*+& *+'8%*2)+& '" -2.%$".4$"5 6+ *"#R.4-+ S &%$: "#$%&%'.")+ 4,
7$%1% %&.4)+ &+7$% &2&.%,"& )2'G,2*+&5 6+ *"#R.4-+ T C "#$%&%'.")" " -2.%$".4$" %>2&.%'.%
<4% 0+*" '+ %&.4)+ )2'G,2*+ )% "-.+/0"-"'.%&5 6+ *"#R.4-+ U C "#$%&%'.")" " 0+$,4-"(3+ )+
,+)%-+ )" *"2>" "*?&.2*" % " ,%.+)+-+;2" %,#$%;")"5 6+ *"#R.4-+ V &%$3+ "#$%&%'.")+&
+& $%&4-.")+& )%&.% %&.4)+5 M D'"-,%'.% '+ *"#R.4-+ W &%$3+ "#$%&%'.")"& "& *+'*-4&@%&
)%&.% .$"7"-8+ % "#+'.")+& .$"7"-8+& 04.4$+&5
!
! "#$!%&"#"'$( (#($)!*+',-+.!
"#$%# &'()%*+, $#-. '(-#$#/%'0' *1' 2-#3# #4(+5&'67, $,2-# , $,18 9#-7, 0#$&-5%'$
%'12:1 '$ ('-%#$ 0# *1 '+%,;<'+'/%# 2#1 &,1, , $#* <*/&5,/'1#/%,8 =,- >1? $#-7,
'(-#$#/%'0,$ ,$ 0,5$ 1,0#+,$ 1'5$ *%5+5@'0,$ /' +5%#-'%*-' ('-' ' -#(-#$#/%'67, 0#
'+%,;<'+'/%#$8
!" # $#%
A (-,0*67, 0, $,1 $# 0. (#+' 352-'67, 0# &,-(,$ #1 '+B*1 1#5, #+.$%5&, &,1,?
(,- #4#1(+,? , '-8 C*'/0, *1 &,-(, $# 1,351#/%' #1 *1' 05-#67,? #+# #1(*--' *1'
&'1'0' 0# '-? &,1(-515/0,;' # '*1#/%'/0, $*' 0#/$50'0#8 "#$%# 1,351#/%,? ' (-#$$7,
/' &'1'0' #1(*--'0' : 1'5,- D*# /,$ $#*$ '--#0,-#$8 A$ 1,+:&*+'$ 0#$%' &'1'0' %#/0#1
' #1(*--'- '$ 1,+:&*+'$ 0'$ &'1'0'$ 35@5/E'$? D*# 37, %-'/$15%5/0, #$%' &,1(-#$$7, F$
&'1'0'$ '0G'&#/%#$ # $*2$#DH#/%#$8 C*'/0, , &,-(, $# 1,351#/%' /' 05-#67, ,(,$%'?
0#$&,1(-51# 'D*#+' &'1'0' 0# '-? &'*$'/0, ' 0515/*567, 0' 0#/$50'0#? <,-1'/0, *1'
@,/' -'-#<#5%' ,* 0# 0#(-#$$7, 0# 0#/$50'0#8 A 05<#-#/6' 0# (-#$$7, #/%-# *1' @,/'
&,1(-5150' # *1' @,/' -'-#<#5%' (,0# $#- #4(-#$$' #1 +52-'$ (,- (,+#B'0' D*'0-'0' ,*
#1 05/'$ (,- &#/%)1#%-, D*'0-'0, I2.-5'J8 A $*&#$$7, 0# @,/'$ &,1(-5150'$ # -'-#<#5%'$
<,-1' , D*# &E'1'1,$ 0# (#-%*-2'67, 0# (-#$$7,? $# (-,('B'/0, &,1, *1 1,351#/%,
,/0*+'%K-5,8 L $,1 : 0#>/50, (,- D*'+D*#- (#-%*-2'67, 0# (-#$$7, 0, '- D*# $#G' &'('@
0# $#/$525+5@'- , ,*350, E*1'/,8 M#-'+1#/%#? %'5$ (#-%*-2'6N#$ #/&,/%-'1;$# /' <'54' 0#
<-#D*O/&5' D*# 3'5 0,$ PQ R@ ',$ PQQQQ R@8
S'0' ' 2-#3# #4(+5&'67, D*'+5%'%53' 0, &,/&#5%, 0# $,1? (,0#;$# 1,0#+'- %'+
<#/T1#/, <)$5&, 0# *1' <,-1' 1'%#1.%5&' &,1, *1 <#/T1#/, ,/0*+'%K-5,8 A ,2%#/67, 0'
#D*'67, 0' ,/0' $# 0. (#+' &,125/'67, 0' $#B*/0' +#5 0# "#U%,/ ('-' , 1,351#/%,? 0'
+#5 0,$ B'$#$ # 0' +#5 0# &,/$#-3'67, 0'$ 1'$$'$8 S#(,5$ 0# '+B*1'$ 1'/,2-'$ '+B:2-5&'$?
#$$' &,125/'67, -#$*+%' /' $#B*5/%# #D*'67, 0' ,/0' #1 $*' <,-1' */5051#/$5,/'+? ('-'
*1' 05$%V/&5' #1 -#+'67, ' <,/%# $,/,-' $*>&5#/%#1#/%# B-'/0# I
1 d2 p
d2 p
= 2 2.
dx2
c dt
!"#$!%? WWXJY
IP8 J
Z1 IP8 J? p : ' (-#$$7, $,/,-'? x : ' 05$%V/&5' &,1 -#+'67, F <,/%# $,/,-'? c : '
3#+,&50'0# 0# (-,('B'67, 0, $,1 /, 1#5, # t , %#1(,8
A #D*'67, 0# (-,('B'67, 0, $,1 : +5/#'-8 =,-%'/%,? '$$*1#;$# D*# 0#(,5$ 0#
0#54'- ' <,/%# $,/,-'? /7, ,&,--#1 05$%,-6N#$ -#+#3'/%#$8 A, &,/%-.-5,? $'2#;$# D*# %'5$
05$%,-6N#$ ,&,--#1 /, $5$%#1' (-,0*%,- 0# $,1 #? #1 ('-%5&*+'-? /,$ '+%,;<'+'/%#$8 S#
<'%,? ,$ 1,351#/%,$ 5/%-)/$#&,$ ', <*/&5,/'1#/%, 0,$ '+%,;<'+'/%#$ $7, /7,;+5/#'-#$ #?
!
"#$%&'( #")%'(" *#+#&)#", +-.)/+("0 1-/" +(23(*)-2#&)(" &4( 5/&#-*#" '/")(*+#2 ( "(2
(*/$/&-50
!
"#$%&'$()*$+' ,' (-+'."(-($+* *-*+/',&$0)&%'
6" -5)(78-5-&)#" "4( 3-*)# 8%&'-2#&)-5 '# 9%-59%#* "/")#2- '# "(&(*/:-;4(0 <%-
-35/+-;4( = '-'-, &- 2-/(*/- '-" >#:#", '#&)*( '# +-/?-" -+@")/+-"0 A")-" +-/?-" -+@")/+-"
"4( -5/2#&)-'-", &(*2-52#&)#, -)*->=" '# %2 3-* '# B(", 3(* %2 -235/B+-'(* 9%#
)*-&"2/)# - /&8(*2-;4( "(&(*- -)*->=" '# %2- +(**#&)# -5)#*&-'-0
C#&)*( '- +-/?-
-+@")/+-, ( "/&-5 #5=)*/+( = '/>/'/'( #2 '%-" (% 2-/" 8-/?-" '# 8*#9DE&+/-" 3#5( '/>/"(*
'# 8*#9DE&+/-0 F(* #?#235(, &%2- +-/?- +(2 )*E" >/-" G%2 8-5-&)# '# $*->#", %2 '#
2='/(" # %2 '# -$%'("H, ( '/>/"(* '/")*/I%/ -" 8*#9DE&+/-" 2-/" I-/?-" 3-*- ( 8-5-&)# '#
$*->#", -" 2-/" -5)-" 3-*- ( 8-5-&)# '# -$%'(" G)J##)#*H # -" 8*#9DE&+/-" /&)#*2#'/K*/-"
3-*- ( 8-5-&)# '# 2='/("0 6" -235/B+-'(*#", 3(* "%- >#:, *#+#I#2 - /&8(*2-;4( "(&(*'# %2- 8(&)# +(2( %2 LC 35-M#*, 3(* #?#235( GB$%*- N0 H0
O/$%*- N0 P C/-$*-2- '# %2 "/")#2- '# "(2 IK"/+(0
C# %2- 2-&#/*- $#*-5, 9%-&)( 2#&(*#" 8(*#2 -" 8*#9DE&+/-" Q" 9%-/" ( -5)(78-5-&)#
#")K '#")/&-'(, 2-/(* ( "#% '/R2#)*(0 S-", /&'#3#&'#&)#2#&)# '/")(, ( -5)(78-5-&)# =
%2 #9%/3-2#&)( #5#)*(72#+R&/+(72-$&=)/+(, 9%# )*-&"8(*2- #&#*$/- #5=)*/+- #2 #&#*$/-+@")/+-0 T#")# )*-I-5U(, #")K "#&'( +(&"/'#*-'( ( -235/B+-'(* +(2( /'#-5 # 9%# ( '/>/"(*
'# 8*#9%E&+/-" &4( /&V%#&+/- &( "/")#2- '( -5)(78-5-&)#, "#&'( '#"3*#:-'( 3-*- B&" '#
2('#5-$#2 # "/2%5-;4(0 W9%/ "#*K -3*#"#&)-'( -3#&-" ( #")%'( #2 -5)(78-5-&)#" '#
8*#9%E&+/-" $*->#"0 T- B$%*- N0N 3('#7"# /'#&)/B+-* 2-/" 8-+/52#&)# -" 3-*)#" '( -5)(
8-5-&)#0
!
"#$%&' ()(* +#'$&',' -. %, '/0123'/'40. 5
!"#$%&'6
77 8)
9&:; ;#;0.,'; -#;0#401;6 ,'; #40.&/#$'-1; .40&. ;#6 1<.&', ., %4=;;141 ., %,
'/0123'/'40. ./.0&1-#4>,#?1* 1 ;#;0.,' ,101&6 1 -#'3&'$,' . ' ;%;<.4;@1 5
!"#$%&'6
77 8)
A B#;0.,' C101& 2 D ?1,<1;01 -. ?#4?1 <'&0.;) E '&&%./' 3&140'/ . ' <.F' <1/'&6
31&,'4-1 1 .40&.3.&&16 1 #,@6 ' G1G#4' . ' '&&%./' 0&';.#&') H1 .40&.3.&&16 14-. D 3.?I'-1
1 ?#&?%#01 ,'$4D0#?16 D $.&'-1 %, ?',<1 ,'$4D0#?1 J%'4-1 %,' ?1&&.40. '/0.&4'-'
5#431&,'F@1 ;141&'8 D '</#?'-' 4' G1G#4' ,KL./) M%'4-1 ' ?1&&.40. .;0#L.& 41 ;.,# 2
?#?/1 <1;#0#L16 1 ?14. ;. -.;/1?' ., %,' -#&.F@1) M%'4-1 ' ?1&&.40. .;0#L.& 41 ;.,# 2
?#?/1 4.$'0#L16 1 ?14. ;. -.;/1?' 4' -#&.F@1 1<1;0') N'&' J%. #;01 1?1&&' ?1, ,'.;0&#'6
' G1G#4' -.L. ,1L#,.40'&2;. ;#,.0&#?',.40. 4'; -%'; -#&.FO.;) P;0' ;#,.0&#' -.<.4-.
-1 ?',<1 ,'$4D0#?16 -.L.4-1 ;.& 1 ,'#; %4#31&,. <1;;=L./ <'&' .L#0'& -#;01&FO.; 41
,1L#,.401 -1 -#'3&'$,')
A +#'3&'$,' 2 D ?1,<1;01 41&,'/,.40. <1& %, ?14. . %, -1,1 <&10.01&) A -1,1
0., ' 3%4F@1 -. <&10.$.& 1 .;<'F1 .40&. ' G1G#4' . 1 .40&.3.&&16 <'&' J%. '/# 4@1 ;.Q'
-.<1;#0'-1 4.4I%, 0#<1 -. ;%Q.#&') R14.; -. '/0123'/'40.; ;@1 ?1,%,.40. .S</#?'-1;
'0&'LD; -' '4'/1$#' ?1, %, <#;0@16 #4T4#0',.40. &=$#-16 ?1,<&#,#4-1 1 '&) N1&D,6 4'
<&U0#?'6 ?14.; 4@1 ;@1 #4T4#0',.40. &=$#-1;) +.<.4-.4-1 -. ;%' 31&,' . -1 ,'0.&#'/ -1
J%'/ 31&', ?14;0&%=-1;6 -#L.&;'; -.31&,'FO.; <1-., 1?1&&.&) P;0'; -.31&,'FO.; L'&#',
?1, ' 3&.JV:4?#'6 . #4W%.4?#', -#&.0',.40. 4' &.;<1;0' ., 3&.JV:4?#' -1; '/0123'/'40.;)
E B%;<.4;@1 2 D ?1,<1;0' <./1 '4./ -' ;%;<.4;@1 . ' '&'4I') 9.?4#?',.40.6
.;0.; ./.,.401; ;@1 &.<&.;.40'-1; ;.,<&. <./' 3'?#/#-'-. '1 ,1L#,.4016 -.41,#4'-1
?1,</#>4?#' 5#4L.&;1 -' &#$#-.X8) H1 ?';1 -' ,'#1&#' -1; '/0123'/'40.;6 ' '&'4I' ?140&#G%#
?1, ?.&?' -. YZ[ -' ?1,</#>4?#' 010'/6 . 1 '4./ ?1, 1; &.;0'40.; (Z[ 5
!"#$%&'6
!
"" #$ % &'() (*+(,'- .-/( 0(, 1(2+- /( 3-,,&45&6 (0.78& -7 +(42/- +,&+&/-$ 90+- &:7/&
& 8&'+(, - 4-'( 4('+,&/- ( ;&,&'+( 0(7 8-<28('+- (8 +-,'- /( 07& .-02=>- /( ,(.-70-$
?/242-'&)8('+(6 ()( @ ,(0.-'0A<() .-, &8-,+(4(, -0 8-/-0 /( <23,&=>- '& 3-,/& (*+(,'&
/- 4-'($ B 20+- 1&C 4-8 D7( 2'E7('42( /2,(+&8('+( '& ,(0.-0+& /- &)+-F1&)&'+($ ? &,&'5&
@ '-,8&)8('+( 4-'1(442-'&/& /( +(42/- ,(1-,=&/- ( 4-,,7;&/-$ G(7 .&.() @ 8&'+(, &
3-32'& 4('+,&/& '& .(=& .-)&, ( +&83@8 &:7/& & 4-)-4&, - 4-'( /( <-)+& & .-02=>- /(
,(.-70-$ B)& +&83@8 @ ,(0.-'0A<() .-, 1-,'(4(, & 8&2-, .&,+( /& 1-,=& ,(0+&7,&/-,&
H/(0)-4&8('+-I4-8.)2J'42&# &- &)+-F1&)&'+($
?;-,& D7( &0 .&,+(0 .,2'42.&20 /-0 &)+-F1&)&'+(0 1-,&8 &.,(0('+&/&06 .-/(F0(
('+('/(, - 0(7 17'42-'&8('+-6 D7( 0( 3&0(2&6 (8 07& ;,&'/( 8&2-,2&6 '- 8-<28('+- /( 78
/2&1,&;8& H-7 4-'(#$ K7&'/- &.)24&8-0 78& 4-,,('+( &)+(,'&/& 4-8 2'1-,8&=>- 0-'-,&
'& 3-32'&6 D7( (0+A 28(,0& /('+,- /( 78 28> /( .-)&,2/&/( .(,8&'('+(6 78 -7+,- 4&8.8&;'@+24- @ ('+>- ;(,&/- ( & 3-32'& 4-8(=& & 8-<28('+&,F0( &*2&)8('+( (8 ,()&=>- &28>$ L-8 - 8-<28('+- /& 3-32'&6 +-/- - 4-':7'+- 8M<() H3-32'&6 &,&'5&6 070.('0>( 4-'(# /- &)+- 1&)&'+( 4-)-4&F0( (8 8-<28('+-$ L-8 200-6 & 4-,,('+( <&,2A<() 1&C 4-8
D7( - 4-'( <23,( '&0 8(08&0 1,(DNO'42&0 0-'-,&0 /& 1-'+(6 4-8.,282'/- ( ,&,(1&C('/- &, P 07& 1,('+(6 20+- @6 (82+2'/- 0-8$ ? 2'+('02/&/( 0-'-,& (0+A /2,(+&8('+( )2;&/& P
D7&'+2/&/( /( &, 8-<28('+&/&6 D7( <&,2& /(.('/('/- /- /2J8(+,- /- &)+-F1&)&'+( ( /&
.-+O'42& &.)24&/& .()- &8.)2Q4&/-,$
R&,& & 8&*282C&=>- /& Q/()2/&/(6 &0 4&,&4+(,S0+24&0 /( 78 &)+-F1&)&'+( /(<(8 0(,
&0 8&20 )2'(&,(0 .-00S<(20$ B'+,(+&'+-6 '& .,A+24&6 20+- 0( &.,(0('+& 4-8- 78 ('-,8(
/(0&Q-$ ? .,(0('=& /( 4&,&4+(,S0+24&0 '>-F)2'(&,(0 '- 17'42-'&8('+- /( 78 &)+-F1&)&'+(
4&70& /20+-,=T(0 .-, &.&,(428('+- /( 8-/-0 /( <23,&=>- 2'&/(D7&/-0 2'+(,&;2'/- 4-8 &0
1,(D7O'42&0 0-'-,&06 /(+7,.&'/-F&0$ B'+,( &0 .-00S<(20 4&70&0 /( '>-F)2'(&,2/&/(0 (0+>- &
1&)+& /( ,2;2/(C /- 4-'( ( /& 070.('0>- 4-8- 78 +-/-$
!" #$%&'$( #)*&#+*,-$( .)/) )'*$01)')2*&(
G(,>- &.,(0('+&/-0 &D72 /-20 8-/()-0 &8.)&8('+( 7+2)2C&/-0 .&,& & ,(.,(0('+&=>-6
8-/()&;(8 ( 17'42-'&8('+- /( &)+-F1&)&'+(0$ % .,28(2,- /()(0 1&C 78& &'&)-;2& /&0
.&,+(0 ()@+,24&F8(4J'24&F&4U0+24& /- &)+-F1&)&'+( 4-8 42,472+-0 ()@+,24-0 (D72<&)('+(06 (
0(,A &.,(0('+&/- &.('&0 .&,& Q'0 /( 2)70+,&=>-6 .-20 '>- 0(,A - &/-+&/- '- .,(0('+(
+,&3&)5-$ % 8-/()- & 0(, 7+2)2C&/- '(0+( +,&3&)5- @ - 0(;7'/-6 D7( 1&C &'&)-;2& &020+(8& 8&00&F8-)&$ G(,A 7+2)2C&/- - 8-/()- &'A)-;- &- 020+(8& 8&00&F8-)& .-20 (0+(
+,&3&)5- (0+A 0('/- 1-4&/- '- (0+7/- /& /2'J824& 8(4J'24& /- &)+-F1&)&'+($
V& .,28(2,& 8-/()&;(86 /&0 .&,+(0 ()@+,24&F8(4J'24&F&4U0+24& /- &)+-F1&)&'+(6 @
'(4(00A,2- +,&'01-,8&, &0 .&,+(0 8(4J'24& ( &4U0+24& (8 42,472+-0 (D72<&)('+(0$ R&,&
20+-6 '& Q;7,& W$X 0>- &.,(0('+&/&0 &0 (D72<&)O'42&0 ('+,( ()(8('+-0 ()@+,24-06 &4U0+24-0 (
8(4J'24-0 H
!" #6
"YX#$
rE
rA
rR
m
CM
I
rM
M
L
CE
CA
CR
rA
rE
Evc (s) = [RE + Red (ω) + sLe (ω)]Ivc (s) + βlV (s),
1
+ Sd2 ZA (s)),
βlIvc = V (s)(Rms + sMms +
sCms
!
• Evc (s) " #$%#$&$'() ) ($'&*+ )%,-.)/) ') 0+0-') /+ ),(+ 1),)'($2
• RE " ) %)#.$,) .+'&()'($ $3 #$,)4*+ 5 1#$678'.-) /) #$&-&(8'.-) /) 0+0-') /+
),(+"1),)'($2
• Red (ω) " ) %)#.$,) /) #$&-&(8'.-) /) 0+0-') 9)#-:9$, .+3 ) 1#$678'.-)2
• Le (ω)" ) -'/7(;'.-) /) 0+0-') /+ ),(+"1),)'($< 67$ = 9)#-:9$, .+3 ) 1#$678'.-)2
• Ivc (s) " ) .+##$'($ 67$ .-#.7,) ') 0+0-') /$9-/+ ) Evc (s)2
• βl " + %#+/7(+ $'(#$ ) /$'&-/)/$ /$ >7?+ 3)@'=(-.+ %$,+ .+3%#-3$'(+ /) 0+0-')
/$'(#+ /+ .)3%+ 3)@'=(-.+2
• Rms " ) #$&-&(8'.-) 3$.;'-.) /+ ),(+"1),)'($2
• Mms " ) 3)&&) 3A9$, /+ ),(+"1),)'($2
• Cms " ) .+3%,-;'.-) /+ ),(+"1),)'($2
• V (s) " ) 9$,+.-/)/$ /$ /$&,+.)3$'(+ /+ .+'$ /+ ),(+"1),)'($2
• Sd " ) :#$) $1$(-9) /+ .+'$2
• ZA (s) " ) -3%$/;'.-) ).B&(-.) /+ 3$-+2
• U (s) " ) .+##$'($ 67$ .-#.7,) ') -3%$/;'.-) ).B&(-.)C
D$&($ 3+/$,+< Evc (s)< RE < Red (ω)< Le (ω) $ βlV (s) #$%#$&$'()3 ) %)#($ $,=(#-.) /+
),(+ 1),)'($< +& -($'& Rms < Mms < Cms $ βlIvc (s) #$%#$&$'()3 ) %)#($ 3$.;'-.) $ ZA (s)
#$%#$&$'() ) %)#($ ).B&(-.) /+ ),(+"1),)'($C E ).+%,)3$'(+ $'(#$ ) %)#($ $,=(#-.) $ ) %)#($
3$.;'-.) = 1$-() )(#)9=& /$ 73 @-#)/+#< 67$ %+/$ &$# /$F'-/+ .+3+ 73 67)/#-%+,+ +'/$
) ($'&*+ $3 73 %)# /$ ($#3-')-& = /-#$()3$'($ %#+%+#.-+'), 5 .+##$'($ '+ +7(#+ %)#
/$ ($#3-')-&C E ).+%,)3$'(+ $'(#$ )& %)#($& 3$.;'-.) $ ).B&(-.) = 1$-() )(#)9=& /$ 73
(#)'&1+#3)/+# -/$), .+3 73) #$,)4*+ /$ $&%-#)& Sd : 1 G !"#$< HHIJC
K#$67$'($3$'($< )/+()"&$ 73) )'),+@-) .+3 + &-&($3) 3)&&)"3+,) 1+#4)/+
)3+#($.-/+ .+3+ 3+/$,+ &-3%,-F.)/+ %)#) 73 ),(+"1),)'($ G %$&'!$< LMMNJC O+3 $,$<
&$ $&(7/) ) /-';3-.) /+ 17'.-+')3$'(+ 3$.;'-.+ /+ ),(+"1),)'($ $3 +%+&-4*+ )+ %#-3$-#+
3+/$,+< 67$ &$ %#$&() 3)-& %)#) + $&(7/+ $,$(#+/-';3-.+ /+ 3$&3+C E &-&($3) 3)&&)
]< %#$&) /$ 73
3+,) .+3 )3+#($.-3$'(+ -/$), .+'&-&($ /$ 73) 3+,) .+3 .+'&()'($ k[ N
m
,)/+ ) 73) %)#$/$ F?) $< /+ +7(#+< ) 73) 3)&&) M [Kg]C P)#),$,+ 5 3+,)< %#$&+ ') 3$&3)
%)#$/$ $ ') 3$&3) 3)&&) M [Kg]< $?-&($ 73 )3+#($.$/+# .+3 .+'&()'($ /$ )3+#($.-3$'(+
c[ Nms ]C Q&($ &-&($3) $&(: $&67$3)(-R)/+ ') F@7#) LCN< +'/$ &$ %$#.$0$ 73) 1+#4) $?($#')
)+ &-&($3)< #$%#$&$'()/) %+# F [N ]C
Fs = −kx;
Fd = −cv = −c
dx
.
dt
x[m]
v[ ms ]
Fs [N ]
Fd [N ]
Fm = M a = M
d2 x
.
dt2
a[ sm2 ]
M [Kg]
x[m]
t[s]
F
Fm = Fs + Fd + F ;
d2 x
dx
M 2 + c + kx = F.
dt
dt
ω0 [ rad
]
s
!
"#$%&'()*+ $, %&$ -$(.$ + /0)*1+# (+ 0&+ ,23*,+ +,41*-&.$5 6/ 0*0-$,+ ,+00+7,/1+8 $0-+
"#$%&'()*+ 9 .+.+ 4$1/ ,/:*,$(-/ ./ 0*0-$,+ %&+(./ +0 "/#;+0 $3-$#(+ $ .$ +,/#-$)*,$(-/
0</ (&1+05 =+> #$0&1-+ %&$ -+1 "#$%&'()*+ 9 .+.+ 4/#?
ω0 =
r
k
.
M
@ 5AB
C0-$ ,/.$1/ 4/.$ 0$# &-*1*D+./ -+,E9, 4+#+ .$0)#$:$# + .*(F,*)+ .$ &, 4'(.&1/
G+#,H(*)/ "/#;+./ $ +,/#-$)*./5 6$0-$ 0*0-$,+8 &,+ ,+00+ 9 4$(.&#+.+ 4/# &, I/ $, &,
4/(-/ I3/5 6$0-$ 0*0-$,+8 + 4/0*;</ .$ $%&*1>E#*/ 0$ .2 %&+(./ / I/ 0$ $()/(-#+ (+ :$#-*)+15
J,+ :$D .$01/)+.+ .$0-+ 4/0*;</8 + K#+:*.+.$ +K$ .$ "/#,+ #$0-+&#+./#+8 4#/:/)+(./ &,
,/:*,$(-/ /0)*1+-L#*/ +(21/K/ +/ 0*0-$,+ ,+00+7,/1+ 1*:#$ .$ "/#;+0 $3-$#(+05 M19,
.+ K#+:*.+.$8 4/.$70$ )/(0*.$#+# $"$*-/0 .$ +,/#-$)*,$(-/0 $ "/#;+0 $3-$#(+08 .$ ,/./
+(21/K/ +/ 0*0-$,+ ,+00+7,/1+ +,/#-$)*./ $ "/#;+./5 6/ )+0/ ./ 4'(.&1/8 (+ $%&+;</
@ 5NB8 x #$4#$0$(-+ / .$01/)+,$(-/ .+ ,+00+ )/, #$1+;</ O 4/0*;</ .$ $%&*1>E#*/ $ k +
+;</ .+ K#+:*.+.$5
M 4+#-*# .+ $%&+;</ @ 5NB 4/.$70$ "+D$# &,+ +(+1/K*+ )/, / 0*0-$,+ ./ +1-/7"+1+(-$8
/(.$ $0-+,/0 *(-$#$00+./0 (/ .$01/)+,$(-/ ./ .*+"#+K,+ $, #$1+;</ O 0&+ 4/0*;</ .$
$%&*1>E#*/ @+&0'()*+ .$ )/##$(-$ +1-$#(+.+B8 .+./ 4/# x[m]5 C0-+ +(+1/K*+ 4/.$ 0$# :*0-+
$, @
!"#$!8
!!PB5 M00*,8 4/.$,/0 &-*1*D+# + $%&+;</ .*"$#$()*+1 .$ 0$K&(.+ /#.$,
@ 5Q!B .$4$(.$(-$ ./ -$,4/8 .+.+ 4/# @ %!&"'($)* +!,$)8 !!RB?
Mef f
dx
d2 x
+ Rm
+ kef f (x)x = βlI0 cos(ωt),
2
dt
dt
/(.$?
• x[m] 7 .$01/)+,$(-/ ./ .*+"#+K,+ $, #$1+;</ O 4/0*;</ .$ $%&*1*E#*/S
• Mef f [Kg] 7 ,+00+ ,L:$1S
• Rm [ Kg
] 7 +,/#-$)*,$(-/ ./ 0*0-$,+8 .+./ 4$1+0 4$#.+0 ,$)F(*)+0S
s
] 7 #*K*.$D .+.+ 4$1/ +($1 .+ 0&04$(0</ $ 4$1+ +#+(G+S
• kef f [ N
m
• β[T ] 7 )+,4/ ,+K(9-*)/8 -/,+./ )/,/ G/,/K'(*/ ($0-$0 $0-&./0S
• l[m] 7 -+,+(G/ ./ I/ .+ E/E*(+S
• I0 [A] 7 )/##$(-$ +1-$#(+.+ .$ $3)*-+;</ )/, + *("/#,+;</ 0/(/#+S
• f [Hz] 7 "#$%&'()*+ .+ )/##$(-$ .$ $3)*-+;</ I0 S
• ω = 2πf 5
@ 5Q!B
!
"#$%# &'(#)'* +'(#,$# '-$#./0. 12# #34$%# 2&0 5'6%# (# #374%089'* (0(0 +'.
βlI0 cos(ωt)* .#$2)%0(' (0 7'..#6%# 0)%#.60(0 0+)470(0 6' 0)%',50)06%# &2)%4+)470(0 +#)'
70&+' :#.0(' 0 +0.%4. (' 4&9 +#.&06#6%#; < .4:4(#= (0(0 +#)' 06#) (0 $2$+#6$9' #
+#)0 0.06>0* 0124 .#+.#$#6%0(0 +'. kef f (x)* 7'&+?# 2& 4&+'.%06%# 4%#& 6#$%# &'(#)';
@'. #3#&+)'* $# 0+.'34&0.&'$ ' 06#) (0 $2$+#6$9' # 0 0.06>0 +'. (20$ &')0$* kef f (x)
$2.:# 7'&' 2& +0.A&#%.' .#+.#$#6%06(' #$%0$ (20$ &')0$; B# 0)C& (4$$' )#/0.&'$ #&
7'6$4(#.089' 0 #)0$%474(0(# (' 7'6#* 0)C& (0 0.06>0 # (' 06#) (0 $2$+#6$9'* kef f (x) +0$$0
0 .#+.#$#6%0. ' 7'6D26%' (#$$#$ '$74)0('.#$* ' 12# 02&#6%0 046(0 &04$ 0 7'&+)#34(0(# ('
&'(#)';
E$%2('$ .#7#6%#$ FF !"#$%& '( )*+* GGHI*F,-./!0 '( )*+* GGJ0I # F,-./!0 '( )*+*
GGJ-II &'$%.0& 0 #34$%K6740 (# 69',)46#0.4(0(#$ #& kef f (x)* (#7'..#6%#$ (' 50%' (#
12# ' 7'6#* 0 0.06>0 # ' 06#) (0 $2$+#6$9' +'(#& $'5.#. 2&0 1206%4(0(# 7'6$4(#.L/#)
(# .#)#/06%#$ (#5'.&08?#$ #)L$%470$; M'&' $#.L &'$%.0(' 0(406%#* 0$ 69',)46#0.4(0(#$
46%.N6$#70$ 0 kef f (x) $2:#.#& 0 #34$%K6740 (# 2&0 (46A&470 70O%470 #& $4$%#&0$ (#
0)%',50)06%#$ F !"#$%& '( )*+* GGHI* $#6(' 2&0 4&+'.%06%# 5'6%# (# (4$%'.8?#$ 6' $'&
'.4:460);
!"!#$%&! '"()%"*+!
! "#$%$&'$ ()"*'+,- ./%) ) $,+(/0)# ) '$-#/) %-1#$ %/%'$2)% 0/&32/(-%4 1$2 (-2)"#$%$&')# $ 0/%(+'/# )% 5$##)2$&')% 6+$ 5-#)2 +'/,/7)0)% ")#) ) "#-0+89- 0-% #$%+,')0-%
0$%'$ '#)1),:-;
<=&0+,-% 5-#8)0-% $ )2-#'$(/0-%4 (-2- - ),'->5),)&'$ $ - %/%'$2)
2)%%)>2-,) 5-#8)0- $ )2-#'$(/0-4 %9- #$"#$%$&')0-% "-# $6+)8?$% 0/5$#$&(/)/% $ %9")%%*.$/% 0$ "-%%+/#$2 0/&32/()% (-2",$@)%;
<-# /%%- "$#'$&($2 ) A#$) 0$ $%'+0-%
0-% %/%'$2)% 0/&32/(-%; B/"/()2$&'$ %/%'$2)% 0/&32/(-% (-2 0/&32/()% (-2",$@)% %9)&),/%)0-% (-2 - +%- 0$ 5$##)2$&')% (-2- -% 0/)C#)2)% 0$ 1/5+#()8?$% $ -% 2)")% 0$
<-/&()#D ")#) 0/%(#$'/7)89- 0$%'$% %/%'$2)%; E-2- .$#$2-% &- ()"/'+,- F4 $%'+0-% #$($&'$%
+'/,/7)2 ')/% 5$##)2$&')% ")#) - $%'+0- 0) 0/&32/() 0$ ),'->5),)&'$%; <)#) ) #$0)89- 0$%'$
!!4 HIIJK4 G"#$%&$'("$'')')* +')%
G)&&#, % *- ./04 HIILK $ G)1)'1)2$& *- ./04 HIIMK;
()"*'+,- 5-#)2 +'/,/7)0)% )% #$5$#=&(/)% G
4 HIIFK4
!" #$%&'()*+'
N/%'$2)% 0/&32/(-% %9- %/%'$2)% 6+$ .)#/)2 &- '$2"-;
<-#')&'- ) "#/&(/"),
()#)('$#*%'/() D ) "#$%$&8) 0) .)#/A.$, '$2"- &)% $6+)8?$% 2)'$2A'/()% 6+$ -%
#$"#$%$&')2; O2 $@$2",- 0$%'$% %/%'$2)% %9- )% $6+)8?$% 0) 2$(3&/()4 (-2- ) 0)
"-%/89- /&%')&'3&$) 0$ +2 -1P$'- $2 2-./2$&'- $2 5+&89- 0- '$2"-; Q) &)'+#$7)
$&(-&'#)2>%$ .A#/-% -+'#-% $@$2",-% 0$ %/%'$2)% 0/&32/(-%; ! 2-./2$&'- 0-% (-#"-%4
) '$2"$#)'+#) ,-(), 0) )'2-%5$#)4 - R+@- 0$ +2 #/- $ - (#$%(/2$&'- 0$ +2 %$# ./.- %9),C+&% 0$%'$% $@$2",-%;
!% %/%'$2)% 0/&32/(-% "-0$2 %$# 0/./0/0-% $2 0$'$#2/&*%'/(-% $ $%'-(A%'/(-%; !%
%/%'$2)% 0$'$#2/&*%'/(-% %9- )6+$,$% #$C/0-% "-# $6+)8?$% 0$'$#2/&*%'/()% $&6+)&'- 6+$
-% $%'-(A%'/(-% %S "-0$2 %$# $%'+0)0-% "-# %+)% "#-"#/$0)0$% "#-1)1/,*%'/()%; !% %/%'$2)%
0/&32/(-% "-0$2 %$# )/&0) 0/./0/0-% $2 0/%(#$'-% $ (-&'*&+-%; T%'$% 0-/% U,'/2-% %$#9)"#$%$&')0-% &)% %$8?$% J; $ J;J4 #$%"$('/.)2$&'$;
!, -#-%./0- (#-1&.%'N/%'$2)% 0/&32/(-% 0/%(#$'-% %9- %/%'$2)% -&0$ ) .)#/A.$, '$2"-#), )%%+2$ .),-#$%
0$ 5-#2) 0/%(#$'); <-# $@$2",-4 6+)&0- %$ '$2 +2 2)"$)2$&'- 0$V&/0- $2 +2 (-&P+&'-
A ⊂ Rm
f : A → A ( "!'&0(+,$ 1$*"!"#. .' 1$*"!2./(/ ("
!#./(+3." 2. f ( 4(/#!/ 2. &'( 1$*2!+,$ !*!1!(0 x0 ∈ A5 6""!' $7#%'8". ( 9/7!#( (""$1!(2(
( ."#( 1$*2!+,$ !*!1!(0 (x0 , x1 , . . . , xn , . . .) $*2.:
!"#$ %
&'( )&*+,$ -.#$/!(0
xn = f (xn−1 ) = f n (x0 ), n = 1, 2, ...
;<5=>
!
"#$%&'#()* +#$# *,*$-.# % /0&12#3 f : R → R; f (x) = 2x − 14 5* x0 < 13 6*7*$#)
limn→−∞ f n (x0 ) = −∞4 8%)# x0 > 13 limn→+∞ f n (x0 ) = +∞4 90%&'# x0 = 13 )*$-7*
6*7*$#) f n (x0 ) = 14 :%7% *)6* +%)#3 -#'*$#) %;7$%7 <0* x0 = 1 = 0$ -#&6# ;,#3 >)6#
=3 f (x) = x4 ?%>) %>&'%3 *)6* -#&6# ;,# = 7*-0.)#73 -#>) -%7% <0%.<0*7 @%.#7 >&>+>%.3 62#
-7A,>$# <0%&6# )* <0*>7% '* x0 3 % /0&12# f (x) )*$-7* >7B 6*&'*7 -%7% >&;&>6#4 C '>&D$>+%
'* )>)6*$%) .>&*%7*)3 '# 6>-# f (x) = ax + b, ∀a, b ∈ R, a ∈
/ 0; 1 3 = E%)6%&6* )>$-.*)4 F)6*)
)>)6*$%) -#))0*$ 0$ -#&6# %67%6#7 <0%&'# |a| < 13 #0 7*-0.)#7 <0%&'# |a| > 13 *$
b
x = (1−a)
4 8#&60'#3 )>)6*$%) /G)>+#) 7*%>) &% )0% H7%&'* $%>#7>% &2# )2# .>&*%7*) * -#'*$
%-7*)*&6%7 '>&D$>+%) )>$-.*) #0 %.6%$*&6* +#$-.*,%)4
I$ '#) $%>) >$-#76%&6*) *,*$-.#) $%6*$B6>+#) '* '>&D$>+% &2# .>&*%7
0&>'>$*&)>#&%. = % /%$G.>% '%) /0&1J*) <0%'7B6>+%)3 '*;&>'%) +#$#3 Fµ : R → R; Fµ (x) =
µx(1 − x)3 -%7% µ > 14 F)6% /%$G.>%3 6%$E=$ +#&K*+>'% +#$# /%$G.>% .#HG)6>+%3 =
%E#7'%'% &% $%>#7>% '* .>@7#) <0* 67%6%$ '* '>&D$>+% &2# .>&*%7 '* )>)6*$%) '>)+7*6#)4
8#$# )*7B @>)6# E7*@*$*&6* % )*H0>73 &*)6* 6>-# '* /0&12# -#'*$#) #E)*7@%7 $0>6#) '#)
7*H>$*) -7*)*&6*) *$ )>)6*$%) '>&D$>+#) *$ H*7%.3 >&+.0>&'# % '>&D$>+% '#) %.6#(/%.%&6*)3
@%7>%&'#()* %-*&%) # -%7D$*67# µ4
L*)#.@*&'# % *<0%12# Fµ (x) = x3 #E6*$#)3 +#$# -#&6#) ;,#) '% /0&12# Fµ , xp1 = 0
* xp2 = (µ−1)
4 M%N*&'# 0$% %&B.>)*3 )* µ > 13 # -#&6# ;,# xp2 -*7$%&*+*7B &# >&6*7@%.#
µ
[0; 1] * # -#&6# ;,# xp1 = 0 )*7B 7*-0.)#7 )* µ > 14 O -#&6# ;,# xp2 )*7B 7*-0.)#7 )* µ > 3
* %67%6#7 )* 1 < µ < 34
P# +%)# *$ <0* 1 < µ < 33 * 6#$%&'# +#$# -#&6# >&>+>%. x0 ∈ (0; 1)3 % /0&12#
Fµ (x) 6*&'*7B % xp2 4 5* x0 ∈
/ [0, 1]3 % /0&12# Fµ (x) 6*&'*7B % −∞4
:%7% 3 < µ < 4 # -#&6# xp2 '*>,% '* -#))0>7 %-*&%) 0$ @%.#7 * +#$*1% % #)+>.%73
#0 )*Q%3 )07H*$ &#@#) -#&6#) -*7>A'>+#) *)6B@*>)4 F)6*) &#@#) -#&6#) -*7>A'>+#) *)6B@*>)
%0$*&6%$ <0%&'# µ @%7>% *$ '>7*12# % 44 90%&'# # )>)6*$% %6>&H* *)6* *)6BH># '*
+#$-.*,>'%'*3 % /0&12# *,>E* )*&)>E>.>'%'* '%) +#&'>1J*) >&>+>%>) -%7% %.H0&) @%.#7*) '*
µ3 *&<0%&6# <0* -%7% #067#) # $%-% $#)67% +#$-#76%$*&6# -*7>A'>+#4 90%&'# µ > 43 #
7*H>$* 6#7&%()* '*;&>6>@%$*&6* +#$-.*,# R+%A6>+#S4
M%N*&'# 0$ *)60'# &% 7*H>2# *$ <0* µ > 43 +#$ +#&'>1J*) >&>+>%>) /#7% '# >&6*7@%.#
[0; 1]3 Fµ >7B '>@*7H>7 -%7% −∞3 #0 )*Q%3 limn→∞ Fun (x0 ) = −∞4 8%)# %) +#&'>1J*)
>&>+>%>) *)6*Q%$ '*&67# '# >&6*7@%.# [0, 1]3 *,>)6>72# >&;&>6#) )0E>&6*7@%.#) Ik ⊂ [0, 1] +0Q#)
-#&6#)3 )* 6#$%'#) +#$# +#&'>1J*) >&>+>%>)3 /%72# +#$ <0* % /0&12# Fµ 6%$E=$ 6*&K%
0$ +#$-#76%$*&6# '>@*7H*&6*4 5* 7*6>7%7$#) '# >&6*7@%.# [0, 1] 6#'#) #) )0E>&6*7@%.#) '*
Ik 3 6*7*$#) 0$ +#&Q0&6# Λ (Λ = [0, 1] − Ik )3 <0* )*7B >&@%7>%&6* *$ 7*.%12# T '>&D$>+%4
U)6# <0*7 '>N*7 <0*3 6#$%&'#()* <0%.<0*7 +#&'>12# >&>+>%. x0 ∈ Λ3 % A7E>6% %))#+>%'%
% *)6* -#&6# 6%$E=$ *)6%7B +#&6>'% *$ Λ4 Λ = 0$ +#&Q0&6# '* -#&6#) 7*-0.)#7*)4 V
-#))G@*. -7#@%7 <0* *)6* +#&Q0&6# -#))0> 0$% >&;&>'%'* '* -#&6#) -*7>A'>+#)3 #&'* Fµ
*,>E* )*&)>E>.>'%'* T) +#&'>1J*) >&>+>%>)3 * <0* *,>)6* %# $*&#) 0$% A7E>6% <0* @>)>6% +%'%
7*H>2# *)+#.K>'% '* Λ3 -#7 $*&#7 <0* *.% )*Q% R !"#$!%3 WXYXS4
!
"#$%&' ()*+ ,&-./0 10 2'3' 40$567#/0 /02 1 < µ < 3)
879 :;7<0= 7&'70%>6: '3:;'6 1: 6#67:2'6 %;#1#2:;6#0;'#6= 0% 6:?'= /02 '3:;'6 %2'
@'&#-@:4) A0;7%10= 301:>6: 7&'7'& 6#67:2'6 /02 2'#6 1#2:;6B:6) C2 :D:2340 /0;E:/#10
1: 1#;F2#/' G#1#2:;6#0;'4 9 0 6#67:2' 1: H:;0;) I 6#67:2' 1: H:;0; 9 J0&2'10 30&
1%'6 :K%'LB:6 '/034'1'6 M() N : M()(N= : 1%'6 @'&#-@:#6 x : y = 0;1: a : b 6<0 3'&F2:7&06+
xn+1 = 1 − ax2n + byn ,
M() N
yn+1 = xn .
M()(N
8 1#J:&:;L' 1:67: 6#67:2' 3'&' 0 2'3' 40$567#/0 9 K%: '$0&' 06 202:;706 6<0
&:3&:6:;7'106 30& %2 3'& 0&1:;'10 (x, y) : ' O&G#7' '660/#'1' P /0;1#L<0 #;#/#'4 (x0 , y0 )
9 %2' 6:K%Q;/#' 1: 3'&:6 0&1:;'106 ((x0 , y0 ), (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ), . . .))
I 6#67:2' 1: H:;0; 301: 7:& ' 6%' 1#;F2#/' &:3&:6:;7'1' :2 %2 34';0 Oxy M:63'L0
1: J'6:N= 0;1: /'1' 30;70 (xn , yn ) 9 '660/#'10 '0 6:% 6%/:660& (x(n+1) , y(n+1) )= 6:$%#;10
'6 :K%'LB:6 M() N : M()(N) R:2>6: '66#2 ' &:3&:6:;7'L<0 10 2'3' 1: H:;0;) S'&#';10
06 @'40&:6 1: a : b= 7:2>6: @'&#'LB:6 ;06 &:$#2:6 := 3'&' '4$%;6 1:66:6 @'40&:6= 0 6#67:2'
'3&:6:;7'&- %2 /0230&7'2:;70 /0234:D0)
T0&92= K%'4K%:& 30;70 #;#/#'4 (x0 , y0 )= 1:;7&0 1: /:&7' &:$#<0 10 :63'L0 1: J'6:=
&:6%47' :2 %2' O&G#7' K%: 7:;1: 6:23&: ' %2 2:620 /0;?%;70 #;@'&#';7: 1: 30;706=
'7&'70& 10 6#67:2' M.$%&' () N) U67' .$%&'= /020 @:&:206 ;0 3&OD#20 /'357%40= 9 ';-40$'
' .$%&' !)V= :;/0;7&'1' 30& M !"#$%& '( )*+= WWVN :2 6:% 7&'G'4E0 7:O&#/0 1: 201:4'$:2
1#;F2#/' 1: '470>J'4';7:6)
!
"#$% &'()*($' +$,+$', -% .'($'# -' #/#$%0+ $%0 ' 0%#0' .+.%1 2*% ' &'()*($'
3,%.*1#',4 Λ (' &+#' -' 0+.+ 1'56#$/&'7 (' #%($/-' -% 2*% +08'# ,%5%0 + -/(90/&+ -'
#/#$%0+ %0 #*+# .,':/0/-+-%#; < #/#$%0+ -% =%('( %:/8% #%(#/8/1/-+-% ># &'(-/?@%#
/(/&/+/# -%($,' -% #*+ 8+&/+ -% +$,+?A';
B/5*,+ C; D E+&/+ -% +$,+?A' .+,+ ' #/#$%0+ -% =%('( .+,+ !"#$ % %!$#&;
!
"#"$%&'" ()*$+*,)"
F%G(%H#% *0 #/#$%0+ -/(90/&' &'0' &'($6(*' 2*+(-' + I+,/JI%1 $%0.' K &'($6(*+;
L0 %:%0.1' -% #/#$%0+ &'($6(*' 8/-/0%(#/'(+1 % (A' 1/(%+, K ' 0'-%1' -% *0 .M(-*1'
N+,0O(/&' P',?+-' +0',$%&/-'; "#$% #/#$%0+ K +(J1'5' +' #/#$%0+ 0+##+ 0'1+ P',?+-' %
+0',$%&/-' ,%.,%#%($+-' (+ G5*,+ ;!; Q'0' I/#$' (' &+./$*1' 7 %#$% $/.' -% #/#$%0+
,%.,%#%($+ 8%0 + -/(90/&+ -% +1$'HP+1+($%#;
R' 0'-%1' -' .M(-*1' N+,0O(/&' +0',$%&/-' + 0+##+ &'0%?+,J + '#&/1+, #'8 +?A' -+
5,+I/-+-% +##/0 2*% P', +.1/&+-+ *0+ P',?+ %:$%,(+ .%,/S-/&+ (' $%0.'; T%0H#% $+08K0
+ P',?+ -% P,/&?A' 3'* -/##/.+?A'4 &'0 ' 0%/' 3+,47 .,'.',&/'(+1 > I%1'&/-+-% +(5*1+, -%
'#&/1+?A';
U+,+ 2*% ' .M(-*1' /(/&/% ' #%* 0'I/0%($' '#&/1+$S,/'7 K (%&%##J,/' +.1/&+, *0+
P',?+ /(/&/+1 (+ 0+##+; V% (A' N'*I%, 0+/# (%(N*0+ P',?+ .%,/S-/&+ +.1/&+-+ +' #/#$%0+7
' 0'I/0%($' -' .M(-*1' $%(-%,J + +0.1/$*-%# &+-+ I%W 0%(',%#7 $%(-%(-' + W%,'; X##' #%
-%I% +' P+$' -%7 #'8 +?A' -+ 5,+I/-+-%7 %:/#$/, *0+ -/##/.+?A' -%I/-' > P,/&?A' &'0 ' +,;
R%#$% &+#'7 .+,+ 2*% ' .M(-*1' &'($/(*% %0 0'I/0%($'7 K (%&%##J,/' +-/&/'(+, %(%,5/+ +'
#/#$%0+7 '* #%)+7 -%I%,J #%, +.1/&+-+ (' #/#$%0+ *0+ P',?+ .%,/S-/&+;
Y.1/&+(-' *0+ P',?+ /(/&/+1 (+ 0+##+ % &'(#/-%,+(-' ' +$,/$' % P',?+# .%,/S-/&+#
+.1/&+-+# +' #/#$%0+ &'0' (*1+#7 ' .M(-*1' $%,J *0 0'I/0%($' 1/I,% -% 0'-' ,%.%$/$/I'
3.%,/S-/&'47 &'0' ' .M(-*1' -% *0 ,%1S5/';
U+,+ '# &+#'# %0 2*% %:/#$%0 P',?+# -% +$,/$' % .%,/S-/&+# +.1/&+-+# > 0+##+7 .'-%0
%:/#$/, &'0.',$+0%($'# .%,/S-/&'# % &+S$/&'#; R+ G5*,+ C;C .'-%H#% I%, *0 0+.+ -% P+#%
x I%,#*# y -% *0 #/#$%0+ .%,/S-/&'7 '(-% x K ' -%#1'&+0%($' -+ 0+##+ &'0 ,%1+?A' +
!
"#$%&'# () *+( # ,# "-($# . (/. (01#02-.3$( 0. "#$%&'# 4(-2%1./5 ( y 6 . 4(/#1%7.7( 7(
7($/#1.)(02# 7. ).$$.8
9%:+-. ;8;< =). >-?%2. "(-%>7%1. 0# ($".&# 7( @.$( 7( +) "A07+/# B.-)C0%1# @#-&.7# (
.)#-2(1%7#8 D (%E# x 6 # 7($/#1.)(02# 7. ).$$. 1#) -(/.&'# . "#$%&'# () *+( # ,# "-($#
. (/. (01#02-.3$( 0. "#$%&'# 4(-2%1./ ( # (%E# y -("-($(02. . 4(/#1%7.7( 7( 7($/#1.)(02#
7. ).$$.8
F.-. ./:+0$ 4./#-($ 7( ".-G)(2-#$ # $%$2(). *+( )#7(/. # )#4%)(02# 7# "A07+/#
B.-)C0%1# @#-&.7# .)#-2(1%7# "#7( ."-($(02.- 1#)"#-2.)(02#$ 1.>2%1#$8 H#)# (E()"/#5
"#7(3$( .0./%$.- # ).". 7( @.$( 0. ,:+-. ;8I8
9%:+-. ;8I< J-?%2. 1.>2%1. 0# ($".&# 7( @.$( 7( +) "A07+/# B.-)C0%1# @#-&.7# (
.)#-2(1%7#8 D (%E# x 6 # 7($/#1.)(02# 7. ).$$. 1#) -(/.&'# . "#$%&'# () *+( # ,# "-($#
. (/. (01#02-.3$( 0. "#$%&'# 4(-2%1./ ( # (%E# y -("-($(02. . 4(/#1%7.7( 7( 7($/#1.)(02#
7. ).$$.8
K. ,:+-. ;8I "#7(3$( #?$(-4.- +) 1#)"#-2.)(02# ?() 7%@(-(02( 7. ,:+-. ;8;8
L$2. 7%@(-(0&. 7( 1#)"#-2.)(02# $( 7(+ "(/. $%)"/($ )#7%,1.&'# 7#$ ".-G)(2-#$8
!
"#$%&'&()# &* +,-'&* ./. 0 ./12 %#)03*0 (#4&' 5-0 (& +,-'& ./. & )6(7$68& )# %9()-:# ;
%0'6<)68&2 %#';$ (& +,-'& ./1 ; 0=4'0$&$0(40 )6>?86: #@*0'A&' 0*4& %0'6#)686)&)0/ B&*0&)#
&%0(&* (&* +,-'&* )0 $&%& )0 >&*02 (C# %#)03*0 8#(8:-6' *0 & )6(7$68& )# %9()-:# ;
'0&:$0(40 8#$%:0=& D8&<468&E #- %0'6<)68& 8#$ -$ %0'?#)# @&*4&(40 ,'&()0/ F&*2 *0
0*%0'&'$#* 40$%# *-+860(402 4#)& & '0,6C# 6(40'(& )& +,-'0 ./1 *0 $#*4'&'G %'00(8H6)&2
0 6*4# 6()68& >#'40$0(40 & 0=6*49(86& )0 '0,6$0* 8&<468#* (# *6*40$&/
!" #$%$& '( %)*+,$-. ( '*$/-$#$& '( 0*12-,$34(&
I6%68&$0(40 *6*40$&* 8#(4?(-#* *C# 8#$%:0=#* )0 *0'0$ &(&:6*&)#* )6'04&$0(40/
J**6$2 -$& *#:-KC# >'05-0(40 (# 0*4-)# )0 *6*40$&* )6(7$68#* ; & -46:6L&KC# )#* $&%&*
)0 M#6(8&'; %&'& & )6*8'046L&KC# )0 *6*40$&* 8#(4?(-#*/ N*4& >0''&$0(4&2 8#$# A0'0$#*
(# 8&%?4-:# O2 *0'G -46:6L&)& (0*40 4'&@&:H# %&'& # 0*4-)# )& )6(7$68& )0 &:4#3>&:&(40*/
P$ $&%& )0 M#6(8&'; ; *0$%'0 &**#86&)# & -$ *6*40$& )0 m 05-&KQ0* )6>0'0(86&6*
f : Rm → Rm ; dx
= f (x)/ I'&4&3*0 )0 -$ $&%0&$0(4# )040'$6(&)# %0:& 6(40'*08KC#
dt
)&* <'@64&* )# *6*40$& 8#$ -$ 8#(R-(4# &@0'4# Σ )0 -$ 0*%&K# A04#'6&: )0 8#3)6$0(*C#
S )# 0*%&K# )0 >&*0 Rm / N*40 $&%0&$0(4# ; )0+(6)# )& *0,-6(40 $&(06'&T π : Σ →
Σ; x(n+1) = π(xn ), n = 0, 1, 2, . . . , n2 #()0 x0 ∈ Σ 0 π(xn ) ; )0+(6)# 8#$# & %'<=6$&
6(40'*08KC# DU(68&E )& <'@64& 5-0 %&**& %#' xn 8#$ & *0KC# Σ2 8#$# 6:-*4'&)# (& +,-'&
./V/ W 6$%#'4&(40 #@*0'A&' 5-0 Σ )0A0 *0' 4#$&)& %05-0(& # *-+860(40 %&'& 5-0 dxdtn 0
dx(n+1)
>#'$0$ -$ 7(,-:# $0(#' 5-0 XY ,'&-*/
dt
Z6,-'& ./VT N=0$%:#* )0 &%:68&KQ0* )0 M#6(8&'; %&'& -$ *6*40$& @6)6$0(*6#(&: D0*5-0')&E
0 %&'& -$ *6*40$& 4'6)6$0(*6#(&: D)6'064&E/
P46:6L&()# $&%&* )0 M#6(8&';2 # 0*4-)# )0 *6*40$&* 8#(4?(-#* *0 4#'(&
>'05-0(40$0(40 *6$%:6+8&)#/ F&%&* )0 M#6(8&'; '0)-L0$ -$ *6*40$& 8#(4?(-# & -$
*6*40$& )6*8'04# 8#$ -$& )6$0(*C# & $0(#*2 4#'(&()#3# $&6* *6$%:0*/ M#' 0=0$%:#2
!
"#$ %&'() *& +),-.$/0 1,*,#&-%,)-$2 3/$-%4&/%$2 $) .$#,-5) *$% 6/1,3$% 7/)*"8 "# #$7$
1,*,#&-%,)-$29 :)# ,%3); "#$ 6/1,3$ 7&/,6*,.$ *) %,%3&#$ .)//&%7)-*& $ "# 7)-3) <=)
*$ /&%7&.3,4$ $72,.$'() *& +),-.$/09 > 7)%%?4&2 7/)4$/ @"& 3)*$% $% 7/)7/,&*$*&% -()
3/,4,$,% *)% .$#,-5)% *$% 6/1,3$% %() #$-3,*$% 7&2) %&" #$7$ *& +),-.$/09
:)-%,*&/$-*) %,%3&#$% -()A$"3B-)#)%
= f (x, t); .)-%,*&/$A%& 3$#10# $ %&'()
*& +),-.$/0 -) &%7$') *& C$%& &%3&-*,*) (x, t) ∈ Rm+1 9 D&%3& .$%) $% %&'E&% %()
mA*,#&-%,)-$,% & $ *,/&'() 3/$-%4&/%$2 0 *&<-,*$ 7)/ t9 F 4$/,G4&2 3&#7)/$2 7)%%",
-$3"/&8$ ,//&4&/%?4&2; 7)/3$-3) $% 6/1,3$% -) &%7$') &%3&-*,*) -() /&3)/-$# H %&'()9
+)/ ,%%); -&%3&% .$%)%; *,C&/&-3&#&-3& *)% %,%3&#$% $"3B-)#)% dx
= f (x) 0 .)-%,*&/$*$
dt
m
"#$ %&@"I-.,$ *& %&'E&% Σ0 , Σ1 , Σ2 , . . . , Σn , . . . ⊂ R ; 3)*$% 7&/7&-*,."2$/&% H *,/&'()
*& t9 J$*) x0 ∈ Σ0 ; *&<-,#)% xn+1 = π(xn ), n = 0, 1, 2, . . .; .)#) %&-*) $ 7/)K&'() *$
,-3&/%&'() *$ 6/1,3$ *&<-,*$ 7)/ (x(tn ), tn ) ∈ Rm+1 .)# Σn+1 7/)K&3$*$; %&L"-*) t; %)1/&
Σ0 9 M &%7$'$#&-3) &-3/& $% %&'E&% Σn 0; &# L&/$2; "-,C)/#& & &%3G /&2$.,)-$*$ .)# $
#$-&,/$ &# @"& t $7$/&.& &=72,.,3$#&-3& -) %,%3&#$9 +)/ &=&#72); @"$-*) f 0 7&/,6*,.$
&# t .)# 7&/?)*) T ; $% %&'E&% %() 3)#$*$% &# ,-3&/4$2)% *& tn+1 − tn = T N !"#$"
%&'&( )*+!; OP!QR9
+$/$ /&7/&%&-3$/ ) #$7$ *& C$%& 1,*,#&-%,)-$2; -() %& .)-%,*&/$ ) &,=) 3&#7);
/&7/&%&-3$-*) 3)*)% )% 7)-3)% *& "#$ 6/1,3$ &# "# #&%#) 72$-)9 D) .$%) *& "#
#$7$ *& C$%& 3/,*,#&-%,)-$2; )-*& 0 /&7/&%&-3$*) ) 3&#7); &%3& *&4& %&/ 7&/7&-*,."2$/
$) 72$-) Oxy 9 +)/ .)-%&@"I-.,$; $% %&'E&% *& +),-.$/0 *& #$7$% *& C$%& 3/,*,#&-%,)-$,%
%&/() 1,*,#&-%,)-$,%9 M %,%3&#$ *) 7I-*"2) 5$/#B-,.) C)/'$*) $#)/3&.,*); 7)/ 7)%%",/
$ *&7&-*I-.,$ &=72?.,3$ -$ 4$/,G4&2 3&#7); 0 "# &=&#72) *& %,%3&#$ -() $"3B-)#)9 D$
<L"/$ S9T; %() &=,1,*$% $% $72,.$'E&% *& +),-.$/0 7$/$ "# #$7$ *& C$%& *&%3& %,%3&#$9
D) 2$*) &%@"&/*) *$ <L"/$ S9T; 3&#)% "#$ %&3$ ,-*,.$-*) "# 7)-3) ) @"$2 /&7/&%&-3$
"#$ 6/1,3$ 7&/,6*,.$9 D) 2$*) *,/&,3) *$ <L"/$ S9T; 3&#)% "#$ %&'() *& +),-.$/0 7$/$
"# %,%3&#$ &# /&L,#& .$63,.)9
dx
dt
U,L"/$ S9TV W$7$% *& +),-.$/0 *) %,%3&#$ *) 7I-*"2) 5$/#B-,.) C)/'$*) $#)/3&.,*)9 D$
&%@"&/*$ "#$ 6/1,3$ 7&/,6*,.$; -$ *,/&,3$ "# /&L,#& .$63,.)9
F <L"/$ S9T 0 $-G2)L$ $ <L"/$ X9! @"& /&7/&%&-3$ ) #$7$ *& 7),-.$/0
$7/&%&-3$*) 7)/ N $',%-) ./ 01&; YY!R &# %"$ #)*&2$L&# *& $23)AC$2$-3&%9 Z&'E&% *&
!
"#$%&'() *+# %#(,'-,.%/. 0/$-$1'2'* ., ,#2.-#* 30. .4$5., &#,6#(/',.%/# 6.($72$&#8
30'*.96.($72$&# #0 &'7/$&#: ;#%*/$/0.,9*. '**$,8 &#,# 0,' <.((',.%/' $,6#(/'%/. %'
$%=.*/$>'?+# 2. '6-$&'?@.* 2. *$*/.,'* 2$%A,$&#* &#%/B%0#*: C*/' <.((',.%/' *.(D 0/$-$1'2'
6'(' ' 6(#20?+# 2#* (.*0-/'2#* 2.*/. /('5'-E#:
F0/(' <.((',.%/' 5'*/'%/. 0/$-$1'2' %# .*/02# 2. *$*/.,'* 2$%A,$&#* ) # 2$'>(','
2. 5$<0(&'?@.*:
C*/. 2$'>(',' &#%*$*/. ., '%'-$*'( # &#,6#(/',.%/# 2'* 7(5$/'* '
&'2' $/.('?+# %#* 6'(A,./(#* 2# *$*/.,': F ,'6' -#>B*/$&# Fµ (x) = µx(1 − x) '/$%>.
$%*/'5$-$2'2.* &#, ' ='($'?+# 2# 6'(A,./(# µ8 *0(>$%2# %#='* 7(5$/'* %# *$*/.,': C*/'*
$%*/'5$-$2'2.* 6#2., *.( (.6(.*.%/'2'* '/('=)* 2. 0, 2$'>(',' 2. 5$<0(&'?+#8 &#,#
'6(.*.%/'2# %' G>0(' H:I:
J$>0(' H:IK L$'>(',' 2. 5$<0(&'?+# 6'(' # ,'6' -#>B*/$&#:
F 2$'>(',' 2. 5$<0(&'?+# 6#2. *.( #5/$2# ' 6'(/$( 2' '%D-$*. 2#* 6#%/#* G4#* .
6.($72$&#* xpi &#, ' ='($'?+# 2#* 6'(A,./(#* 2# *$*/.,': M# &'*# 2# ,'6' -#>B*/$&#8
&#%*$*/. ., '%'-$*'( # *0(>$,.%/# 2. /'$* 6#%/#* 6'(' 2$<.(.%/.* ='-#(.* 2# 6'(A,./(# µ:
M' G>0(' H:I 6#2.,#* =.( 30. &#, µ > 38 # ,'6' -#>B*/$&# 6'**' ' /.( 0, &#,6#(/',.%/#
,0-/$6.($72$&#: N 6'(/$( 2. µ > 3, 58 =O9*. .%/+# 0,' '-/.(%A%&$' 2. (.>$,.* &'7/$&#*
. &#.(.%/.* P%+# &'7/$&#*Q8 '/) µ = 4: R, &#,6#(/',.%/# 2.*/. /$6# %# .*/02# 2.
0, *$*/.,' 30'-30.( *0>.(. <#(/.,.%/. &#,6#(/',.%/#* &'7/$&#* 2# ,.*,#: N G>0('
H:I ) '%D-#>' ' G>0(' S:H8 30. <#$ #5/$2' ' 6'(/$( 2' ,.2$?+# 2. 0, '-/#9<'-'%/. '# '(
-$=(. . <#$ '6(.*.%/'2# ., P
!"#$% &' ()*8
TTI5Q: ;#,# =.(.,#* %# 6(74$,# &'6B/0-#
&#,6#(/',.%/#* 2. 5$<0(&'?@.* <#(', #5*.(='2#* ., ,.2$2'* .46.($,.%/'$* <.$/'* .,
'-/#9<'-'%/.* '# '( -$=(. . %# =D&0# $%2$&'%2# <#(/.,.%/. &#,6#(/',.%/#* &'7/$&#*:
!
!" #$#%&'(# )$*+'$,-# ,(.%$,-#
"#$ %&'(% )*+#* #(%,%-.'/ % 0%-%1(% 2%'+ #*3 +45' 6+%5% +*3 63% 5*7)489' 0(*24+%:
;9' *<4+#* 63% 6)42% 5*7)489' 0%(% 54)=342% 2%>#42%:
?)#(*#%)#'/ 63% 5*7)489'
3%#*3@#42% 0%(% +4+#*3%+ 54)=342'+ 2%>#42'+ 54+2(*#'+ %30-%3*)#* %2*4#' $ 5%5% 0'(
A',*(# B: C*1%)*D/ *3 +*6 -41(' E.%'#42 CD)%342%- FD+#*3+ G
!"#$!%/ HIJIK:
F*&6)5'
C*1%)*D/ 63 3%0% f : V → V $ 54#' 2%>#42' +* %#*)5*( L+ +*&64)#*+ 2')548M*+N
H: F* f 0'++64( +*)+4,4-45%5* L+ 2')548M*+ 4)424%4+O
P: F* f 0'++64( #(%)+4#4145%5* #'0'->&42%O
: Q 2')R6)#' 5'+ 0')#'+ 0*(4>542'+ 5* f S'( 5*)+' *3 V :
" +*)+4,4-45%5* L+ 2')548M*+ 4)424%4+ 430-42%(@ *3 430(*14+4,4-45%5*/ '6 +*R%/ 0%(%
5'4+ 0')#'+ T6%4+T6*(/ %0>+ 63 #*30' +6724*)#*3*)#* &(%)5*/ % 54+#=)24% *)#(* *-*+ )9'
+*(@ % 3*+3% T6* % 4)424%-: Q S%#' 5* f 0'++64( #(%)+4#4145%5* #'0'->&42% +4&)472% 54U*(
T6*/ 4#*(%)5'V+* 63 )W3*(' +6724*)#* 5* 1*U*+/ #'5'+ '+ *+#%5'+ 0'++X1*4+ 0%(% ' +4+#*3%
+*(9' %0('<43%5%3*)#* %#4)&45'+: Q6 +*R%/ 0%(% 63 #*30' +6724*)#*3*)#* &(%)5*/ %
>(,4#% 1%4 %2%,%( 0%++%)5'/ #9' 0*(#' T6%)#' +* T6*4(%/ 5* T6%-T6*( 0')#' )% (*&49'
')5* % S6)89' *+#@ 5*7)45%: C*145' % 4+#'/ )9' 0'5*3'+ 5*2'30'( 63 +4+#*3% 2%>#42'
*3 +6,+4+#*3%+ 4)1%(4%)#*+ *Y'6 4)5*0*)5*)#*+: ? % 2')5489'/ 54U*( T6* ' 2')R6)#' 5*
0')#'+ 0*(4>542'+ 5* f $ 5*)+' *3 V / +4&)472% 54U*( T6* 0%(% ' 2')R6)#' 5'+ 0')#'+
0*(4>542'+ 2')#45'+ )' 2')R6)#' #'#%- 5* 0')#'+ 54+0')X1*4+/ T6%-T6*( 0')#' )' 2')R6)#'
5'+ 54+0')X1*4+ * 2'3 63% 14U4).%)8% #9' 0*T6*)% T6%)#' +* T6*4(%/ 4(@ *<4+#4( 63 0')#'
0*(4>542' )*+#% 14U4).%)8%: ?+#% 2')5489' R6)#%3*)#* 2'3 % #(%)+4#4145%5* #'0'->&42%
430-42% *3 63% (*&6-%(45%5* 430-X24#% )' +4+#*3%/ ' T6* %2%((*#% *3 63 2'30'(#%3*)#'
5* Z*#*()' (*#'()'Z% #'5'+ '+ *+#%5'+ 0'++X1*4+: [%- S*)\3*)' $ 5*)'34)%5' (*2'((])24%:
?+#%+ #(]+ 2')548M*+ (*+6-#%3 )%+ 2%(%2#*(X+#42%+ 24#%5%+ )'+ *<*30-'+ 5%+ +*8M*+ :P/ :
* :^:
C* 63% 3%)*4(% &*(%-/ 63 +4+#*3% 54+2(*#' f : A → A $ 2%>#42' +* *<4+#* 63
+6,2')R6)#' V ⊂ A 26R% (*+#(489' f : V → V 5* f % V $ 2%>#42'O * 54U*3'+ T6* 63
+4+#*3% 54)=342' 2')#X)6' $ 2%>#42' +* .'61*( 63% %0-42%89' 5* _'4)2%($ 2%>#42% % *-*
%++'24%5%: ;' 2%+' 5% S6)89' -'&X+#42% Fµ (x) = µx(1 − x)/ T6%)5' µ > 4/ 5*7)45% )'
2')R6)#' Λ ⊂ [0; 1]/ 2')+#4#64 63 +4+#*3% 54+2(*#' 6)4543*)+4')%- *3 (*&43* 2%>#42' 0'4+
$ 0'++X1*- 0('1%( T6*/ )*++%+ 2')548M*+/ % S6)89' %#*)5* L+ #(]+ 2')548M*+ 5* C*1%)*D:
F4+#*3%+ 2%>#42'+ 0'5*3 %0(*+*)#%( 364#' 5' +*6 2'30'(#%3*)#' T6%-4#%#41'
%#(%1$+ 5% #'0'-'&4% 5' 2')R6)#' (*&*)#* G%#(%#'( '6 (*06-+'(K 5* +6%+ >(,4#%+: _%(%
4++' *<4+#*3 541*(+%+ #$2)42%+/ 2'3' % %0-42%89' 5% &*'3*#(4% S(%2#%-: Q6#(% S*((%3*)#%
6#4-4U%5% $ % %1%-4%89' T6%-4#%#41% 5% +*)+4,4-45%5* 5%+ 2')548M*+ 4)424%4+ %#(%1$+ 5'
2@-26-' 5'+ *<0'*)#*+ 5* BD%06)'1: _%(% *+#65%( % 54)=342% 5'+ %-#'VS%-%)#*+ 5* S'(3%
!
"#$%&'$'&($) *+,-. #'&%&/$0.* . 1$2$ 0+ 3.&45$,6 + . 0&$7,$1$ 0+ 8&9#,5$:-.;
<$&*
9+,,$1+4'$*) 5.1. (+,+1.* 4. 5$2&'#%. $ *+7#&,) *-. $12%$1+4'+ #'&%&/$0.* 2$,$ . +*'#0.
0$ 0&4=1&5$ 0+ $%'.>9$%$4'+*;
!
!"#$%"&' &'()"&' *% '+),-.'+'#)*/
"#$%# &'()%*+, $#-., '(-#$#/%'0,$ ,$ %-'1'+2,$ (*1+3&'0,$ /' 4-#' 0# $3$%#5'$
03/653&,$ # 7*# *%3+38'5 ,$ '+%,9:'+'/%#$ &,5, '+;, 0# #$%*0,< = #$%*0, 0# 03/653&'$
&'>%3&'$ #5 '+%,9:'+'/%#$ ? *5 '$$*/%, -#&#/%#@ 0'0'$ ' 7*'/%30'0# # 0'%'$ 0,$ '-%3A,$
(*1+3&'0,$< B$$, 5,$%-' &,5, #$%# '$$*/%, '3/0' ? (,*&, #C(+,-'0,< D $#A*3- $#-.,
'/'+3$'0,$ 1-#;#5#/%# '+A*/$ 0#$%#$ '-%3A,$<
D #;30E/&3' 0' #C3$%E/&3' 0# (,$$);#3$ -#A35#$ &'>%3&,$ #5 '+%,9:'+'/%#$ :,3 ,1$#-;'0'
(#+' (-35#3-' ;#8 (,- F
!" #$ %&'@
GHIJK@ ,/0# :,3 /,%'0, , '('-#&35#/%, 0# 03;#-$,$
$*12'-5L/3&,$ /, $3$%#5' *%3+38'/0,9$# 03;#-$'$ :-#7*E/&3'$ # %#/$M#$ 0# #C&3%'N.,< O5
F
()*+ #$ %&'@ GHHGK :,3 30#/%3P&'0, &',$ '%-';?$ 0' 5#03N., 0# #C(,#/%#$ 0# QR'(*/,; #
035#/$M#$ :-'&%'3$< O/%-#%'/%,@ ,$ -#$*+%'0,$ /., (,0#5 $#- &,/$30#-'0,$ ;4+30,$@ (,3$
, '*%,- /., &,+,&' &,5, :,-'5 5'/3(*+'0,$ ,$ 0'0,$@ /., (#-53%3/0, ' -#(#%313+30'0#
0,$ #C(#-35#/%,$< =*%-,$ 0,3$ '-%3A,$ %-'%'5 0, '/%39&,/%-,+# # 0' $3/&-,/38'N., 0, &',$
F
+!# ,!-@ !SST'K # F+!# ,!-@ !SST1K #5 $3$%#5'$ 0# '+%,9:'+'/%#$@ 5'$ #$%#$ /., #$%.,
/, :,&, 0#$%# %-'1'+2,@ (,3$ %-'%'5 0# $3$%#5'$ 0# '+%,9:'+'/%#$ # /., 0, '+%,9:'+'/%#
3/03;*0*'+5#/%#<
"#$%# %-'1'+2, , 3/%#-#$$# ? /' #C3$%E/&3' 0# -#A35#$ &'>%3&,$ /, :*/&3,/'5#/%, 0#
'+%,9:'+'/%#$< "#$%' 03-#N., ,$ '-%3A,$ 5'3$ &,/%*/0#/%#$ $.,U F
F
./-0!1 #$ %&'@ !SSV'K@
./-0!1 #$ %&'@ !SSV1K@ F0!"22 #$ %&'@ !SSIK # F3!()2"4 #$ %&'@ !SSIK<
O5 F./-0!1 #$ %&'@ !SSV'K #C3$%# ' %#/%'%3;' 0# &,5(-,;'N.,
0' #C3$%E/&3' 0#
&',$ #5 '+%,9:'+'/%#$ 0# A-';#$ '%-';?$ 0# #C(#-35#/%,$ (-4%3&,$< "'7*#+# %-'1'+2,@ $.,
'(-#$#/%'0,$ ,$ -#$*+%'0,$ 0# #C(#-35#/%,$ &,5 -#+'N., W ;'-3'N., 0' &,--#/%# '+%#-/'0'
#5 %,-/, 0' :-#7*E/&3' 0# -#$$,/6/&3' 0# *5 '+%,9:'+'/%# 0# A-';#$< = 7*# $# ,1$#-;'
(-35#3-'5#/%# ? *5 &,5(,-%'5#/%, 0# 0#:,-5'N., /' &*-;' 0# 35(#06/&3' # :'$# 0,
'+%,9:'+'/%#@ /' -#A3., (->C35' W :-#7*E/&3' 0# -#$$,/6/&3' FPA*-' T<GK<
!"#$%& '()* +$%,&- ./ %/--01213"& 4&%& ."5/%/16/- 30%%/16/- 78 ,&9/1.0 ): ; / '<= >/.".&10 &%( < 9"1?& 3016@1$& %/4%/-/16& & 3$%,& .& ">4/.213"& / & 9"1?& 6%&3/A&.& %/4%/-/16&
& 3$%,& ./ 5&-/ .0 &960B5&9&16/( < 5%/C$D13"& ./ %/--01213"& E 0 40160 01./ & ">4/.213"&
40--$" >&"0% ,&90% 7 !"#$% &' ()*: ;FFG&=(
H &$60% /-6$.& 30> >&"- ./6&9?/ & %/#"I0 C$/ 40--$" /-6/ 30>40%6&>/160 &6@4"30
./ ./50%>&JI0( K 30903&.0 $> -"1&9 10 &960B5&9&16/ 01./ & 5%/C$D13"& ,&%"& /16%/ 'L Hz
/ LLHz : C$/ E 4%MN">0 .& 5%/C$D13"& ./ %/--01213"&( O/-6/ -"1&9 & 30%%/16/ &96/%1&.& E
"13%/>/16&.& #%&.$&9>/16/ / & %/-40-6& /> 5%/C$D13"& .0 &960B5&9&16/ E >/.".&(
!
"#$%&' !()* +,-./0&1 2. 3&.4%56/#', -'&' 2#3.&.60., /1&&.60., 789 :.2#2', 61 '& 7
&' ()*; )<<='9(
!"#$%
>10'?,. 6' @$%&' !() 4%.; /1: 1 '%:.601 2' /1&&.60.; 61A', B'&:C6#/', ,%&$.: 6'
&.,-1,0' .: 3&.4%56/#' 21 'D01?3'D'60.( E%&$.: #6/D%,#A. 3&.4%56/#', 'F'#G1 2' 3&.4%56/#'
2. &.,,16H6/#' /1: A'D1& 2. f /2 7@$%&' !()F9( +,0', 3&.4%56/#', B'&:C6#/', AI1 ,%&$#621
$&'2'0#A':.60.; '0J 4%. -&1A1/': %: '%:.601 '-&1G#:'2':.60. %6#31&:. .: 0121
1 .,-./0&1 /16,#2.&'21 7@$%&' !()29( E.$%621 7
!"#$% &' ()*;
)<<=F9; ' @$%&' !()2 J
%: #62#/'0#A1 31&0. 2' .G#,056/#' 2. /'1, .: 'D01?3'D'60.,( K '%01& '@&:' '#62' 4%. 1
'D01?3'D'60. J .G0&.:':.60. 6I1 D#6.'&; 2.A#21 F',#/':.60. L 3'D0' 2. &#$#2.M #60&N6,./'
21 /16.; 21 '6.D 2' ,%,-.6,I1 . 2' '&'6B'(
>1 0&'F'DB1 7
!"#$% &' ()*; )<<=F9; J '-&.,.60'2' ' -1,,#F#D#2'2. 2' .G#,056/#' 2.
/'1, .: 'D01?3'D'60., -&#6/#-'D:.60. 2.A#21 L 3'D0' 2. &#$#2.M 21 /16.( O&#:.#&':.60.
1 '%01& '-&.,.60' 1 2#'$&':' 2. F#3%&/'PI1 1F0#21 '0&'AJ, 2. .G-.&#:.601 7@$%&' !( 9;
162. J '-&.,.60'21 1 2#'$&':' 21 2.,D1/':.601 21 /16. .: 3%6PI1 2' /1&&.60. 'D0.&6'2'(
+,0. .G-.&#:.601 31# 3.#01 61 '& . 1 #6/&.:.601 2' /1&&.60. 31# 2. 20 mA
( +,0. 0#-1 2.
s
!
"#$%&$'$ ()%*&$ +, - #."#/$ $ 01#234./#$ "0 /$52,
6#%*&$ +, 7 8#$%&$'$ "0 9#:*&/$;<5 "0 *' $=35>:$=$.30 '0"#"5 .5 $& (
@AAB9-,
!"#$% &' ()*?
C*$."5 $ /5&&0.30 D #%*$= $ 2, 72A 5 $=35>:$=$.30 /5'0;$ $ E#9&$& 0' "*$2 $'F=#3*"02
"#:0&0.302 0 E$&#$.302 .$ 02/$=$ "0 30'F5, G230 :0.H'0.5 "0 9#:*&/$;<5 $F$&0/0 .5E$'0.30
0' 3, 2A /5' $ $F$&#;<5 "0 I*$3&5 $'F=#3*"02 E$&#$.302 .5 30'F5, J F&5/0225 30&'#.$
0' *' .K'0&5 E#&3*$= "0 #.).#3$2 $'F=#3*"02 E$&#$.302 .5 30'F5,
LFM2 5 &02*=3$"5 F&N3#/5 $F&020.3$"5 .$ )%*&$ +, 5 $*35& #.#/#$ *'$ "#2/*22<5
259&0 $ &#%#"0O "5 /5.0, P&#'0#&$'0.30 0=0 :$O *'$ /&Q3#/$ R '$.0#&$ "0 /5'5 $ &#%#"0O
"0 /5.02 D '0"#"$, G23$ '0"#"$ D :0#3$ 20'F&0 "0 '$.0#&$ 023N3#/$, S5' 023$ '0"#"$
D F522QE0= /5.23&*#& *'$ /*&E$ "0 &#%#"0O 0' :*.;<5 "5 "02=5/$'0.35 x "5 /5.0? 20."5
I*0 023$ /*&E$ $F&020.3$ *'$ :5&'$ I*$"&N3#/$ "5 3#F5 kef f (x) = α + βx + γx2 ? 5."0
α = 480N m−1 ? β = 31N m−2 0 γ = 7, 5N m−3 F$&$ 5 $=35>:$=$.30 "52 01F0&#'0.352 0'
( !"#$% &' ()*? @AAB9-, T*923#3*#."5 kef f .$ 0I*$;<5 (@,UA- 2*&%0 *' 30&'5 /K9#/5?
I*0 20%*."5 5 $*35&? :$O /5' I*0 0230 3#F5 "0 :*.;<5 .<5 F&5E5I*0 /$52 0' 2#230'$2
"#.V'#/52, S5' #225? 5 $*35& 20%*0 0' 9*2/$ "0 .<5>=#.0$&#"$"02 .$ &#%#"0O "5 /5.0
I*0 20W$' "#.V'#/$2 $5 =5.%5 "5 30'F5? 5* 20W$? I*0 .<5 30.X$' $ '02'$ :5&'$ "$2
$F&020.3$"$2 .$2 '0"#;Y02 023N3#/$2, S5'5 /5./=*2<5 5 $*35& 2*%0&0 I*0 $ &#%#"0O "5
/5.0 F5"0 $F&020.3$& *'$ :5&'$ 0235/N23#/$,
Z5 3&$9$=X5 (#$+,, &' ()*? @AA[- 5 $*35& $.$=#2$ "$"52 /5=03$"52 01F0&#'0.3$='0.30
"0 *' $=35>:$=$.30 0 *3#=#O$ "#E0&2$2 3D/.#/$2 "0 305&#$ "5 /$52 0 $.N=#202 "0 2D&#02 30'F5&$#2
F$&$ '523&$& I*0 &0$='0.30 52 $=35>:$=$.302 $F&020.3$' /$52, \0'F52 "0 $3&$25? "#'0.2<5
:&$/3$= 0 01F50.302 "0 ]^$F*.5E :5&$' "030&'#.$"52 $ F$&3#& "52 "$"52 01F0&#'0.3$#2,
Z$ )%*&$ +,+ F5"0>20 5920&E$& "*$2 20;Y02 "0 P5#./$&D 593#"$2 $ F$&3#& "52 "$"52
01F0&#'0.3$#2, G23$2 )%*&$2 2*%0&0' /5'F5&3$'0.352 /$M3#/52, T0%*."5 5 $*35&? $
"#.V'#/$ /$M3#/$ 0' $=35>:$=$.302 20 "N 0' F0I*0.$ "#'0.2<5 0 0230 /5'F5&3$'0.35 20
'523&$ $"0I*$"5 F$&$ #'F=0'0.3$;<5 "0 3D/.#/$2 "0 /5.3&5=0,
!
"#$%&' ()(* +%', ,-./-, 0- 12#34'&5 6'&' 2 '7829:'7'38-) ;' 6'&8- <'= ' ,-.>2 :2# :-#8'
32, ?'72&-, @AB#@2, 02 ,#,8-@' - 3' 6'&8- <C= ' ,-.>2 :2# :-#8' 32, ?'72&-, @D3#@2, 02
,#,8-@' < !"## $% &'(E FGGH=)
I@ <
!"## $% &'(E FGGH=E ?A&#', -,8#@'8#?', 6'&' ' 0#@-3,>2 :&'48'7 :2&'@ 8-,8'0',)
J6-,'& 0- ,- 2C8-& %@ C2@ &-,%78'02 K%'7#8'8#?2E &-,%78'02, K%'38#8'8#?2, 6'&' '
0#@-3,>2 :&'48'7 3>2 :2&'@ ,'8#,:'8L&#2, 62& %@' ,5&#- 0- K%-,8/-,) M,82 620- ,-& -B67#4'02
-@ 6'&8- 6-7' 6-K%-3' K%'38#0'0- 0- 62382, @-0#02,E K%- 8'@C5@ ,- @2,8&'&'@ C',8'38&%#02,2,E 2 K%- :'N 42@ K%- 2, 4A74%72, 6'&' 0#@-3,>2 :&'48'7 ,- 82&3-@ #3-O4'N-,)
12& 2%8&2 7'02E ' -,8#@'8#?' 02 -B62-38- 0- PQ'6%32? ,- @2,8&2% 67-3'@-38,'8#,:'8L&#'E @2,8&'302 &-,%78'02, 423,#,8-38-, 32, 0#:-&-38-, @58202, %8#7#N'02, 6'&'
-,8#@'.>2) I,8-, &-,%78'02, '6&-,-38'&'@ ' -?#0R34#' 0- %@ -B62-38- 0- PQ'6%32?
62,#8#?2E #30#4'302 :2&8-@-38- 2 4'2,)
"#3'7@-38- :2&'@ -,8#@'02, 2, '%82?'72&-, - 2, '%82?-82&-, ',,24#'02, 42@ ',
L&C#8', 6-&#L0#4', 3>2 -,8A?-#,) J,,#@E 42@ ' '67#4'.>2 0- 0#?-&,', 8543#4',E 2 '%82&
,%$-&- :2&8-@-38- ' -B#,8R34#' 0- 0#3S@#4' 4'L8#4' -@ '7829:'7'38-,)
J85 -,8- @2@-382 :2&'@ '3'7#,'02, '6-3', 8&'C'7T2, K%- '6&-,-38'@ &-,%78'02, 0-
-B6-&#@-382, 6&A8#42, 3' -B#,8R34#' 0- 4'2, -@ '7829:'7'38-,) ;2 8&'C'7T2 < )!*+#",
&'(E
$%
FGGH= ,>2 '6&-,-38'0', ', 6&#@-#&', 8-38'8#?', 0- @20-7'$-@ 02, &-$#@-, 4'L8#42,
-34238&'02, 3' 7#8-&'8%&') ;-,8- '&8#$2 2 '%82& %8#7#N' 2 @20-72 02 '7829:'7'38- '3A72$2
'2 02 6R30%72 T'&@U3#42 :2&.'02 - '@2&8-4#02 <F)VG=) ;-,8- @20-72 2 6'&S@-8&2 kef f
!
"#$"#%#&'( (% &)*+,-&#("-.(.#% .* /*&#0 .( ("(&1( # .* (&#, .( %2%$#&%)*3 4*"560 (
6*.#,(7#6 .#%'(% &)* ,-&#("-.(.#% %# 6*%'"( 8(%'(&'# /*6$,#9(0 $*-% * /*&#0 ( ("(&1( #
* (&#, .( %2%$#&%)* $*.#6 %*:"#" 26( ;2(&'-.(.# /*&%-.#"<=#, .# $#;2#&(% .#:*"6(>?#%3
@ (2'*" #&')* %27#"# 26( &)*+,-&#("-.(.# 6(-% %-6$,#% $("( "#$"#%#&'(" kef f 0 ($"#%#&'(.(
&( A72"( B3C3
D-72"( B3CE F)*+,-&#("-.(.# (%%26-.( $("( kef f $"*$*%'* #6 G !"#$%& '( )*+0 HIIJK3
L(.( ( %27#%')* .# &)*+,-&#("-.(.# $"*$*%'( &( A72"( B3C0 * (2'*" %#72# #6 82%/(
.* /*6$*"'(6#&'* /(M'-/* &#%'# 6*.#,*0 2'-,-N(&.* /*6* 8(%# *% "#%2,'(.*% $"<'-/*%
*8'-.*% #6 G,!%$$ '( )*+0 HIIJK3 O6$*"'(&'# *8%#"=(" &( A72"( B3C ;2# kef f 5 /*&%'(&'#
$("( =(,*"#% .# .#%,*/(6#&'* .# −5 < x < 5 6-,-6#'"*%0 *2 %#P(0 ( &)* ,-&#("-.(.#
/*6#>( ( %# 6(&-:#%'(" $("( 7"(&.#% .#%,*/(6#&'*% .* /*&#3 F(% A72"(% B3Q # B3! * (2'*"
($"#%#&'( * 7"<A/* .# =#,*/-.(.# # .#%,*/(6#&'* #6 "#,(>)* R /*""#&'#0 # * .-(7"(6( .#
8-:2"/(>)*0 "#%$#/'-=(6#&'#3 S68*% :*"(6 *8'-.*% ( $("'-" .* 6*.#,* .( #;2(>)* GH3TIK
# .( &)*+,-&#("-.(.# .( A72"( B3C $("( kef f 3 @% $("U6#'"*% 2'-,-N(.*% &( #;2(>)* %)*
Mef f = 16g, Rm = 0, 41 kg
# βl = 5, 5T m3 S :"#;2V&/-( .( :*&'# .# #9/-'(>)* :*- .# B Hz
s
# ( =("-(>)* &( /*""#&'# I0 :*- .# H ( S3 F* .-(7"(6( .# 8-:2"/(>)* GA72"( B3!K &)* 5
/,("* * 6*6#&'* #6 ;2# *% $*&'*% $#"-M.-/*% $#".#6 #%'(8-,-.(.# $("( * ($("#/-6#&'*
.# &*=(% M"8-'(%3 4*"560 * ;2# %# =V &( A72"( B3! 5 ;2# $("( =(,*"#% .# /*""#&'# H0CC0 H0Q
# H0QC0 %27#"# :*"'#6#&'# 8-:2"/(>?#% (&<,*7(% R;2#,(% =-%'(% $("( * 6($( ,*7W%'-/* GA72"(
3!K3 F#%'# /(%*0 * ;2# %# =V $("( *% .#6(-% =(,*"#% .# /*""#&'# 6*%'"(.*% &(;2#,( A72"(0
%27#"-"-(6 ( $"#%#&>( .# /(*%3 S,56 .-%%*0 ( A72"( B3J '(6856 %27#"# :*"'#6#&'# *
/*6$*"'(6#&'* /(M'-/*3 L# :('*0 ( A72"( B3J $*%%2- 62-'(% %#6#,1(&>(% /*6 ( A72"( 3Q3
X*.* #%%# /*6$*"'(6#&'* .-&U6-/* .* 6*.#,* #%'< #6 /*&/*".U&/-( /*6 *% "#%2,'(.*%
#9$#"-6#&'(-% .*% ;2(-% "#%2,'*2 ( A72"( B3 3
!
"#$%&' ()*+ ,&-./0 1' 2340/#1'13 5 v[m/s]6 3 13740/'839:0 5x[m]6 38 &34';<0 ' /0&&39:3
5I0 [A]6 ='&' 0 801340 =&0=07:0 38 5 !"#$%& '( )*+> ?@@!6)
"#$%&' ()A+ B#'$&'8' 13 C#D%&/';<0 ='&' 0 801340 =&0=07:0 38 5 !"#$%& '( )*+> ?@@!6)
"#$%&' ()!+ E3;<0 13 F0#9/'&G ='&' D&3H%I9/#' 13 &37709J9/#' f = 53Hz 3 /0&&39:3 13
3K/#:';<0 I0 = 2.7A ='&' 0 801340 =&0=07:0 38 5 !"#$%& '( )*+> ?@@!6)
!
!"#$%&'()" *" $"*+&"
"# $%&'&()& *+$,)-.# '&%/ +$%&'&()+0+ + 1#0&.+2&1 $%#$#')+ $&.# +-)#% (&')&
)%+3+.4# $+%+ # )&%1# 0+ *+56+ +*7')5*+8 9-& : *#1# + 1+5#%5+ 0#' +.)#;<+.+()&' '=#
-)5.5>+0#'? @ $+%)5% 0+ 1#0&.+2&1 -)5.5>+0+ $#% A !"#$%& '( )*+8 BCCDE $+%+ # +.)#;<+.+()&8
'&%/ +05*5#(+0# # )&%1# %&.+)5F# G *+56+ +*7')5*+ '&.+0+? H&%/ +$%&'&()+0+ )+13:1 +
1&)#0#.#25+ + '&% &1$%&2+0+ $+%+ # 0&'&(F#.F51&()# & 2&%+I=# 0#' %&'-.)+0#' 0# )%+3+.4#?
"# )%+3+.4# A !"#$%& '( )*+8 BCCDE : &65350+ -1+ )&()+)5F+ 0& 1#0&.+2&1 0#
'5')&1+ 05(J15*# 0& -1 +.)#;<+.+()& 0& 2%+F&'8 +)%+F:' 0# 1#0&.# 0# $K(0-.# 4+%1L(5*#
+1#%)&*50# AB?MCE? "&''+ 1#0&.+2&18 3&1 *#1# #' %&'-.)+0#' &6$&%51&()+5' +$%&'&()+0#'
&1 A,!%$$ '( )*+8 BCCDE8 : +''-150# 9-& # +.)#;<+.+()& &(*#()%+;'& 0&()%# 0& -1+ *+56+
'&.+0+ '&1 9-+.9-&% +3&%)-%+ #(0& $#0&;'& *#()%#.+% + $%&''=# 5()&%(+8 *4&2+(0# +): #
F/*-# 9-+(0# (&*&''/%5#? N#%:18 (=# : 0&')+ 1+(&5%+ 9-& #' +.)#;<+.+()&' '=# -)5.5>+0#'
(+ 1+5#%5+ 0+' '-+' +$.5*+IO&'8 & '51 &1 *+56+' +*7')5*+' AP2-%+ Q?ME?
R52-%+ Q?MS "+ &'9-&%0+ + %&$%&'&()+I=# 0# 1#0&.# 0& A !"#$%& '( )*+8 BCCDE & 0#'
&6$&%51&()#' 0& A,!%$$ '( )*+8 BCCDE? "+ 05%&5)+ -1+ *+56+ +*7')5*+ -'-+.?
T-+(0# .&F+;'& &1 *#('50&%+I=# + 1+(&5%+ (+ 9-+. &'*-)+;'& #' +.)#;<+.+()&'8 #- '&U+8
&1 *+56+' +*7')5*+'8 0&F&;'& .&F+% &1 *#('50&%+I=# +' <#%I+' %&')+-%+0#%+' +# 1#F51&()#
0# *#(& 9-& '-%2&1 (&')& +135&()&? V')+' <#%I+' (=# <#%+1 *#('50&%+0+' (#' )%+3+.4#'
0& A !"#$%& '( )*+8 BCCDE & A,!%$$ '( )*+8 BCCDE?
W#1 # 1#F51&()# 0# *#(& 0# +.)#;<+.+()&8 '-%2& -1+ F+%5+I=# 0& $%&''=# 0&()%#
0+ *+56+ +*7')5*+? V')+ $%&''=# +-1&()+ *#1 # 1#F51&()# 0# *#(& $+%+ 0&()%# 0+ *+56+8
& 5')# <+> *#1 9-& '-%U+ -1+ <#%I+ %&')+-%+0#%+ +25(0# (# *#(& 0& <#%1+ + <#%I/;.# +
'-+ $#'5I=# 0& &9-5.,3%5#? X 1&'1# +*#()&*& 9-+(0# # *#(& '& 1#F51&()+ $+%+ <#%+ 0+
*+56+8 0515(-5(0# + $%&''=# 5()&%(+ 0+ *+56+8 <+>&(0# '-%25% -1+ <#%I+ +25(0# (# '&()50#
0& )%+>&% # *#(& (#F+1&()& G '-+ $#'5I=# 0& &9-5.,3%5#?
@''518 $#0&;'& +05*5#(+% -1 )&%1# F G &9-+I=# AB?MCE8 *#1# '&(0# &''+ <#%I+
%&')+-%+0#%+ 0&F50# G9-&.+' F+%5+IO&' 0& $%&''=# (# 5()&%5#% 0+' *+56+' +*7')5*+'? X-
!
"#$%& % #'(%)*+ ,-./!0 1 2+3456%3% 3% "#7(489# :+;2%<
Mef f
d2 x
dx
+
R
+ kef f (x)x + F = βlI0 Cos(ωt).
m
dt2
dt
,=./0
> :+;)% F ?+3# "#; 3#"6;49% 6+2+ % 34:#;#8)% , δP 0 #89;# % ?;#""*+ 489#;8% 3% 6%4@%
%6A"946% # % ?;#""*+ %92+":1;46% P0 2(B94?B46%3% ?#B% C;#% 3+ 6+8# 3+ %B9+D:%B%89# , A0.
E89*+<
F = ∆P A,
,=.-0
+83# % C;#% A 3+ 6+8# 3+ %B9+D:%B%89# 1 3%3% 8% #'(%)*+ ,=.F0& +83# D 1 + 34G2#9;+ 3+
6+8#<
A = π(
D 2
).
2
,=.F0
> ?;#""*+ 489#;8% P 3% 6%4@% %6A"946% ?+3# "#; 3#"6;49% ?#B% B#4 3+" 7%"#" 43#%4"
,=. 0& +83# V [m3 ] 1 + H+B(2# 489#;4+; 3% 6%4@%& n + 8A2#;+ 3# 2+B#" 3+ 7C"& R 1 (2%
J
] # T [K] % 9#2?#;%9(;% 3+ 7C".
6+8"9%89# 3%3% ?+; 8, 314[ (molK)
P V = nRT.
,=. 0
I% ?;#"#89# 2+3#B%7#2& + ?;+6#""+ :J"46+ :+4 6+8"43#;%3+ 4"+91;246+. >""42& %
?;#""*+ 489#;8% 3% 6%4@% %6A"946% P "#;C 3#?#83#89# %?#8%" 3% H%;4%)*+ 3+ H+B(2# 489#;8+
3% 6%4@% %6A"946% 3%3+ ?+; V .
K H+B(2# 489#;8+ V 3% 6%4@% %6A"946% ?+3# "#; 3#"6;49+ 3% "#7(489# 2%8#4;%<
V = V0 + Ax,
,=.=0
+83# V0 ;#?;#"#89% + H+B(2# 48464%B 3% 6%4@% %6A"946%& A % C;#% 3+ 6+8# # x + 3#"B+6%2#89+
3+ 6+8# #2 ;#B%)*+ % "(% ?+"4)*+ 3# #'(4BJL;4+.
>""42& % ?%;94; 3% #'(%)*+ ,=.-0 ?+3#2+" 3#3(M4; % :+;)% ;#"9%(;%3+;% F & 6+2+
F = (P − P0 )A.
O(L"949(483+ P % ?%;94; 3% #'(%)*+ ,=. 0& 9#2+"<
,=.N0
!
F =(
nRT
− P0 )A.
V
!"#$
%&'()*+,-'*& & .,&/+((& /&0& )(&12,0)/&3 1+0&(4
P0 V0 = nRT.
!"5$
6'17&3
F =(
9:;(1)1:)'*&
V
P0 V0
− P0 )A.
V
!"8$
- .-,1), *- +<:-=7& !"!$3 1+0&(4
F =(
P0 V0 − P0 V0 − P0 Ax
)A.
V0 + Ax
!">?$
−P0 A2 x
.
V0 + Ax
!">>$
9+@-3
F =
9:;(1)1:)'*&
,+(1-:,-*&,-
F
A
- .-,1), *- +<:-=7&
!"A$3 B'-C0+'1+ .&*+0&( +(/,+D+, - E&,=-
/&0& (+'*&4
F =−
9:;(1)1:)'*&
F
π 2 ( D2 )4 P0 x
.
V0 + π( D2 )2 x
'- +<:-=7& !">$3 1+0G(+ & 0&*+C& -*&1-*& '+(1+ 1,-;-CH&"
π 2 ( D2 )4 P0 x
dx
d2 x
+ kef f (x)x −
Mef f 2 + Rm
= βlI0 Cos(ωt).
dt
dt
V0 + π( D2 )2 x
I-,- B'( *+ ()0:C-=7&3 - +<:-=7&
!">A$
!">A$ E&) *+(0+0;,-*- '-( <:-1,& +<:-=J+(
*)E+,+'/)-)( *+ .,)0+),- &,*+0 -;-)K&3 &'*+
,+(.+/1)D-0+'1+"
!">F$
.
u
+
.
v
,+.,+(+'1-0 :0 (+'& + :0 /&((+'&3
!
.
x= y,
.
y=
"#$% &
.
π 2 ( D2 )4 P0 x
βlI0 v
Rm y
kef f (x)x
−
−
−
,
Mef f
Mef f
Mef f
V0 + π( D2 )2 xMef f
.
.
"#$%#&
u= u + ω v −u(u2 + v 2 ),
"#$%'&
.
"#$%(&
v= −ωu + v − v(u2 + v 2 ).
)**+,- ./*0/ 01232456- /*07 */.86 91696*06 6 /*0:86 82 8+.;,+<2 /, 2406=>242.0/*
4/?2.86 /, <6.*+8/12@A6 2* >61@2* 1/*02:128612* 91/*/.0/* B:2.86 82 *:2 294+<2@A6 /,
<2+C2* 2<D*0+<2* */4282*$ E212 /*0/ /*0:86 >6+ :0+4+F286 6 ,68/46 291/*/.0286 .2 /B:2@A6
"#$%G&- 961 */1 2 ,2+* *+,94/* >61,2 8/ 4/?21 /, <6.*+8/12@A6 6* />/+06* 82 <2+C2 2<D*0+<2
*/4282$ H/**/ ,68/46 968/=*/ 63*/1?21 B:/ B:2.86 V0 >61 ,:+06 I12.8/- 6: 0/.8/1 2
+.J.+06- 6 */: <6,96102,/.06 0/.8/17 26 82 /B:2@A6 "!$%K&$ L6, 6 0/1,6 82 <2+C2
2<D*0+<2- 6 ,68/46 M ,68+J<286- */.86 28+<+6.286 ,2+* :, 0/1,6 .A6 4+./21- 6 B:/
8/?/17 0/1 <6.*/B:N.<+2* +,96102.0/* .6* 1/I+,/* <2O0+<6* /.<6.01286* 961 "
!"## $% &'(-
!KKP&$
E212 6 /*0:86 86 ,68/46 "#$%G& "6: "#$% & 2 "#$%(&&- >6+ :0+4+F282 :,2 ,/068646I+2
2.746I2 Q *:I/1+82 961 " )!*+#",
$% &'(-
!KKP&$ L6,6 291/*/.0286 .6 <29R0:46 G- /*02
,/068646I+2 M 2,942,/.0/ :0+4+F282 .2 4+0/120:12 9212 2.74+*/ 8/ *+*0/,2* 8+.;,+<6*$
S*02 ,/068646I+2 <6.*+*0/ /, 2.24+*21 6* 1/01206* 8/ >2*/* / +,94/,/.021 6* 8+2I12,2*
8/ 3+>:1<2@A6 <6, 2 ?21+2@A6 86* 921;,/016* 1/4/?2.0/*- .6 <2*6- 2 <611/.0/ I0 / 6
?64:,/ V0 $ E212 6 <74<:46 8/*0/* 8+2I12,2* >612, 8/*/.?64?+86* / +,94/,/.0286* 92<60/*
.6* *6>0T21/* U205/,20+<2 / U20423$ E212 2 ?24+82@A6- /*0/* 92<60/* >612, :0+4+F286*
.6 *+*0/,2 8/ :, 9N.8:46 521,V.+<6 >61@286 / 2,610/<+86- *+*0/,2 /*0/ <6.5/<+86 /
2,942,/.0/ /*0:8286 .2 4+0/120:12 8/ *+0/,2* 8+.;,+<6*$
)**+,- >6+ 63*/1?286 6 <6,96102,/.06 86 ,68/46 82 /B:2@A6 "#$%G& 9212 8+>/1/.0/*
?2461/* 8/ V0 / I0 - +.+<+2.86=*/ <6, ?2461/* 2406* / 8+,+.:+.86=*/ 20M ?2461/* 8/ ?64:,/
B:/ */W2, >2<0R?/+* <6, 2 +.*0242@A6 86* 2406=>242.0/* <6.*+8/1286* .6 +.0/1+61 82 <2+C2
2<D*0+<2$
!
!"#$%&'()#
"#$# % &'(')*%+*,-').% &'(.' .$#/#+0%1 2%$#- 3.,+,4#&%( %( 5#$6-'.$%( &%
#+.%72#+#).' 3.,+,4#&% 5%$ 8 !"#$%& '( )*+1 9::;<1 %)&' %( 5#$6-'.$%( (=%> Mef f =
16g, Rm = 0, 41 kg
1 βl = 5, 5T m1 &,6-'.$% &' ?@ cm ' # 2$'A3B)C,# &# 2%).' &' 'DC,.#E=% 2%,
s
&' ! Hz F G )=%7+,)'#$,&#&' kef f (x) 3.,+,4#&# 2%, # -'(-# 3.,+,4#&# 5%$ 8 !"#$%& '( )*+1
9::;<1 ' '(.H $'5$'(').#&# )# IJ3$# FKF G&,C,%)#+-').'1 2%, C%)(,&'$#&% '(.' #+.%72#+#).'
,)(.#+#&% '- 3-# C#,D# #CL(.,C#1 C%)2%$-' #5$'(').#&% )% C#5M.3+% KF G( 2'$$#-').#(
3.,+,4#&#( 5#$# % &'(')*%+*,-').% 2%$#- % &,#J$#-# &' /,23$C#EN'(1 % $'.$#.% &' 2#('(
8#.$#.%$'(< ' %( '(5'C.$%( &' 2$'A3B)C,#(F O(.#( 2'$$#-').#( (=% #-5+#-').' &,23)&,&#(
)% '(.3&% &' (,(.'-#( C#P.,C%( ' 2%$#- #5$'(').#&#( ' &,(C3.,&#( )% C#5M.3+% !F
O(.' C#5M.3+% ('$H &,*,&,&% '- &3#( (3/('EN'(F G 5$,-',$# &'+#( ,$H .$#.#$ &#
,)Q3B)C,# &% *%+3-' &# C#,D# #CL(.,C# )# &,)6-,C# &% #+.%72#+#).' ' # ('J3)&# ,$H .$#.#$
&# ,)Q3B)C,# &% *%+3-' &# C#,D# #CL(.,C# )%( '(5'C.$%( &' 2$'A3B)C,#F
!" #$%&'($)#*
+,
-,&'./
+*
)*#0*
*)123#)*
$*
+#$4.#)* +,2 *&3,5%*&*$3/2
GA3,1 '(.#-%( '(5'C,#+-').' ,).'$'((#&%( )% '(.3&% &# ,)Q3B)C,# &% *%+3-' )#
&,)6-,C# &%( #+.%72#+#).'(F "#$# ,(.%1 ,$'-%( 3(#$ C%-% 2'$$#-').# &' #5$'(').#E=%1 %
&,#J$#-# &' /,23$C#EN'(F "#$# 'D'-5+,IC#$ % 5$%C'((% 5#$# # C$,#E=% &%( &,#J$#-#( &'
/,23$C#EN'(1 ('J3' # IJ3$# @F?F
R,J3$# @F?> S# '(A3'$&# ('J3' # IJ3$# C%- %( $'.$#.%( &' 2#(' &%( #.$#.%$'( &% #+.%72#+#).'
,)(.#+#&% )# C#,D# #CL(.,C#F T ',D% y $'5$'(').# # *'+%C,&#&' &' &'(+%C#-').% &% C%)' 'm
1 % ',D% x $'5$'(').# % &'(+%C#-').% &% C%)' '- m ' % ',D% V0 $'5$'(').# % *%+3-' &#
s
C#,D# #CL(.,C# '- m3 F S# &,$',.# % &,#J$#-# &' /,23$C#E=% %/.,&% # 5#$.,$ &# IJ3$# &#
'(A3'$&#F T ',D% V0 $'5$'(').# % *%+3-' &# C#,D# #CL(.,C# '- m3 ' % ',D% x $'5$'(').#
% &'(+%C#-').% &% C%)' '- mF G C%$$').' &' 'DC,.#E=% 3.,+,4#&# 2%, &' @GF
!" #$%&" '" ()*%(&'" (+ ,-. / 0(12" " )(345 '( 65178"&/9 '18%21'" 75 8":;2%<5 =
( (>(+:<1#8"'" 7" #$%&" =-?9 57'( 5) :5725) '5 "2&"25& *%( 172(&8(:2"+ ()2" )(345 )45
:<52"'5) 7" #$%&" '" '1&(12"- @ )(345 %21<1A"'" :"&" $(&"& 5) &()%<2"'5) 051 '( x +"15& *%(
B ( " C(<581'"'( y (72&( DB9. [ ms ] ( B9. [ ms ]- E5+5 C1+5) 7" )(345 =- 9 ") )(3F() '( 65178"&/
172(&8(:2"+ ") G&H12") 8572;7%")9 ( 5 &()%<2"'5 '()2" 172(&)(8345 )45 5) '"'5) *%( )(&45
%21<1A"'5) :"&" " 857)2&%345 '5) '1"$&"+") '( H10%&8"345- 65&/+9 85+:%2"8157"<+(72(9
1)25 / 1+:5));C(< :(<5 0"25 '( 85+:%2"'5&() )(&(+9 :5& 7"2%&(A" '1)8&(25)- I +/25'5
"'52"'5 :"&" )5<%8157"& ()2( :&5H<(+" / "5 17C/) '( )( %21<1A"& %+" )(345 85+ y = 0
J'( ():())%&" BK9 "'52"&D)( %+" )(345 85+ %+" ():())%&" '10(&(72( '( B9 85+5 5 8")5 '(
−0, 1 < y < 0, 19 ( 857)1'(&"7'5 25'5) 5) :5725) 7()2( 172(&C"<5 :"&" " 857)2&%345 '5)
'1"$&"+") '( H10%&8"345!5) '1"$&"+") '( H10%&8"3459 )(+:&( *%( x 21C(& " :5))1H1<1'"'( '( "))%+1& +"1)
'( %+ C"<5& J+%<21:(&15'181'"'( '( G&H12")K9 )1$71#8" '1A(& *%( '1)25&3F() ()245 )(7'5
$(&"'") 75 )5+ (+121'5 :(<5 "<25D0"<"72(- @))1+9 ()2( 21:5 '( '1"$&"+" &(:&()(72" H(+
85+5 0%78157" " '17L+18" '5 "<25D0"<"72( :"&" '10(&(72() :"&L+(2&5)- M( 0"259 *%"7'5
%+" )(7G1'( :%&" / ":<18"'" (+ %+ "<25D0"<"72(9 ():(&"D)( *%( 5 '()<58"+(725 x '5 857(
'5 "<25D0"<"72( )(N" O71856"&" ()2( ()2%'5 05&"+ 857)2&%;'5) .. '1"$&"+") '( H10%&8"345 (+ 0%7345 '5 C5<%+(
'" 8"1>" "8O)218" )(<"'"- M10(&(72() '1"$&"+") '( H10%&8"345 05&"+ 857)2&%;'5) :"&"
'1C(&)5) C"<5&() '( 85&&(72( '( (>812"345- @) C"&1"3F() '( 85&&(72( 05&"+ '()'( B9?@ "2/
.B@- !" :&P218"9 7()2( "<25D0"<"72(9 %+" 85&&(72( '( .B@ / 857)1'(&"'" (>2&(+"+(72(
"<2"9 :5&/+ ()2( ()2%'5 051 7(8())P&15 :"&" "7"<1)"& 5) (>2&(+5) 75 85+:5&2"+(725 '5
)1)2(+" "<25D0"<"72( ( 8"1>" "8O)218"- @ C"&1"345 '5 C5<%+( '" 8"1>" "8O)218" 051 0(125 "
:"&21& '( B9BB. m3 9 *%( / 5 C5<%+( +;71+5 '" 8"1>" "8O)218" :"&" *%( 5 "<25D0"<"72( (+
()2%'5 :5))" )(& 17)2"<"'5- I C"<5& '5 C5<%+( '" 8"1>" "8O)218" 2(C( (7245 178&(+(725)
'( B9BB. m3 - @ :"&21& '( B9. m3 9 5) &()%<2"'5) )( +"721C(&"+ *%"<12"21C"+(72( 1'Q72185)
"5 8")5 '5 C5<%+( '( 8"1>" "8O)218" 17#71259 ( :5& 1))5 745 )(&45 (>:<1812"'5) 7") #$%&")
'()2( 2&"H"<R5-
!
"#$%&' ()*+ ,#'$&'-' ./ 0#1%&2'345 6'&' %-' 25&&/78/ ./ /92#8'345 ./ :;!<) = /#95 V0
&/6&/>/78' 5 ?5@%-/ .' 2'#9' '2A>8#2' /- m3 ) = /#95 x &/6&/>/78' 5 ./>@52'-/785 .5
257/ /- m)
B5 .#'$&'-' .' C$%&' ()* 65./D>/ 50>/&?'& E%/; 6'&' %- ?'@5& ./ 25&&/78/ ./
/92#8'345 257>#./&'.5 0'#95 F I0 G:;!<H; 745 /9#>8/- #7.I2#5> ./ 25-65&8'-/785> 2'J8#25>)
B/>8' C$%&'; K 65>>I?/@ 50>/&?'& E%/ 6'&' 2'#9'> '2A>8#2'> 25- ?5@%-/> -/75&/> 5 257/
8/7./ ' 8/& 5 >/% ./>@52'-/785 @#-#8'.5 /- &/@'345 L -/>-' 25&&/78/ ./ /92#8'345 6'&'
?5@%-/> -'#5&/>)
"#$%&' ()M+ ,#'$&'-' ./ 0#1%&2'345 6'&' %-' 25&&/78/ ./ /92#8'345 ./ N<) = /#95 V0
&/6&/>/78' 5 ?5@%-/ .' 2'#9' '2A>8#2' /- m3 ) = /#95 x &/6&/>/78' 5 ./>@52'-/785 .5
257/ /- m)
B5 .#'$&'-' .' C$%&' ()M; 25- I0 GN<; 65./D>/ 50>/&?'& E%/ 6'&' ?5@%-/>
./ 2'#9' '2A>8#2' -/75&/> E%/ :;:! m3 ; 745 /9#>8/- #7.I2#5> ./ 25-65&8'-/785>
-%@8#6/&#J.#25>) O%'7.5 5 ?5@%-/ .' 2'#9' '2A>8#2' V0 />8P /78&/ :;:! m3 / :;:Q m3 ;
>%&$/- J&0#8'> 6/&#J.#2'> 25- 6/&I5.5 #$%'@ ' 8&R>) O%'7.5 5 ?5@%-/ .' 2'#9' '2A>8#2'
'8#7$/ %- ?'@5& -'#5& E%/ :;:Q m3 ; >%&$/- J&0#8'> 6/&#J.#2'> 25- 6/&I5.5> $&'7./>;
>%$/&#7.5 25-65&8'-/785 2'J8#25) S5-5 25-/78'.5 '78/&#5&-/78/; 25-65&8'-/785> ./
-%@8#6/&#5.#2#.'./ $/&'- .#>85&3T/> 75 >5- 6&5.%U#.5 6/@5 '@85D1'@'78/)
!
"#$%&' !( ) *#'$&'+' ,- .#/%&0'123 4'&' %+' 03&&-56- -70#6'123 ,- 89( : -#73 V0
&-4&-;-56' 3 <3=%+- ,' 0'#7' '0>;6#0' -+ m3 ( : -#73 x &-4&-;-56' 3 ,-;=30'+-563 ,3
035- -+ m(
?3 ,#'$&'+' ,' @$%&' !( A 4'&' I0 B89A 43,-C;- 3.;-&<'& D%-A 4'&' ' +'#3&#' ,3;
<3=%+-; ,- 0'#7' '0>;6#0' ;#+%=',3;A 3 ;#;6-+' ';;%+- 03+43&6'+-563; 03-&-56-; E523
0'F6#03;G( H+' -70-123 ;%&$- D%'5,3 3 <3=%+- ,' 0'#7' '0>;6#0' ';;%+- <'=3&-; -56&- IAI8
m3 - IAIJ m3 A 5-;6- #56-&<'=3 3 ;#;6-+' '4&-;-56' 03+43&6'+-563; ,- +%=6#4-&#3,#0#,',-(
"#$%&' !(K) *#'$&'+' ,- .#/%&0'123 4'&' %+' 03&&-56- ,- -70#6'123 ,- J9( : -#73 V0
&-4&-;-56' 3 <3=%+- ,' 0'#7' '0>;6#0' -+ m3 ( : -#73 x &-4&-;-56' 3 ,-;=30'+-563 ,3
035- -+ m(
?3 ,#'$&'+' ,' @$%&' !(KE I0 BJ9GA 3 ;#;6-+' ';;%+- %+ 03+43&6'+-563 03-&-564'&' 63,3; 3; <3=%+-; ,' 0'#7' '0>;6#0' V0 ;#+%=',3;(
!
"#$%&' ()(* +#'$&',' -. /#0%&1'234 5'&' %,' 14&&.67. .81#7'234 -. 9) : .#84 V0
&.5&.;.67' 4 <4=%,. -' 1'#8' '1>;7#1' ., m3 ) : .#84 x &.5&.;.67' 4 -.;=41',.674 -4
146. ., m)
?' @$%&' ()(A 14, I0 B 9A 4 14,54&7',.674 -4 ;#;7.,' ;. ,'67C, 14.&.67. 5'&'
<4=%,.; -. 1'#8' '1>;7#1' ,.64&.; D%. EAEFG m3 ) 9; /#0%&1'2H.; 14,.2', ' ;%&$#& D%'6-4
4 <4=%,. -' 1'#8' '1>;7#1' V0 C ,'#4& D%. EAEF m3 A ;%&$#6-4 .6734 I&/#7'; 5.&#I-#1';
-. 5.&J4-4 -4#;) 9 .8.,5=4 -'; 14&&.67.; -. .81#7'234 ,.64&.; D%. 9A 5'&' <4=%,.;
'5&48#,'-',.67. ,.64&.; D%. EAEG m3 4 '=74K0'='67. 1467#6%' 7.6-4 14,54&7',.674;
14.&.67.;)
"#$%&' ()!* +#'$&',' -. /#0%&1'234 5'&' %,' 14&&.67. -. .81#7'234 -. L9) : .#84 V0
&.5&.;.67' 4 <4=%,. -' 1'#8' '1>;7#1' ., m3 ) : .#84 x &.5&.;.67' 4 -.;=41',.674 -4
146. ., m)
?4
-#'$&','
-'
@$%&'
()!A
14,
I0 BL9A ;%&$., 14,54&7',.674; -.
,%=7#5.&#4-#1#-'-. M5.&J4-4 7&N;O 5'&' <4=%,.; -. 1'#8' '1>;7#1' ,'#4&.; D%. EAEP m3 )
Q;7. 14,54&7',.674 ;. .;7.6-. 5'&' 4 &.;7'67. -4; <4=%,.; ;#,%='-4;) R4,4 14,.67'-4
'67.&#4&,.67.A 14,54&7',.674; -. ,%=7#5.&#4-#1#-'-. $.&', -#;74&2H.; 64 ;4, $.&'-4
5.=4 '=74K0'='67.A 4 D%. -.<. ;.& .<#7'-4) R4,5'&'6-4K;. ' @$%&' ()! 14, ' @$%&' ()(A
647'K;. D%. 64 &',4 #60.&#4& ;%&$. %,' /#0%&1'234 14, I&/#7'; 5.&#I-#1'; -. 5.&#4-4 -4#;A
!
"#$%&' $ ('))*%+* &* *,(-+$./' $#0*%+$1
2-3#)$ 41!5 6-$3)$0$ &* 7-8#)($./' 9$)$ #0$ ('))*%+* &* *,(-+$./' &* 4:1 ; *-,' V0
)*9)*<*%+$ ' ='>#0* &$ ($-,$ $(?<+-($ *0 m3 1 ; *-,' x )*9)*<*%+$ ' &*<>'($0*%+' &'
('%* *0 m1
@' &-$3)$0$ &$ A3#)$ 41!B ('0 I0 C4:B *,-<+*0 D)7-+$< 9*)-D&-($< ('0 9*)E'&' #0
9$)$ ='>#0*< <#A(-*%+*0*%+* 9*"#*%'< F V0 ≤ 0, 005 m3 G &* ($-,$< $(?<+-($<1 H')I0B
9$)$ "#$<* +'&'< '< ='>#0*< &* ($-,$ $(?<+-($ <-0#>$&'< 9$)$ *<+$ ('))*%+* &* *,(-+$./'B
<#)3*0 D)7-+$< 9*)-D&-($< &* 9*)E'&' +)J<1 @' &-$3)$0$ &$ A3#)$ 41KB ' ('09')+$0*%+'
&* 0#>+-9*)-'&-(-&$&* I "#$>-+$+-=$0*%+* -3#$> $' &$ A3#)$ 41!1 ; &-$3)$0$ &$ A3#)$
41LMB I <*0*>N$%+* $' &$< A3#)$< 41! * 41KB '%&* ' ('09')+$0*%+' &* 0#>+-9*)-'&-(-&$&*
<* 0$%+I0 9$)$ "#$<* +'&'< '< ='>#0*< &* ($-,$ $(?<+-($ <-0#>$&'<1
2-3#)$ 41K5 6-$3)$0$ &* 7-8#)($./' 9$)$ #0$ ('))*%+* &* *,(-+$./' &* O:1 ; *-,' V0
)*9)*<*%+$ ' ='>#0* &$ ($-,$ $(?<+-($ *0 m3 1 ; *-,' x )*9)*<*%+$ ' &*<>'($0*%+' &'
('%* *0 m1
!
"#$%&' ()*+, -#'$&'.' /0 1#2%&3'456 7'&' %.' 36&&0890 /0 0:3#9'456 /0 ;<) = 0#:6 V0
&07&0>089' 6 ?6@%.0 /' 3'#:' '3A>9#3' 0. m3 ) = 0#:6 x &07&0>089' 6 /0>@63'.0896 /6
3680 0. m)
"#$%&' ()**, -#'$&'.' /0 1#2%&3'456 7'&' %.' 36&&0890 /0 0:3#9'456 /0 !<) = 0#:6 V0
&07&0>089' 6 ?6@%.0 /' 3'#:' '3A>9#3' 0. m3 ) = 0#:6 x &07&0>089' 6 /0>@63'.0896 /6
3680 0. m)
B6 /#'$&'.' /' C$%&' ()**D 36. I0 E!<D 6 >#>90.' '7&0>089' F&1#9'> 70&#F/#3'> /0
70&G6/6 A8#36 7'&' ?6@%.0> /0 3'#:' '3A>9#3' .086&0> H%0 +D++I m3 ) -076#> /#>>6D 6
>#>90.' '7&0>089' F&1#9'> 70&#F/#3'> /0 70&G6/6 /6#> '9J 6 ?6@%.0 ?'@0& '7&6:#.'/'.0890
+D+*; m3 D H%'8/6 6 &'.6 >%70&#6& >0 1#2%&3' 0 6 #820&#6& /K >#8'#> /0 .%@9#70&#6/#3#/'/0)
L>96 636&&0 '9J 6 ?6@%.0 /' 3'#:' '3A>9#3' ?'@0& '7&6:#.'/'.0890 +D+ I m3 ) M'&' 6 ?'@6&
/0 ?6@%.0 /' 3'#:' '3A>9#3' /0 +D+II m3 D 36.04' ' >%&$#& %.' .#>9%&' 86> /6#> &'.6>)
N%'8/6 6 ?6@%.0 /' 3'#:' '3A>9#3' ?'@0 089&0 +D+II m3 0 +D+! m3 '3689030. O'80@'>
/0 6&/0. 0 #8/G3#6> /0 3'6>D '>>#. 36.6 '3689030 86 /#'$&'.' /0 1#2%&3'456 /6 .'7'
@6$G>9#36 PC$%&' Q)RS) "#8'@.0890 H%'8/6 6 ?6@%.0 /' 3'#:' '3A>9#3' J .'#6& H%0 +D+!
m3 D 6 >#>90.' >0 9&'8>26&.'D '7&0>089'8/6 %. 36.76&9'.0896 H%0 #8/#3' 3'6>)
!
"#$%&' ()*+, -#'$&'.' /0 1#2%&3'456 7'&' %.' 36&&0890 /0 0:3#9'456 /0 *!;) < 0#:6 V0
&07&0=089' 6 >6?%.0 /' 3'#:' '3@=9#3' 0. m3 ) < 0#:6 x &07&0=089' 6 /0=?63'.0896 /6
3680 0. m)
A6 /#'$&'.' /' B$%&' ()*+C 36. I0 D*!;C 7'&' >6?%.0= .086&0= E%0 !C!! m3 C 6
=#=90.' '7&0=089' F&1#9'= 70&#F/#3'= /0 70&G6/6 %.) H%'8/6 6 >6?%.0 /' 3'#:' '3@=9#3' I
.'#6& E%0 !C!! m3 0 .086& E%0 !C!* m3 C =%&$0. F&1#9'= 70&#F/#3'= /0 70&#6/6 /6#=) J'&'
>6?%.0= /0 3'#:' '3@=9#3' 089&0 !C!* m3 0 '7&6:#.'/'.0890 !C!+K m3 C 6 &'.6 =%70&#6&
=62&0 %.' 86>' 1#2%&3'456C 08E%'896 E%0 6 &'.6 #820&#6& /L =#8'#= /0 .%?9#70&#6/#3#/'/0)
"#8'?.0890C ' 7'&9#& /0 %. >6?%.0 /0 3'#:' '3@=9#3' /0 '7&6:#.'/'.0890 !C!+K m3 C 6
=#=90.' '7&0=089' %. 36.76&9'.0896 /0 .%?9#70&#6/#3#/'/0 #8/#3'8/6 26&90.0890 3'6=)
;8'?#='8/6M=0
6=
&0=%?9'/6=C
76/0.6=
61=0&>'&
E%0
6
=#=90.'
'==%.0
36.76&9'.0896= /#20&0890= 7'&' '= >L&#'= 36&&0890= /0 0:3#9'456 =#.%?'/'= 0 0=9'=
/#20&084'= 089&0 '= =#.%?'4N0= 856 76/0. =0& 9&#>#'?.0890 '8'?#='/'=) O=96 =%$0&0 %.'
/#2G3#? 36836&/P83#' 089&0 36&&0890= /0 0:3#9'456 '7?#3'/'= 0 >6?%.0= /0 3'#:' '3@=9#3')
Q'=C /0 %. .6/6 $0&'?C 7'&' 36&&0890= /0 0:3#9'456 1'#:'=C 6 36.76&9'.0896 /6
'?96M2'?'890 368=#/0&'/6 /0#:' /0 '==%.#& .%?9#70&#6/#3#/'/0 7'&' >6?%.0= 70E%086= /0
3'#:'= '3@=9#3'=) R=90 36.76&9'.0896 76/0 =0& 61=0&>'/6 8'= B$%&'= ()+C ()SC ()T 0 () )
J6&I.C 856 1'=9' 0=36?U0& 6 .086& >6?%.0 76==G>0?C 76#= 7'&' >6?%.0= .%#96
70E%086=C 6 3680 908/0 ' 90& 6 =0% /0=?63'.0896 ?#.#9'/6) O=96 76/0 =0& 61=0&>'/6
7'&' /#20&0890= >'?6&0= /0 36&&0890=C 36.6 8'= B$%&'= ()+ 0 ()*+) J'&' 0:0.7?#B3'& 0=90
36.76&9'.0896C 76/0M=0 61=0&>'& ' =#.%?'456 /' B$%&' ()*S) A0=9' =#.%?'456 U6%>0 '
>'&#'456 /' 36&&0890 /0 0:3#9'456 /0 !C ; '9I *!; 7'&' 6 .G8#.6 >'?6& /0 >6?%.0 /0
3'#:' '3@=9#3' 76==G>0? V!C!!* m3 W) A' B$%&' ()*SC '70='& /0 6 '?96M2'?'890 '7&0=089'& %.
36.76&9'.0896 360&0890 7'&' 96/'= '= 36&&0890= =#.%?'/'=C 6 3680 90. 6 =0% /0=?63'.0896
0:9&0.'.0890 ?#.#9'/6 0. 36.7'&'456 X B$%&' ()*TC 76& 0:0.7?6C 68/0 6 '?96M2'?'890
0=9L '6 '& ?#>&0) Y#.#9'& 6 /0=?63'.0896 /6 3680 /6 '?96M2'?'890 3'%=' .%#9' 70&/' /0
0B3#Z83#'C E%0 856 I %. 36.76&9'.0896 /0=0[L>0? '6 =#=90.')
!
"#$%&' ()!*+ ,#'$&'-' ./ 0#1%&2'345 6'&' 5 758%-/ ./ 9:99! m3 ) ; /#<5 I0 &/6&/=/>?' '
25&&/>?/ ./ /<2#?'345 /- A) ; /#<5 x &/6&/=/>?' 5 ./=852'-/>?5 .5 25>/ /- m)
;0=/&7'>.5@=/ >57'-/>?/ '= A$%&'= ./ ()B '?C ()!B: 65./@=/ /8/$/& 758%-/= />?&/
9:9! m3 / 9:9B m3 25-5 5= 758%-/= ./ 2'#<' '2D=?#2' E%/ '6&/=/>?'- %- -/8F5&
25-65&?'-/>?5 .#>G-#25 6'&' '= 25&&/>?/= =#-%8'.'= / E%/ 8#-#?'- 65%25 5 ./=852'-/>?5
.5 25>/ .5 '8?5@1'8'>?/)
H==#-: 65./@=/ '6&/=/>?'& %-' 25-6'&'345 />?&/ .%'=
=#-%83I/=: 5>./ /- %-' ./8'= 5 '8?5@1'8'>?/ /=?J '5 '& 8#7&/ K2'#<' '2D=?#2' 25- 758%-/
#>A>#?5LKA$%&' ()!ML: / >' 5%?&' 5 '8?5@1'8'>?/ /=?J #>=?'8'.5 /- %-' 2'#<' '2D=?#2' ./
758%-/ #$%'8 ' 9:9! m3 KA$%&' ()! L)
"#$%&' ()!M+ ,#'$&'-' ./ 0#1%&2'345 6'&' 5 '8?5@1'8'>?/ '5 '& 8#7&/) ; /#<5 I0 &/6&/=/>?'
' 25&&/>?/ ./ /<2#?'345 '68#2'.' /- A) ; /#<5 x &/6&/=/>?' 5 ./=852'-/>?5 .5 25>/ /m)
N5 .#'$&'-' .' A$%&' ()!M: 5>./ 5 '8?5@1'8'>?/ /=?J '5 '& 8#7&/: 5 '8?5@1'8'>?/
'6&/=/>?' O&0#?'= 6/&#O.#2'= ./ 6/&P5.5 D>#25 6'&' 25&&/>?/= ./ /<2#?'345 '?C BH)
Q%'>.5 ' 25&&/>?/ ./ /<2#?'345 C -'#5& E%/ BH / -/>5& E%/ (H: =%&$/- 6/&P5.5= ./
-%8?#6/&#5.#2#.'./) R=?/= 25-65&?'-/>?5= =%$/&/- 15&?/-/>?/ 2'5=) Q%'>.5 ' 25&&/>?/
./ /<2#?'345 C -'#5& E%/ (H / -/>5& E%/ SH: =%&$/ %-' T'>/8' ./ 5&./- 5>./ '= O&0#?'=
6/&#O.#2'= '==%-/- 6/&P5.5 ?&U=) Q%'>.5 ' 25&&/>?/ ./ /<2#?'345 C -'#5& E%/ S: H:
!
"#$%&'"(' # %)(#*+%)%"(' %,-'.'"(% /#&,#-(%&'"(#. .01'-2"3# +#-('&'"(' /%#.4
5210-% 647 8 92%1-%&% 3' :2+0-/%;<# ,%-% # $#)0&' 3' /%2=% %/>.(2/% 3' ?@?7 m3 4 A '2=#
I0 -',-'.'"(% % /#--'"(' 3' '=/2(%;<# %,)2/%3% '& A4 A '2=# x -',-'.'"(% # 3'.)#/%&'"(#
3# /#"' '& m4
A 32%1-%&% 3% B10-% 647 @ #"3' # %)(#*+%)%"(' '.(C 2".(%)%3# '& 0&% /%2=% %/>.(2/%
3' ?@?7 m3 @ ,#3' .'- /#"+-#"(%3# /#& % .'D0E"/2% 3' B10-%. ,%-% 32+'-'"('. /#--'"('.
3' '=/2(%;<# F3' 64! %(G 647!H@ #"3' #. ,#"(#. 3% B10-% 647 ,#3'& .'- '"/#"(-%3#. "%.
B10-%. 64! %(G 647! D0%"3# # $#)0&' 3% /%2=% %/>.(2/% $%)' ?@?7 m3 4 I# 1-CB/# 3% B10-%
647 @ ,#3'*.' %"%)2.%- D0' # /#&,#-(%&'"(# 3# %)(#*+%)%"(' G /#'-'"(' ,%-% /#--'"('.
:%2=%.@ ' .' &#.(-% )2"'%- %(G % /#--'"(' 3' J4 K0%"3# % /#--'"(' 3' '=/2(%;<# G &%2#D0' J@ .0-1' 0&% :2+0-/%;<#@ D0' .' '.('"3' %(G # $%)#- 3' /#--'"(' 3' '=/2(%;<# 3'
6J4 L#&,%-%"3#*.' '.(' &'.&# 2"('-$%)# "% B10-% 647M@ ,#3'*.' "#(%- D0' "% B10-% 647
N#0$' % 32&2"02;<# 3# /#&,#-(%&'"(# 3' &0)(2,'-2#32/23%3'4 I% B10-% 647 @ D0%"3# %
/#--'"(' 3' '=/2(%;<# G &%2#- D0' 6J@ .0-1' 0& /#&,#-(%&'"(# 3' &0)(2,'-2#32/23%3' D0'
.' '.('"3' %(G 7?J4 O.(' &'.&# /#&,#-(%&'"(# '=2.(' "% B10-% 647M D0%"3# % /#--'"('
3' '=/2(%;<# G &%2#- D0' 6J ' &'"#- D0' PJ4 I% B10-% 647 "<# '=2.('& 2"3Q/2#. 3'
/#&,#-(%&'"(#. /%R(2/#.@ /#&# "% 647M4 L#&,%-%"3#*.' # 1-CB/# 3% B10-% 647 /#& #
3% B10-% 647M@ G /)%-# # :'"'B/2# 3% /%2=% %/>.(2/% "% 32"S&2/% 3# %)(# +%)%"('@ #"3'
%. R-:2(%. 3# .2.('&% .' %,-'.'"(%& &%2. 3'B"23%. ' /#'-'"('. ,%-% % &%2#- ,%-(' 3#
2"('-$%)# 3' /#--'"('. 3' '=/2(%;<#4
9' 0& &#3# 1'-%)@ %"%)2.%"3#*.' (#3#. #. 1-CB/#. "#$%&'"('@ ' $'-2B/%"3# #
/#&,#-(%&'"(# 3#. $#)0&'. 3' /%2=% %/>.(2/% ,%-% 32+'-'"('. /#--'"('. 3' '=/2(%;<#@
,#3'*.' /#"/)02- D0' ,%-% $#)0&'. 3' /%2=% %/>.(2/% ,'D0'"#.@ # %)(#*+%)%"(' %,-'.'"(%
0& :#& /#&,#-(%&'"(# "% &%2#-2% 3#. /%.#.@ /'..%"3# %. :2+0-/%;T'.@ ' 32&2"02"3# %.
32.(#-;T'. "# .#& 1'-%3# ,')# %)(#*+%)%"('4 A -'.0)(%3# %032(2$# .' (#-"% %2"3% &')N#- .'
'.(' $#)0&' 3% /%2=% %/>.(2/% +#- './#)N23# /#& /023%3#@ "<# )2&2(%"3# '& 3'&%.2%3# #
3'.)#/%&'"(# 3# /#"' 3# %)(#*+%)%"('4 A0 .'U%@ D0%"(# &'"#. :2+0-/%;T'. +#-'& 1'-%3%.@
&'"#-'. .'-<# %. 32.(#-;T'. ' &%2#- .'-C % B3')23%3' 3# .2.('&% %)(#*+%)%"(' &%2. /%2=%
!
"#$%&'#"(
!" #$%&'($)#*
+,
-,&'./
+*
)*#0*
*)123#)*
$*
4/25,23* /. %4/6'($)#* +,2 *&3,7%*&*$3/2
)"*" +, "-&./0"-"1&2 '32"-4 %2 " 0*25+61#'" 3" #.**21&2 32 27#'&"89. : #.1%&"1&24
&2*'"/%2 1" *2%;.%&" 2, 0*25+61#'" ";21"% " 0*25+61#'" 3" 0.1&2 32 21&*"3"4 %2, 121<+,"
.+&*" 0*25+61#'" 2%;$*'"( ).*:,4 2, +, "-&./0"-"1&2 *2"-4 : 1"&+*"- 5+2 0*25+61#'"%
3'02*21&2% 3" '1=2&"3" ;2-" 0.1&2 %+*=", 1" *2%;.%&" 2, 0*25+61#'"( >2%&" %289. 2%&",.%
'1&2*2%%"3.% 1. ;";2- 3. ?.-+,2 3" #"'7" "#$%&'#" %2-"3" 1" *2%;.%&" 2, 0*25+61#'" 3.
"-&./0"-"1&2( @%&2 2%&+3. : ',;.*&"1&2 ;.'% ,.%&*" #.,. . "-&./0"-"1&2 %2*A 202&'?",21&2
;2*#2B'3. "+3'&'?",21&2( C% 3'%&.*8D2% E2*"3"% ;2-.% "-&./0"-"1&2% 32?2,/%2 ".% 0"&.*2%
19./-'12"*2% 21?.-?'3.% 1" %+" #.,;.%'89.(
)"*" 2%&2 2%&+3. 0.*", %',+-"3.% .% 2%;2#&*.% 32 0*25+61#'"% ;"*" 3'?2*%.% ?.-+,2%
32 #"'7" "#$%&'#" #., " #.**21&2 32 27#'&"89. #.1%&"1&24 2 ;"*" 3'?2*%"% #.**21&2% 32
27#'&"89. #., . ?.-+,2 3" #"'7" "#$%&'#" #.1%&"1&2( C;*2%21&"*2,.% "5+' .% 27;2*',21&.%
32 ?"*'"89. 32 ?.-+,2 ;"*" " #.**21&2 32 27#'&"89. 32 FGC4 5+2 : . #"%. .132 %+*E2,
3'?2*%.% #.,;.*&",21&.% ;"*" 3'02*21&2% #"'7"% "#$%&'#"%(
>" HE+*" I(FI4 ;.32/%2 .B%2*?"* .% *2%+-&"3.% 3"% *2%;.%&"% 2, 0*25+61#'" 3.
"-&./0"-"1&2 ;"*" 3'02*21&2% ?.-+,2% 3" #"'7" "#$%&'#"( )"*" 2%&" %',+-"89. " #.**21&2
32 27#'&"89. : #.1%&"1&2 #., ?"-.* 32 FGC 2 " 0*25+61#'" 32 27#'&"89. 32%&" #.**21&2 :
J!Hz (
!
"#$%&' ()*(+ ,'- .$%&'- /' 0-1%0&/' 2345-0 6- &02&'26- /0 7'-0 8'&' /#70&0920- :';6&0/0 :6;%40) < 0#=6 y &08&0-092' ' :0;6>#/'/0 /0 /0-;6>'40926 /6 >690 04 ms ) < 0#=6 x
&08&0-092' 6 /0-;6>'40926 /6 >690 04 m) ,' /#&0#2' 2345-0 '- .$%&'- >6&&0-869/0920'6- &02&'26- /0 7'-0 8'&' /#70&0920- :6;%40-) < 0#=6 f &08&0-092' ' 7&01%39>#' 04 Hz ) <
0#=6 ?48;#2%/0 &08&0-092' ' '48;#2%/0 /'- 7&01%39>#'- 04 dB ) @6/'- '- .$%&'- 76&'4
$0&'/'- 8'&' %4' >6&&0920 /0 0=>#2'AB6 >69-2'920 /0 *C?)
,' .$%&' ()*(D 86/05-0 6E-0&:'& >;'&'40920 ' 7&01%39>#' /0 !F Hz G7&01%39>#' /0
&0--69H9>#' 0 /' 76920I -6E&0--'#9/65-0 04 &0;'AB6 '- 6%2&'- 7&01%39>#'-) ?;J4 /#-26D
86/05-0 6E-0&:'& %4 '%40926 /' '48;#2%/0 /0 26/'- '- 7&01%39>#'- 1%'9/6 6 :6;%40 /'
>'#=' '>K-2#>' :';0 CDCF m3 ) L-20 >6486&2'40926 -%$0&0 76&2040920 >'6-D >646 -%$0&#/6
2'4EJ4 8'&' 6 40-46 :6;%40 9' .$%&' ()*M) @'#- >6486&2'40926- >'N2#>6- #48;#>'4
04 /#-26&AO0-D 0 #-26 86/0 -0& :#-26 4'#- >;'&'40920 90-2' .$%&'D PQ 1%0 -%&$04 /#:0&-'6%2&'- 7&01%39>#'- /#70&0920- /' '8;#>'/' 80;' 76920 /0 0=>#2'AB6)
< >6486&2'40926 46-2&'/6 9' .$%&' ()*( >6&&0-869/0 '6 0-2%/6 /#9H4#>6
46-2&'/6 9' .$%&' ()*M) R0 7'26D -0 6E-0&:'&46- 6 :6;%40 /' >'#=' '>K-2#>' /0 CDCC*
m3 D 9' .$%&' ()*M 6 -#-204' '8&0-092' %4 >6486&2'40926 ;#90'&D 0 9' .$%&' ()*(D 86%>'6%2&'- 7&01%39>#'- -%&$04 ';J4 /' 7&01%39>#' /' 76920 /0 0=>#2'AB6) S'&' 6 :6;%40 /'
>'#=' '>K-2#>' /0 CDC* m3 D 9' .$%&' ()*M -%&$04 N&E#2'- 80&#N/#>'- /0 80&T6/6 /6#-D 0
9' .$%&' ()*( 96:'- 7&01%39>#'- '8'&0>04) "#9';40920 1%'9/6 6 :';6& /' >'#=' '>K-2#>'
'2#9$0 %4 :';6& /0 CDCF m3 D 9' .$%&' ()*M -%&$04 >6486&2'40926- /0 4%;2#80&#6/#>#/'/0
-%$0&#9/6 %4 >6486&2'40926 >'N2#>6D 0 9' .$%&' ()*( 1%'-0 26/'- '- 7&01%39>#'- 234 6
:';6& /0 '48;#2%/0 '%4092'/6) < >6486&2'40926 /0 '%40926 /'- '48;#2%/0- /0 /#:0&-'7&01%39>#'- /' .$%&' ()*(D 1%'9/6 6 :6;%40 :';0 CDCF m3 D J '9Q;6$6 '6 '8&0-092'/6 9'
.$%&' !)M)
<%2&6 709U4096 1%0 86/0 -0& 6E-0&:'/6 9' .$%&' ()*( J 6 '8'&0>#40926 /0
!"#$%&'()*+ ,*"-.')(*+ /0# 12 Hz 3 (4'!4"-# 4 546%-# 0* (*)7* *(8+9)(* 5*) *%-#'9*'04:
;+9*+ ,*"-.')(*+ +%"<#- (4'!4"-# 4+ =#">404+ 0*+ ?"@)9*+ 5A4 *%-#'9*'04:
B*++*'04 *<4"* =*"* 4+ #7=#")-#'94+ 0# 5*")*CA4 0# (4""#'9# 0# #7()9*CA4D
(4'+)0#"4%E+# 4+ 546%-#+ 0# (*)7* *(8+9)(* 0# FDFG m3 D =4" +# 9"*9*" 0# %- 546%-# 0# @40#+#-=#',4 0)'H-)(4D # FDFI m3 D =4" +# 9"*9*" 0# %- 546%-# (4- 0#+#-=#',4 0)'H-)(4
0#J()#'9#D #- 9#"-4+ 0# 0)+94"CK#+ 4@+#"5*0*+ '4+ 0)*<"*-*+ 0# @)!%"(*CA4: L* J<%"*
M:GID 4'0# 9#-E+# %- 0#+#-=#',4 0)'H-)(4 'A4 +*9)+!*9?")4D =40#E+# 4@+#"5*" * "#+=4+9*
#- !"#$%&'()* =*"* 4 546%-# 0* (*)7* *(8+9)(* (4'+9*'9# #- FDFI m3 : L#6*D =40#E+#
4@+#"5*" $%# =*"* * (4""#'9# 0# 2N *+ !"#$%&'()*+ 0)!#"#'9#+ 0* !"#$%&'()* 0* !4'9# 0#
#7()9*CA4 (4-#C*- * *%-#'9*" +%*+ *-=6)9%0#+D *6O- 04 +%"<)-#'94 0# ,*"-.')(*+: B*"*
* (4""#'9# 0# #7()9*CA4 0# IN #+9*+ !"#$%&'()*+ 94"'*-E+# -*)+ ="4'%'()*0*+: P)'*6-#'9#
#- GFN * <"*'0# -*)4")* 0*+ !"#$%&'()*+ *++%-# <"*'0#+ *-=6)9%0#+D (4-4 =40# +#"
4@+#"5*04 '* J<%"* M:GID 0*'04 )'0)()4+ 0# (4-=4"9*-#'94 (*?9)(4D *'Q64<4 *4 4(4"")04
'* J<%"* M:GM:
R- (4-=4"9*-#'94 )'9#"#++*'9# $%# =40# +#" 4@+#"5*04 '* J<%"* M:GI O 4
+%"<)-#'94 0# ,*"-.')(*+ =*"* 5*64"#+ 0# (4""#'9#+ 0# #7()9*CA4 $%# 'A4 *="#+#'9*(4-=4"9*-#'94+ 0# -%69)=#")40)()0*0# '4+ 0)*<"*-*+ 0# @)!%"(*CA4:
S+94 =40# +#"
4@+#"5*04 $%*'04 * (4""#'9# 5*6# 2ND '* J<%"* M:GI +%"<#- 5Q")*+ ,*"-.')(*+D #'$%*'94
$%# '* J<%"* M: D 4 (4-=4"9*-#'94 O (4#"#'9#: T +%"<)-#'94 0#+9*+ ,*"-.')(*+ +# 0#5# U+
'A4E6)'#*")0*0#+ 0*+ #$%*CK#+ $%# (4-#C*- * +# -*')!#+9*"D #-@4"* #+9*+ 'A4E6)'#*")0*0#+
*)'0* 'A4 +#V*- +%J()#'9#+ =*"* (*%+*"#- @)!%"(*CK#+:
!
"#$%&' !()*+ ,'- .$%&'- /' 0-1%0&/' 2034-0 5- &02&'/5- /0 6'-0 7'&' /#60&0820- 9':5&0/0 ;5&&0820 /0 0<;#2'=>5( ? 0#<5 y &07&0-082' ' 90:5;#/'/0 /0 /0-:5;'30825 /5 ;580 03
m
( ? 0#<5 x &07&0-082' 5 /0-:5;'30825 /5 ;580 03 m( ,' /#&0#2' 2034-0 '- .$%&'s
;5&&0-758/0820- '5- &02&'25- /0 6'-0 7'&' /#60&0820- ;5&&0820- /0 0<;#2'=>5( ? 0#<5 f
&07&0-082' ' 6&01%@8;#' 03 Hz ( ? 0#<5 A37:#2%/0 &07&0-082' ' '37:#2%/0 /'- 6&01%@8;#'03 dB ( B5/'- '- .$%&'- 65&'3 $0&'/'- 7'&' %3 95:%30 ;58-2'820 /0 CDC* m3 ( ,0-20
;'-5 5 -#-203' '7&0-082' %3 /0-03708E5 /#8F3#;5 8>5 -'2#-6'2G&#5(
,' .$%&' !()HD 75/04-0 5I-0&9'& ' &0-75-2' 03 6&01%@8;#' 7'&' 5 95:%30 /' ;'#<'
';J-2#;' ;58-2'820 03 CDC) m3 D 1%0 K %3 95:%30 ;58-#/0&'/5 ;53 I53 ;5375&2'30825
/#8F3#;5 ;535 5I-0&9'/5 85- /#'$&'3'- /0 I#6%&;'=>5 35-2&'/5- 8'- .$%&'- !(L ' !()L(
,' .$%&' !()HD 75/04-0 5I-0&9'& 1%0 7'&' 25/'- '- ;5&&0820- -#3%:'/'-D 8>5 -0 203 #8/M;#5/0 ;'5- 0 '- 6&01%@8;#'- -0 35-2&'3 I03 /0.8#/'-( A#8/' 8' .$%&' !()HD 75/04-0 5I-0&9'&
1%0 ' ;'#<' ';J-2#;' /0 95:%30 CDC) m3 #8#I0 5 '7'&0;#30825 /0 E'&3N8#;'- /09#/5 O8>54:#80'&#/'/0- /'- 01%'=P0-D /#60&082030820 /' .$%&' !()*D 58/0 0-2'- '7'&0;03 3'#7&58%8;#'/'-( Q-25 75/0 -0& 5I-0&9'/5 3'#- ;:'&'30820 8' ;5&&0820 /0 0<;#2'=>5 /0 RA(
!
"#$%&' ()*+, -'. /$%&'. 0' 1.2%1&0' 3456.1 7. &13&'37. 01 8'.1 9'&' 0#81&1:31. ;'<7&1.
01 =7&&1:31 01 1>=#3'?@7) A 1#>7 y &19&1.1:3' ' ;1<7=#0'01 01 01.<7='51:37 07 =7:1 15
m
) A 1#>7 x &19&1.1:3' 7 01.<7='51:37 07 =7:1 15 m) -' 0#&1#3' 3456.1 '. /$%&'.
s
=7&&1.97:01:31. '7. &13&'37. 01 8'.1 9'&' 0#81&1:31. =7&&1:31. 01 1>=#3'?@7) A 1#>7 f
&19&1.1:3' ' 8&12%4:=#' 15 Hz ) A 1#>7 B59<#3%01 &19&1.1:3' ' '59<#3%01 0'. 8&12%4:=#'.
15 dB ) C70'. '. /$%&'. 87&'5 $1&'0'. 9'&' %5 ;7<%51 =7:.3':31 01 DED* m3 )
F7016.1 '$7&' =759'&'& '. /$%&'. ()*! 1 ()*+) F'&' ' =7&&1:31 01 1>=#3'?@7 01 DE AE
' /$%&' ()*! '9&1.1:3' %5' '%51:37 01 '9&7>#5'0'51:31 *D dB 15 ;G&#'. 8&12%4:=#'.
':31. 0' 8&12%4:=#' 01 1>=#3'?@7 0' 87:31E 1 0197#. 0' 8&12%4:=#' 01 &1..7:H:=#') -'
/$%&' ()*+E '. 8&12%4:=#'. '9'&1=15 5'#. 01/:#0'. 2%1 :' ()*!) I%':07 ' =7&&1:31 01
1>=#3'?@7 ;'<1 JAE '. 8&12%4:=#'. :' /$%&' ()*+ '9'&1=15 5'#. '31:%'0'. 2%1 :' /$%&'
()*!) I%':07 ' =7&&1:31 01 1>=#3'?@7 ;'<1 *D AE :' /$%&' ()*! 370'. '. 8&12%4:=#'.
97..%15 '59<#3%01. '<3'.E 1:2%':37 2%1 :' /$%&' ()*+E 5%#3'. 8&12%4:=#'. .@7 '31:%'0'.
1 '. K'&5L:#='. .1 .7M&1..'15) N'.1'07 :'. /$%&'. ()*+ 1 ()*!E 97016.1 '/&5'&E 01 %5
5707 $1&'<E 2%1 ;7<%51. 51:7&1. '31:%'5 '. 8&12%4:=#'. 1.9O&#'.E 8'P1:07 =75 2%1 7
.75 .1Q' &19&70%P#07 01 87&5' 5'#. /1< '7 .#:'< 01 1:3&'0')
B:'<#.':076.1 7. &1.%<3'07. 07. 0#'$&'5'. 01 M#8%&='?R1. 1 7. &1.%<3'07. 1.91=3&'#.
9'&' ' &1.97.3' 15 8&12%4:=#'. 07. '<3768'<':31.E 97016.1 01 %5 5707 $1&'< '/&5'& 2%1
2%':37 51:7& 7 ;7<%51 0' ='#>' '=O.3#='E 51<K7& .1&G ' &1.97.3' 07 '<3768'<':31) F7&S5E
' 1.=7<K' 01.31 ;7<%51 01;1 .1& 375'07 =75 =%#0'07) T1 87& =7:.#01&'07 7 51:7& ;'<7&
97..U;1< 01 ='#>' '=O.3#='E 7% .1Q'E DEDD* m3 E :@7 31&157. %5' M7' &1.97.3'E 97#. 1.31
;7<%51 <#5#3' 7 01.<7='51:37 07 =7:1 07 '<3768'<':31E =757 9701 .1& ;#.37 :' /$%&'
()*JE ='%.':07 =759&1..R1.) -' 7%3&' 1>3&15#0'01E 7 ;7<%51 01;1 .1& 51:7& 2%1 DEDV
m3 E 9'&' 1;#3'& '7 5G>#57 .#3%'?R1. 01 5%<3#91&#70#=#0'01) B..#5E 9'&' 7. &1.%<3'07.
7M3#07. :1.31 3&'M'<K7E %5 M75 ;7<%51 9'&' ='#>' '=O.3#=' 01;1 .1& 1.=7<K#07 1:3&1 DED*
!
m3 " #$#% m3 & '()*+,*($ -+)+ ./" ( )"0/1*+2( ,+ 32"142+2" 2" /5+ 6+47+ +680*46+ 0"1+2+
0"9+ :(5$ ; ,"6"00<)4( ./" + 6+47+ +680*46+ +1;5 2" -"./",+$ *",=+ ( 0"/ >(1/5" "06(1=42(
6(5 :+0*+,*" 6/42+2(& ?-"0+) 2" +./4 "0*+)5(0 6(,042")+,2( /5 +1*(@A+1+,*" "0-"6B36($
+ 5"*(2(1(C4+ +./4 +-)"0",*+2+ -(2" 0") 1">+2+ + /5 +1*(@A+1+,*" ./+1./")$ 6(,*)4:/4,2(
+0045 -+)+ /5 5"1=() -)(9"*( 2" 6+47+0 +680*46+0 0"1+2+0&
!
!"#!$%&'" ( )*+,+$-"& .%)%*"&
"# $%&'()$*+,)-+-*# +##&.)+-+# +&# +(/&'0+(+$/*# 0&,+1 ,*.*$/*1*$/* *#/2-+-+# 3&,
4
!"#$% &' ()*5
6778+95 4
!"#$% &' ()*5
6778:95 4#$+,,
&' ()*5
677;9 * 4-$./,+0
&' ()*5
677;9< =*#/*# /,+:+(>&# &# +(/&'0+(+$/*# 0&,+1 *#/2-+-&# #&: & 3&$/& -* ?)#/+ -* #)#/*1+#
-)$@1).&#< A#/*# /,+:+(>&# #2B*,*1 0&,/*1*$/* + 3,*#*$C+ -* .+&# *1 +(/&'0+(+$/*# -*?)-&
+ $%&'()$*+,)-+-*# *1 #*2 #)#/*1+< =&# /,+:+(>&# -* 4
()*5
6778:95 4#$+,,
&' ()*5
677;9 * 4-$./,+0
&' ()*5
!"#$% &' ()*5 6778+95 4 !"#$% &'
677;95 & +(/&'0+(+$/* D *#/2-+-& +& +,
()?,* &2 $& ?E.2&<
=*#/* /,+:+(>& 0&) 3,&3&#/& & *#/2-& -+ -)$@1).+ -&# +(/&'0+(+$/*# (*?+$-& *1
.&$#)-*,+C%& 21 /*,1& +-).)&$+( ,*3,*#*$/+$-& 21+ .+)F+ +.G#/).+ #*(+-+ * ?*,)H.+&2'#*
& #*2 .&13&,/+1*$/&< " 1*/&-&(&B)+ 2/)()I+-+ 0&) +$E(&B+ J 2/)()I+-+ 3&, 4 -$./,+0 &' ()*5
677;95 2/)()I+$-& ,*/,+/&# -* 0+#* * -)+B,+1+# -* :)02,.+C%&5 3&, #*, +13(+1*$/* 2/)()I+-+
3+,+ & *#/2-& -* #)#/*1+# -)$@1).&#< K&) 2/)()I+-+ + 1*#1+ $%&'()$*+,)-+-* 3,&3&#/+ 3&,
4-$./,+0 &' ()*5 677;9<
L# ,*#2(/+-&# +3,*#*$/+-&# $*#/* /,+:+(>& #2B*,*1 21+ 0&,/* )$M2N$.)+ -+ .+)F+
+.G#/).+ $+ -)$@1).+ -&# +(/&'0+(+$/*#< O* 21 1&-& B*,+(5 ?&(21*# 3*P2*$&# 0+I*1 .&1
P2* &# .&13&,/+1*$/&# 12(/)3*,)Q-).&# * .+Q/).&# -)1)$2+1< L# ,*#2(/+-&# #%& +)$-+
1*(>&,*# #* *#/* 3*P2*$& ?&(21* D *#.&(>)-& .&1 .2)-+-&5 3&)# #* 0&,*1 *#.&(>)-&# ?&(21*#
-* .+)F+ +.G#/).+ 12)/& 3*P2*$&#5 & -*#(&.+1*$/& -& .&$* #*,E ()1)/+-& 3,*R2-).+$-& #2+
*H.)N$.)+< " -)1)$2)C%& -&# .&13&,/+1*$/&# #2B*,)$-& .+&# * +/D 1*#1& +# :)02,.+CS*#5
0+I .&1 P2* 0,*P2N$.)+# *#3G,)+# -*)F*1 -* #*, B*,+-+# $+ ,*#3&#/+ -& #)#/*1+5 * )#/&
+21*$/+ + H-*()-+-* -& #)#/*1+ +(/&'0+(+$/* 1+)# .+)F+ +.G#/).+< T )13&,/+$/* &:#*,?+,
P2* + 1*/&-&(&B)+ -* *#.&(>+ -& ?&(21* -+ .+)F+ +.G#/).+ +3,*#*$/+-+ $*#/* /,+:+(>&
3&-* #*, +3().+-+ + P2+(P2*, +(/&'0+(+$/* -* :&:)$+ 1Q?*(5 .&$/,):2)$-& 3+,+ 1*(>&,*#
3,&R*/&# -* .+)F+# +.G#/).+# #*(+-+#<
U&-&# &# ,*#2(/+-&# +3,*#*$/+-&# $*#/* /,+:+(>& 0&,+1 B*,+-&# 2/)()I+$-&'#* +
0,*P2N$.)+ -* ,*##&$@$.)+ -& +(/&'0+(+$/* 4VW Hz 9 .&1& #*$-& + 0,*P2N$.)+ -+ 0&$/* -*
*F.)/+C%&< L *#/2-& 0&) ,*+()I+-& $*#/+ 0,*P2N$.)+ 3&)# D &$-* & +(/&'0+(+$/* 1+$)0*#/+ .&1
1+)# .(+,*I+ #2+# .+,+./*,X#/).+#< Y&,D15 /)3).+1*$/*5 &# +(/&'0+(+$/*# $%& #%& 2/)()I+-&#
*1 #2+# 0,*P2N$.)+# -* ,*##&$@$.)+5 1+# +.)1+ -*(+#5 #*$-& +# 0,*P2N$.)+# -+ .&,,*$/*
-* *F.)/+C%& ()1)/+-+# 3&, -)?)#&,*# -* 0,*P2N$.)+< A$/,*/+$/&5 3&-*1&# ?*,)H.+, $+#
#)12(+CS*# -+# HB2,+#5 8<Z * 8<65 &$-* & +(/&'0+(+$/* D & 1*#1& 2/)()I+-& $*#/* /,+:+(>&
*F.)/+-& .&1 +# .&,,*$/*# *1 [7 Hz 5 P2* &# .&13&,/+1*$/&# &:#*,?+-&# $+ 0,*P2N$.)+
-* ,*##&$@$.)+ /+1:D1 3&-*1 #*, &:#*,?+-&# *1 0,*P2N$.)+# -* /,+:+(>& -& +(/&'0+(+$/*<
\&13+,+$-& +# HB2,+#5 8<Z * 8<6 H.+ .(+,& & :*$*0X.)& -+ 3,*#*$C+ -+ .+)F+ +.G#/).+
#*(+-+5 ,*-2I)$-& &# .&13&,/+1*$/&# 12(/)3*,)Q-).&# * .+Q/).&#<
!
"#$%&' ()*+ ,#'$&'-' ./ 0#1%&2'345 6'&' 5 '78591'7':8/ '5 '& 7#;&/ <;57%-/ #:=:#85>) ?
/#@5 I0 &/6&/A/:8' ' 25&&/:8/ ./ /@2#8'345 '67#2'.' :5 A#A8/-' /- A) ? /#@5 x &/6&/A/:8'
5 ./A752'-/:85 .5 25:/ /- m) B 1&/C%D:2#' %8#7#E'.' 15# ./ ! Hz )
"#$%&' ()F+ ,#'$&'-' ./ 0#1%&2'345 6'&' 5 ;57%-/ ./ !G!* m3 ) ? /#@5 I0 &/6&/A/:8' '
25&&/:8/ ./ /@2#8'345 '67#2'.' :5 A#A8/-' /- A) ? /#@5 x &/6&/A/:8' 5 ./A752'-/:85 .5
25:/ /- m) B 1&/C%D:2#' %8#7#E'.' 15# ./ ! Hz )
H5-5 65AAI;/#A 8&'0'7J5A 1%8%&5A 65./-5A 7#A8'&+
• K5 '&8#$5 < !"#$% &' () (*G *LLM> N /A8%.'.' ' ;'&#'345 .' -'AA' /1/8#;' .5A
25:/A .5A '78591'7':8/A C%':.5 /A8/A /A845 /- 1%:2#5:'-/:85) OA8/ /1/#85 65./&P
8/& #-65&8':8/A &/Q/@5A :5A /A8%.5A 'C%# '6&/A/:8'.5AR
• K/A8/ 8&'0'7J5 15&'- 25:A#./&'.'A '6/:'A 2'#@'A '2SA8#2'A A/7'.'A) T5&N- /@#A8/5%8&5A 8#65A ./ 2'#@'A '2SA8#2'AG 25-5 'A .%8'.'A) OA8%.'& 5 25-65&8'-/:85
.#:U-#25 .5 '78591'7':8/ #:A8'7'.5 /- 2'#@'A '2SA8#2'A .%8'.'A ./;/ A/& 7/;'.5 /25:A#./&'345R
• K5 '&8#$5 ./ <+#& ",- %. /01G F!!V> N '=&-'.5 C%/ :' 6&P8#2' kef f (x) N -/.#.5 /'78591'7':8/A ./ 15&-' /A8P8#2') W'-0N- N '=&-'.5 C%/ /A8' -5./7'$/- :45 N 2'6'E
./ '6&/A/:8'& 25-65&8'-/:85A 2'X8#25A :5A '78591'7':8/A) T5&N-G N #-65&8':8/
!
"#"$%&"' ()$*+' )&," (+-)$".)( -) kef f (x) ) /'+/+' ()$*+'%"& #)&,) (+-)$+ 01)
')/')&)#,)( ()$*+' "& #2+3$%#)"'%-"-)& -+& "$,+34"$"#,)&5
• 6)&,) ,'"7"$*+ 8+#&%-)'"(+& " 4')019#8%" -) ')&&+#:#8%" )&,;,%8"< =+'>(? 8+(+
"/')&)#,"-+ #" )01"@2+ AB<CD? " 4')019#8%" -) ')&&+#:#8%" > -)/)#-)#,) -) k Akef f (x)
#+ 8"&+ -+ (+-)$+ AE<!FDD ) -" ("&&" M < =+',"#,+ " 4')019#8%" -) ')&&+#:#8%"
,"(7>( > -)/)#-)#,) -"& #2+3$%#)"'%-"-)& "/')&)#,"-"& #)&,) ,'"7"$*+ ) ,"(7>(
/+-) "/')&)#,"' G"'%"@H)& #+ 41#8%+#"()#,+ -+ &%&,)("< I&&%(? -)G)3&) )&,1-"' "
G"'%"@2+ -" 4')019#8%" -) ')&&+#:#8%" )( 41#@2+ -"& #2+3$%#)"'%-"-)& #" -%#:(%8"
-+ "$,+34"$"#,)5
• J1,'+ )&,1-+ %#,)')&&"#,) > 8+#&%-)'"' + G+$1() -" 8"%K" "8L&,%8" 8+(+ &)#-+
G"'%;G)$< M&,+ &)'%" /+&&NG)$ &) %#&,"$"'(+& 1( -%"4'".(" ",%G+ )( 1(" 8"%K"
"8L&,%8"? +#-) + 8+#,'+$) -)&,) -%"4'".(" &)O" (+-)$"-+ -) ,"$ ("#)%'" " 8+#,'+$"'
+& 8+(/+',"()#,+& 8+(/$)K+& #+& "$,+34"$"#,)&<
• P&,1-+& 8+#&%-)'"#-+ " .)+(),'%" -"& 8"%K"& "8L&,%8"& ,"(7>( -)G)( &)' $)G"-+&
)( 8+#&%-)'"@2+< 6)&,) ,'"7"$*+ "/)#"& + G+$1() -" 8"%K" 4+% 8+#&%-)'"-+
8+(+ &)#-+ ')$)G"#,)? ) 4+'"( -)&/')Q"-"& "& /+&&NG)%& ')R)KH)& &+#+'"& 01)
-%4)')#,)& .)+(),'%"& -) 8"%K" "81&,%8" /+-)( 8"1&"'< ="'" )&," )K,)#&2+ -)G)(
&)' 8+#&%-)'"-"& ,"(7>( 8+#-%@H)& -) 8+#,+'#+ /"'" "& )01"@H)& -%4)')#8%"%& 01)
(+-)$"( )&,)& &%&,)("&<
• J1,'" (+-)$".)( -) #2+3$%#)"'%-"-) %#,)')&&"#,) > 1(" "7+'-".)( 1,%$%Q"#-+3&)
8+(/+#)#,)& "$)",S'%"&< T) 4",+? + &%&,)(" 8+#&%-)'"-+ #" /';,%8" &+4') %#R19#8%"
-+ "(/$%U8"-+' "01% 8+#&%-)'"-+ %-)"$<
• ="'" " (+-)$".)( -+ ,)'(+ 01) ')/')&)#," " 8"%K" "8L&,%8" &)$"-"? #)&,) ,'"7"$*+ 4+%
8+#&%-)'"-+ + /'+8)&&+ 8+(+ &)#-+ %&+,>'(%8+< =+'>(? 8+( 8%'81$"@2+ -) 8+'')#,) )
+ (+G%()#,+ -" 7+7%#"? &1'.) 1( "01)8%()#,+ %#,)'#+ #" 8"%K" "8L&,%8" 01) -)G)
&)' $)G"-+ )( 8+#&%-)'"@2+ )( (+-)$".)#& 41,1'"&<
• P( (+-)$".)#& 41,1'"&? /+-)3&) 8+#&%-)'"' ,"(7>( +& (",)'%"%& 1,%$%Q"-+& ,"#,+
/"'" " 8+#&,'1@2+ -"& /"',)& -+& "$,+34"$"#,)& 7)( 8+(+ " 8+#&,'1@2+ -) 8"%K"&
"8L&,%8"&< T) 4",+? +& (",)'%"%& "7&+'G)#,)& .)'"$()#,) 1&"-+& #+ %#,)'%+' -"&
8"%K"& "8L&,%8"& (+-%U8"( "& ')R)KH)&? ,)#-+ %(/"8,+ #" )K,)#&2+ ')4)'%-" "8%("?
$)G"#-+3&) "& 8"'"8,)'N&,%8"& .)+(>,'%8"& )( 8+#&%-)'"@2+<
!
! "!#$%&'!&
"#"$#"%&'( )* +* ,*- .$,/0%( 1*- 1/,2$,%3( '* 045678789 :;<=; 78 6 >:;?< :@
AB6:C* ,8D
!!! "#$%&' ()*+,--#%$ .&$&/#%, *
E7CA656>6F( %G( H/"D ( IJJK* L* IM*
"'',300+( N* 1*- /"H&$( 1* +*- O0$N&( G* "*
78%&9#+&' "8-4,9- *
#"''0H( 3*
0123" 2%
%4)*56+4#*% 4*
%=> O:;P( %O( H/"D /Q;789=;( IJJ *
1&%5:**; <*) "*6%5 !%$#%,,)- *
R* =<* 0S@:;<( HND .:A6? E;=CC( !TTK*
#&$"%&N( '* '*
2+*6-4#+-*
U6V4;7<9=( H/"D "A:WC57A6? /:A7=5F :@ "V=;7A6( IJJ *
+&X"%&O( $* '*
2% #%4)*56+4#*% 4* +=&*4#+ 58%&9#+&' -8-4,9 *
!* =<* D "<<7C:8
Y=C?=F( IJKJ*
+,UN"/0%( X*
>=, ?*65-@,&;,) 7,-#$% 0**;:**; *
R* =<* E=5=;4:;:W9B( H/"D "W<7:
"V65=W; E;=CC( IJJI*
+GH$&N( +*- +GH$&N( ,*- E&10/,U( "* UB6:57A C565= 78 68 =?=A5;:<F86V7A
?:W<CQ=6P=;*
A*6)%&' *< 4=, 265#* !%$#%,,)#%$ "*+#,48 (
X7=886( L* I!!( 8* ZTZZ( Q* I[ (
!TTZ*
+GH$&N( +*- +GH$&N( ,*- E&10/,U( "* ,85;78C7A V=V4;68= @;7A57:8 68< :8C=5 :@
AB6:C 78 68 =?=A5;:<F86V7A ?:W<CQ=6P=;*
A*6)%&' *< 4=, 265#* !%$#%,,)#%$ "*+#,48 (
%=> O:;P( L* I!\( 8* Z!MM( Q* I[M( !TTZ*
.,&+'&$].&$$"$"( %*- E$"+0( U* E* U*
0&*- B 69& #%4)*56CD* *
D &<96; #?^AB=;
'1+"( IJJR*
3"$U,"( U*
.*5,'&$,9 , "#96'&CD* *
!* =<* /_: E6W?:( /E( #;6C7?D &<75:;6 <6
H87L=;C7<6<= <= /_: E6W?:( !TTM*
3&( `* 2*- '&H( Y* O* "857]A:85;:? :@ AB6:C :@ 5>:]<=9;==C]:@]@;==<:V ?:W<CQ=6P=;
CFC5=V 68< AB6:C CF8AB;:87a657:8 :@ <7b=;=85 :;<=; CFC5=VC*
F)&+4&'-(
0=&*-E "*'#4*%- &%5
L* !T( !TTR*
3&( `* 2*- '&H( Y* O* UB6:C CF8AB;:87a657:8 68< Q6;6V=5=; 78<=857cA657:8 @:;
?:W<CQ=6P=; CFC5=VC*
0=&*-E "*'#4*%- &%5 F)&+4&'- (
L* !I( !TTR*
G0)%/0%( +* &*- ),'#H$%( G* '*- G0)%/0%( G* $*
+#)+6#4*- ,'H4)#+*- *
F6%5&9,%4*- 5, &%G'#-, 5,
R* =<* %=> G=;C=F( H/"D E;=857A= )6??( IJJT*
N',E/U)( E* ':W<CQ=6P=; <7C5:;57:8* ,8D
()*+,,5#%$- *< 4=,
0*%<,),%+, *% 2+*6-4#+-E "@,,+=E &%5 "#$%&' ()*+,--#%$ *
IJZ *
!!!
%4,)%&4#*%&'
!
"#$%&' %( $() *+",-",.' /( 012345267 2582 29: 72;; 9<9=198251>18; <? @<98
=<A:;382B85;(
!"#$ %&'#&&()#&' *$+#(,- .)#&, ' C( DD' EDDF(
"#$"&' +( G(
/-&01#+02 &02$'#(3( H( 8:( &8I J<5B' &J' K$LM 0( N29 &<;>529:
O<7329P' Q9@' EDR!(
"**' %(
450$3 #& "-&01#+02 3-3,(13( M
O27S51:68 K91C85;1>P T58;;' EDD!(
TL,/%, $(*() O+KL' "( #( O42<;M L >A><512= ?<5 89619885;( Q9M
7%%%( EDUV( C( VF' 9( U(
TL*,"&Q$' %() 0LNQ$' 0(
*$!&" *-3,(1 %&'#&(()#&'( !( 8:( M
.)$+(("#&'3 $6 ,5(
%=;8C185' HWW (
T%*"$QO' L() 0XK,%/' Q() 0XK,%/' 0( L 5<A>8 >< @42<>1@ ;>2>8 <9 29
8=8@>5<:P9271@ =<A:;382B85(
,L**"&' Y(
+$!3,#+ 89 .0)#3' HWWU(
:!&"01(&,$3 "( ;!"#$( H( 8:( OA51>1S2' T,' -52;1=M
Q9?<7A; YA;1@
O89>85 #>:2' HWWV(
,%Q$$' X() 0XK,%/ 0() T%*"$QO' L() 0XK,%/' 0( O<9Z572>1<9 <? @42<; 19 2
=<A:;382B85 ;P;>87 A;196 >178 ;8518; 292=P;1;(
*$+#(,-' $29 G529@1;@<' C( EHF' 9( VF
$Q#NL' +( $(
<$!)&02 $6 ,5( !"#$ %&'#&(()#&'
!' 3( E[EV' HWWU(
&=2#3( ( *>&,(3( "( 2,$?6020&,(3 ( 40#@03 +A3,#+03 B(2$ 1C,$"$ "(
D5#(2(?*1022( E( 8:( ,1< :8 X29815<' ,X' -52;1=M
+( $48=:<9 $85C1\<; :8 Y25B8>196
#>:2' EDD (
*"&.' T() YQL" .() &Q' ]() ]%Q' ,( #P23A9<C 8^3<989>; 29: 689852= :1789;1<9;
<? ;>52968 2>>52@><5 <? 8=8@>5<:P9271@ @<98 =<A:;382B85(
45#&(3( .5-3E F(,,E' C( U'
EDDE(
]%Q' ,( X() *L"' *() &Q' ]( $( -1?A5@2>1<9 29: @42<; <? :158@> 52:12>1<9 =<A:;382B85;(
45#&(3( .5-3E F(,,E' C( !' 3( R
D[RVH' EDU (