Generalized Symmetrical Components for Periodic ... - EPQU Journal

giantsneckspiffyΗλεκτρονική - Συσκευές

13 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

72 εμφανίσεις

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Summary: This paper deals with the generalization of the symmetrical components technique to
periodic non-sinusoidal three-phase currents and voltages. The generalization of the concept of
symmetrical components is discussed together with their derivation in both time- and frequency
domain. The main conclusion of the paper is that an orthogonal decomposition of periodic non-
sinusoidal three-phase signals into positive sequence, negative sequence and zero-sequence
components is not possible, but that an additional current and voltage component should be
introduced which is called the residual component.
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1
0
( ) ( ) ( ) ( ) ( ) ( )
T
a a b b c c
T
f t g t f t g t f t g t dt
∗ = + +

f g


L=￿EIDAI￿6DEI￿HJD￿C￿￿=￿EJOFH￿FAHJOE￿F￿EAIJD=JJDA=?JELA
F￿MAH=￿@JDAIGK=HA￿BJDAH￿IL=￿KA?=￿>A?￿￿FKJA@=I
JDA IK￿ ￿B JDA?￿HHAIF￿￿@E￿C L=￿KAI B￿H JDA IAGKA￿?A
?￿￿F￿￿A￿JI￿ 6DEI EI = LAHO E￿JAHAIJE￿C BA=JKHA MDE?D
?￿￿IE@AH=>￿OIE￿F￿EBEAIJDA=￿=￿OIEI￿BJDHAA￿FD=IAIOIJA￿I￿
1￿JDA)FFA￿@EN=￿=￿C￿HEJD￿EI@AHELA@J￿?￿￿FKJAJDAIA
CA￿AH=￿E￿A@ IAGKA￿?A?￿￿F￿￿A￿JI￿ 1J EI ID￿M￿ JD=J I￿￿A
F=HJ ￿B JDA CA￿AH=￿ FAHE￿@E? JDHAA￿FD=IA IEC￿=￿?=￿￿￿J >A
J=￿A￿?=HA￿B>OJDA=>￿LABH=￿AM￿H￿￿=￿@JD=JJDAHAB￿HA=￿
=@@EJE￿￿=￿?￿￿F￿￿A￿JEIJ￿>A?￿￿IE@AHA@￿)￿I￿MAME￿￿￿￿JE?A
I￿￿A￿JDAH@EBBAHA￿?AIMEJDJDA?￿=IIE?=￿?=IA￿BIE￿KI￿E@=￿
JDHAA￿FD=IA GK=￿JEJEAI￿
!￿,-418)61￿￿￿.60-/-￿-4)￿1￿-,
+￿￿2￿￿-￿65
6DA @AHEL=JE￿￿ ￿B JDA CA￿AH=￿E￿A@?￿￿F￿￿A￿JI =￿@ JDA
K￿@AH￿OE￿C ￿=JDA￿=JE?=￿?￿￿?AFJI =HA @EI?KIIA@ E￿ JDA
)FFA￿@EN￿1JEIID￿M￿JD=JJDA?￿￿F￿￿A￿JI?=￿>A@AHELA@E￿
=HEC￿H￿KI￿=JDA￿=JE?=￿M=O>OBE￿@E￿C?KHHA￿JI￿HL￿￿J=CAI
E￿ JDA JDHAA FD=IAI MDE?D D=LA JDA FH￿FAHJO ￿B = F￿IEJELA
IAGKA￿?A￿ = ￿AC=JELA IAGKA￿?A ￿H = ￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿J￿ MDE?D =HA ￿HJD￿C￿￿=￿ J￿ JDA ￿JDAH IAGKA￿?A
?￿￿F￿￿A￿JI=￿@=HA=I?￿￿IA=IF￿IIE>￿AE￿JDA￿A=IJIGK=HAI
IA￿IA J￿ JDA CELA￿ JDHAA￿FD=IA?KHHA￿JI ￿H L￿￿J=CAI￿
,A￿￿JE￿C JDA?KHHA￿J ￿H L￿￿J=CA E￿ FD=IA = >O B
=
J￿ JDA
?KHHA￿J￿HL￿￿J=CAE￿FD=IA>>OB
>
J￿=￿@JDA?KHHA￿J￿H
L￿￿J=CAE￿FD=IA?>OB
?
J￿JDA￿=JDA￿=JE?=￿=￿=￿OIEI￿A=@I
J￿ JDA B￿￿￿￿ME￿C ANFHAIIE￿￿I B￿H JDA @A?￿￿F￿IEJE￿￿ ￿B JDA
JDHAA￿FD=IA GK=￿JEJEAI￿
￿ JDA?￿￿F￿￿A￿JIMEJD￿AH￿￿IAGKA￿?AIO￿￿AJHO￿
( ) ( )
( ) ( )
( ) ( )
oa o
ob o
oc o
f t f t
f t f t
f t f t
   
   
=
   
   
   
 
MEJDJDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿J￿
( )
1
3
( ) ( ) ( ) ( )
o a b c
f t f t f t f t
= + +
!
￿ JDA?￿￿F￿￿A￿JIMEJDF￿IEJELAIAGKA￿?AIO￿￿AJHO￿
( ) ( )
( ) ( 3)
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pa p
pb p
pc p
f t f t
f t f t T/
f t f t T/
   
   
= −
   
   

   
   
"
MEJDJDACA￿AH=￿E￿A@F￿IEJELAIAGKA￿?A?￿￿F￿￿A￿J￿

1
3
( ) ( ) ( 3) ( 3)
( )
( 2 3) ( 2 3)
a o b o
p
c o
f t f t f t T/f t T/
f t
f t T/f t T/
− + + − +
 
=
 
+ + − +
 
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!￿ JDA?￿￿F￿￿A￿JIMEJD￿AC=JELAIAGKA￿?AIO￿￿AJHO￿
( ) ( )
( ) ( 3)
( ) ( 2 3)
na n
nb n
nc n
f t f t
f t f t T/
f t f t T/
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   
+
   

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MEJDJDACA￿AH=￿E￿A@￿AC=JELAIAGKA￿?A?￿￿F￿￿A￿J￿
1
3
( ) ( ) ( 3) ( 3)
( )
( 2 3) ( 2 3)
a o b o
n
c o
f t f t f t T/f t T/
f t
f t T/f t T/
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 
=
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 
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5E￿?A JDA ANFHAIIE￿￿ ￿B JDA CA￿AH=￿E￿A@ ￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿JEIAN=?J￿OJDAI=￿A=IB￿HJDAIE￿KI￿E@=￿?=IA￿JDA
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ANFHAIIE￿￿I￿>J=E￿A@DAHA=￿@JDA?￿=IIE?=￿ANFHAIIE￿￿IKIA@
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JDAFD=IAGK=￿JEJEAIJ￿?=￿?K￿=JAJDAF￿IEJELAIAGKA￿?A=￿@
JDA ￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI￿ 6DEI EI ￿￿J ￿A?AII=HO
B￿H JDA IE￿KI￿E@=￿?=IA MDAHA JDA I=￿A HAIK￿J EI ￿>J=E￿A@
MEJD ￿H MEJD￿KJ IK>JH=?JE￿C JDA ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿J
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GK=￿JEJEAI￿ MDE?D =HA JDA I=￿A E￿ JDA JDHAA FD=IAI￿ =￿@
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f t f t
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 
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&
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?￿￿F￿￿A￿JI?=￿JDA￿>AANFHAIIA@E￿=I￿￿AMD=JIE￿F￿AH
M=OKIE￿CJDADAJAH￿F￿￿=HGK=￿JEJEAI￿
( )
1
3
( ) ( ) ( 3) ( 2 3)
p a b c
f t f t f t T/f t T/
= + + + +
  
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=￿@￿
( )
1
3
( ) ( ) ( 3) ( 2 3)
n a b c
f t f t f t T/f t T/
= + − + −
  

6DAIA ANFHAIIE￿￿I =HA B￿H￿=￿￿O JDA I=￿A =I E￿ JDA
IE￿KI￿E@=￿?=IA￿
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F￿IEJELA IAGKA￿?A =￿@ JDA ￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI
￿B JDA FD=IA GK=￿JEJEAI =@@ KF J￿ JDA FD=IA GK=￿JEJO￿ 6DA
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?￿￿JH=IJ MEJD JDA IE￿KI￿E@=￿ IEJK=JE￿￿￿ JDEI EI ￿￿J JDA?=IA￿
6DAHAB￿HAEJEI￿A?AII=HOJ￿@ABE￿A=HAIE@K=￿?￿￿F￿￿A￿JMDE?D
EIJDAHA￿=E￿E￿CF=HJ￿EJIFDOIE?=￿￿A=￿E￿CEI@EI?KIIA@E￿5A?JE￿￿
'￿6DAANF￿E?EJANFHAIIE￿￿￿BJDAHAIE@K=￿?￿￿F￿￿A￿JEI￿
( )
1
3
( ) ( ) ( 3) ( 2 3)
ra a a a
f t f t f t T/f t T/
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( )
1
3
( ) ( ) ( 3) ( 2 3)
rb b b b
f t f t f t T/f t T/
= + + + +
  

 
( )
1
3
( ) ( ) ( 3) ( 2 3)
rc c c c
f t f t f t T/f t T/
= + + + +
  

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6DEI ANFHAIIE￿￿ ID￿MI JD=J JDA HAIE@K=￿?￿￿F￿￿A￿JI
?￿￿IJEJKJAFAHE￿@E?BK￿?JE￿￿IMEJDFAHE￿@6￿!￿6DAHAIE@K=￿
?￿￿F￿￿A￿JIL=￿EIDEB=￿@￿￿￿OEB￿
( ) ( 3) ( 2 3)
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( ) ( 3) ( 2 3)
( ) ( ) ( )
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b b b
c c c
a b c
f t f t T/f t T/
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D￿￿@B￿H=￿￿JE￿A

E￿JDAFAHE￿@￿1JEIHA=@E￿O?DA?￿A@JD=JJDA
HAIE@K=￿?￿￿F￿￿A￿JL=￿EIDAIB￿HIE￿KI￿E@=￿JE￿ABK￿?JE￿￿I￿
￿￿JABE￿=￿￿OJD=J=￿￿IO￿￿AJHE?=￿?￿￿F￿￿A￿JI=HAA=IE￿O
@AHELA@E￿JDAJE￿A@￿￿=E￿￿￿=￿E￿IJ=￿J=￿A￿KI>=IEI￿
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6DA FAHE￿@E? ￿￿￿￿IE￿KI￿E@=￿ JDHAA￿FD=IA GK=￿JEJEAI?=￿
>AAN=?J￿O@A?￿￿F￿IA@E￿J￿JDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿JI￿JDACA￿AH=￿E￿A@F￿IEJELAIAGKA￿?A?￿￿F￿￿A￿JI￿
JDA CA￿AH=￿E￿A@ ￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI =￿@ JDA
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 
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 
 
 
 
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 
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6DEI EI?=￿￿A@ JDA @A?￿￿F￿IEJE￿￿ E￿J￿ CA￿AH=￿E￿A@
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI￿6DAE￿F￿HJ=￿JBA=JKHAEIJD=J￿AN?AFJ
B￿HJDAHAIE@K=￿?￿￿F￿￿A￿JI￿JDA=￿=￿OIEI￿B=>￿=￿?A@JDHAA￿
FD=IA IOIJA￿?=￿ >A HA@K?A@ J￿ JDA =￿=￿OIEI ￿B = IE￿C￿A￿
FD=IA IOIJA￿￿
6DAAIIA￿JE=￿@EBBAHA￿?AIMEJDHAIFA?JJ￿JDAMA￿￿￿￿￿M￿
IE￿KI￿E@=￿?=IA=HA
a JDAANEIJA￿?A￿BJDAHAIE@K=￿?￿￿F￿￿A￿JI￿
a JDAB=?JJD=JJDA￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JIID￿K￿@>A
IK>JH=?JA@ B￿H JDA?￿￿FKJ=JE￿￿ ￿B JDA F￿IEJELA =￿@
￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI
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￿AH￿=IEIJDA?=IAB￿H￿E￿A￿J￿￿￿E￿AL￿￿J=CAI￿HB￿HJDA?KHHA￿JI
E￿ = JDHAA￿MEHA JDHAA￿FD=IA IOIJA￿￿ IK?D JD=J JDA ￿AH￿￿
IAGKA￿?A?￿￿F￿￿A￿J EI E@A￿JE?=￿￿O ￿AH￿￿ JDA JDHAA￿FD=IA
?KHHA￿JI=￿@￿￿HL￿￿J=CAI?=￿￿￿J￿A?AII=HE￿O>AHAFHAIA￿JA@
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MEJD JDA IO￿￿AJHO FH￿FAHJEAI IAJ B￿HJD E￿ 5A?JE￿￿!￿ 6DA
HAIE@K=￿?￿￿F￿￿A￿JEI￿A?AII=HO￿
%￿￿460￿/￿￿)￿16;￿.60-+￿￿2￿￿-￿65
)I F￿E￿JA@ ￿KJ E￿ 5A?JE￿￿!￿ JDA ￿HJD￿C￿￿=￿EJO ￿B JDA
?￿￿F￿￿A￿JI￿>J=E￿A@B￿HJDAIO￿￿AJHE?=￿?￿￿F￿￿A￿JIE￿JDA
IE￿KI￿E@=￿?=IA EI = LAHO E￿JAHAIJE￿C FH￿FAHJO￿ 6DA
@A?￿￿F￿IEJE￿￿ ￿B JDA FAHE￿@E? JDHAA￿FD=IA GK=￿JEJEAI E￿J￿
JDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI￿JDACA￿AH=￿E￿A@
F￿IEJELA IAGKA￿?A?￿￿F￿￿A￿JI￿ JDA CA￿AH=￿E￿A@ ￿AC=JELA
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B￿￿￿￿MIBH￿￿JDA=￿=￿OIEIE￿JDA)FFA￿@EN￿
6DEI￿HJD￿C￿￿=￿EJOFH￿FAHJOE￿F￿EAI￿
a 6DA =?JELA F￿MAH?￿HHAIF￿￿@E￿C J￿ = JDHAA￿FD=IA
FAHE￿@E? ￿￿￿￿IE￿KI￿E@=￿ L￿￿J=CA =￿@ = JDHAA￿FD=IA
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F￿MAHI?￿HHAIF￿￿@E￿CJ￿JDA￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI
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IJH￿￿CAHFH￿FAHJEAID￿￿@JD=￿JDA￿HJD￿C￿￿=￿EJOFH￿FAHJOMEJD
HAIFA?JJ￿JDAE￿JAH￿=￿FH￿@K?J@ABE￿A@E￿5A?JE￿￿!￿
a 6DA @￿J FH￿@K?J￿ =I @ABE￿A@ E￿ JDA )FFA￿@EN￿ ￿B JDA
CA￿AH=￿E￿A@ ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿J =￿@ =￿O ￿B JDA
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￿AH￿￿ MDE?D EI = IJH￿￿C B￿H￿ ￿B ￿HJD￿C￿￿=￿EJO￿ ￿K?D
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E￿JAH￿=￿FH￿@K?J￿
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CA￿AH=￿E￿A@ F￿IEJELA IAGKA￿?A?￿￿F￿￿A￿JI￿ JDA
CA￿AH=￿E￿A@ ￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI￿ ￿H JDA
HAIE@K=￿?￿￿F￿￿A￿JI￿EI￿AH￿￿6DEIEI=MA=￿AHB￿H￿￿B
￿HJD￿C￿￿=￿EJOJD=￿JDA￿￿A?￿HHAIF￿￿@E￿CJ￿=￿AH￿@￿J
FH￿@K?J￿ >KJ IJE￿￿ = ￿K?D IJH￿￿CAH B￿H￿ JD=￿ JDA ￿￿A
?￿HHAIF￿￿@E￿CJ￿=￿AH￿E￿JAH￿=￿FH￿@K?J￿
6DAIA IJH￿￿CAH ￿HJD￿C￿￿=￿EJO FH￿FAHJEAI E￿F￿O BA=JKHAI
MEJD HAIFA?J J￿ JDA E￿IJ=￿J=￿A￿KI F￿MAHI =￿@ JDA
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J￿JDA=LAH=CAF￿MAHI=￿@H￿IL=￿KAIHABAHHA@J￿=>￿LA￿
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1JM=IF￿E￿JA@￿KJE￿5A?JE￿￿!JD=J=￿￿JDAH=FFH￿=?DJ￿
JDA =￿=￿OIEI ￿B JDHAA￿FD=IA ￿￿￿￿IE￿KI￿E@=￿ JDHAA￿FD=IA
GK=￿JEJEAI ￿￿ EI J￿ @A?￿￿F￿IA JDA FAHE￿@E? BK￿?JE￿￿I E￿J￿
.￿KHEAHIAHEAI=￿@J￿KIA=@A?￿￿F￿IEJE￿￿E￿J￿IO￿￿AJHE?=￿
?￿￿F￿￿A￿JIB￿HJDAIE￿KI￿E@=￿JDHAA￿FD=IAJE￿ABK￿?JE￿￿I=J
A=?D D=H￿￿￿E? BHAGKA￿?O￿ 1J EI E￿JAHAIJE￿C J￿ IAA D￿M JDEI
@A?￿￿F￿IEJE￿￿HA￿=JAIJ￿JDA@A?￿￿F￿IEJE￿￿E￿J￿CA￿AH=￿E￿A@
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI￿>J=E￿A@E￿JDAFHALE￿KIIA?JE￿￿I￿
6DAB￿￿￿￿ME￿CHAIK￿JI=HAHA=@E￿O￿>J=E￿A@￿
a +￿￿IE@AH=D=H￿￿￿E?￿BBHAGKA￿?O￿H@AH!￿￿￿MEJD￿=
￿￿￿￿AC=JELAE￿JACAH￿6DAF￿IEJELAIAGKA￿?A￿JDA￿AC=JELA
IAGKA￿?A =￿@ JDA ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI ￿B JDA
?￿HHAIF￿￿@E￿C IE￿KI￿E@=￿ JDHAA￿FD=IA BK￿?JE￿￿ D=LA
HAIFA?JELA￿OJDAIO￿￿AJHOFH￿FAHJEAI￿BJDACA￿AH=￿E￿A@
F￿IEJELA IAGKA￿?A￿ JDA CA￿AH=￿E￿A@ ￿AC=JELA IAGKA￿?A
=￿@JDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI￿
a +￿￿IE@AH=D=H￿￿￿E?￿BBHAGKA￿?O￿H@AH!￿￿ ￿MEJD￿=
￿￿￿￿AC=JELAE￿JACAH￿6DAF￿IEJELAIAGKA￿?A￿JDA￿AC=JELA
IAGKA￿?A =￿@ JDA ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI ￿B JDA
?￿HHAIF￿￿@E￿C IE￿KI￿E@=￿ JDHAA￿FD=IA BK￿?JE￿￿ D=LA
HAIFA?JELA￿OJDAIO￿￿AJHOFH￿FAHJEAI￿BJDACA￿AH=￿E￿A@
￿AC=JELA IAGKA￿?A￿ JDA CA￿AH=￿E￿A@ F￿IEJELA IAGKA￿?A
=￿@JDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI￿
a +￿￿IE@AH = D=H￿￿￿E? ￿B BHAGKA￿?O ￿H@AH!￿￿ MEJD ￿ =
F￿IEJELAE￿JACAH￿6DAF￿IEJELAIAGKA￿?A=￿@JDA￿AC=JELA
IAGKA￿?A?￿￿F￿￿A￿JI￿BJDA?￿HHAIF￿￿@E￿CIE￿KI￿E@=￿
JDHAA￿FD=IA BK￿?JE￿￿ @￿ ￿￿J D=LA JDA IO￿￿AJHO
FH￿FAHJEAI￿B=￿O￿BJDACA￿AH=￿E￿A@F￿IEJELAIAGKA￿?A￿
JDACA￿AH=￿E￿A@￿AC=JELAIAGKA￿?A=￿@JDACA￿AH=￿E￿A@
￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI￿￿￿JDA￿JDAHD=￿@JDA￿AH￿￿
IAGKA￿?A?￿￿F￿￿A￿JIEJEI>A?=KIA￿BJDAIAJAH￿IJD=J
JDA CA￿AH=￿E￿A@ ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿J D=I J￿ >A
IK>JH=?JA@B￿H?￿￿FKJE￿CJDACA￿AH=￿E￿A@F￿IEJELA=￿@
￿AC=JELA IAGKA￿?A?￿￿F￿￿A￿JI ￿B JDA?￿HHAIF￿￿@E￿C
IE￿KI￿E@=￿ JDHAA￿FD=IA BK￿?JE￿￿ D=LA JDA IO￿￿AJHO
FH￿FAHJEAI￿BJDACA￿AH=￿E￿A@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI￿
6DEI ID￿MI JD=J JDA CA￿AH=￿E￿A@ F￿IEJELA IAGKA￿?A￿
￿AC=JELAIAGKA￿?A￿=￿@￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI=HA￿￿J
IKBBE?EA￿JJ￿@AI?HE>A=￿￿FAHE￿@E?JE￿AJDHAA￿FD=IABK￿?JE￿￿I￿
)?￿￿F￿￿A￿J ID￿K￿@ >A =@@A@￿ JDA HAIE@K=￿?￿￿F￿￿A￿J￿
MDE?D?￿HHAIF￿￿@IE￿JDA.￿KHEAHANF=￿IE￿￿=FFH￿=?DJ￿JDA
F￿IEJELA=￿@￿AC=JELAIAGKA￿?A?￿￿F￿￿A￿JI￿BJDAD=H￿￿￿E?I
￿B ￿H@AHI MDE?D =HA ￿K￿JEF￿AI ￿B!￿ ￿￿JA JD=J JDA HAIE@K=￿
?￿￿F￿￿A￿JI?￿￿F￿AJA￿O?￿HHAIF￿￿@ J￿ JDA D=H￿￿￿E?I ￿B
￿H@AHIMDE?D=HA=￿K￿JEF￿A￿B!EBJDAIK￿￿BJDAJDHAA￿FD=IA
GK=￿JEJEAI=HA￿AH￿￿=IEIJDA?=IAB￿H￿E￿A￿J￿￿￿E￿AL￿￿J=CAI￿H
B￿HJDA?KHHA￿JIE￿=JDHAA￿MEHAJDHAA￿FD=IAIOIJA￿￿
6DA ￿KJK=￿ ￿HJD￿C￿￿=￿EJO ￿B JDA CA￿AH=￿E￿A@ F￿IEJELA
IAGKA￿?A￿ JDA CA￿AH=￿E￿A@ ￿AC=JELA IAGKA￿?A￿ JDA ￿AH￿
IAGKA￿?A￿ =￿@ JDA HAIE@K=￿?￿￿F￿￿A￿JI￿ E￿ JDA IA￿IA ￿B =
￿AH￿E￿JAH￿=￿FH￿@K?J￿?=￿￿>LE￿KI￿O=￿I￿>A@AHELA@BH￿￿JDA
￿KJK=￿￿HJD￿C￿￿=￿EJO￿BJDAIO￿￿AJHE?=￿?￿￿F￿￿A￿JIE￿JDA
IE￿KI￿E@=￿?=IA =￿@ BH￿￿ JDA ￿KJK=￿ ￿HJD￿C￿￿=￿EJO ￿B
IE￿KI￿E@=￿BK￿?JE￿￿I￿B@EBBAHA￿JD=H￿￿￿E?￿H@AHI￿
1JEIE￿JAHAIJE￿CJ￿￿￿JE?AJD=JB￿H=D=H￿￿￿E?￿B=￿￿H@AH￿
MDE?D EI ￿￿J = ￿K￿JEF￿A ￿B!￿ JDA ￿AC=JELA =￿@ F￿IEJELA
IAGKA￿?A IO￿￿AJHE?=￿?￿￿F￿￿A￿JI B￿H JDEI IE￿KI￿E@=￿
BK￿?JE￿￿ ￿B FAHE￿@ 6￿￿ EI @EBBAHA￿J BH￿￿ JDA CA￿AH=￿E￿A@
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI B￿H JDEI FAHE￿@E? BK￿?JE￿￿ MEJD
BK￿@=￿A￿J=￿FAHE￿@6￿
'￿4-￿)61￿￿6￿,-2-￿*4￿+￿\5)￿)￿;515
1￿ =?￿￿JHE>KJE￿￿ J￿ JDA FHALE￿KI M￿H￿ID￿F ￿!￿ ￿￿
,AFA￿>H￿?￿ FH￿F￿IAI =￿ =￿C￿HEJD￿ J￿ @AHELA CA￿AH=￿E￿A@
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI￿.￿H JDA?=IA ￿B ￿AH￿￿IK￿ JDHAA￿
FD=IA GK=￿JEJEAI DA?￿=E￿I JD=J JDHAA￿FD=IA @EIJ￿HJA@
GK=￿JEJEAI?=￿>A@A?￿￿F￿IA@E￿J￿CA￿AH=￿E￿A@F￿IEJELA=￿@
￿AC=JELAIAGKA￿?A?￿￿F￿￿A￿JI￿-N=￿F￿AI>A￿￿MID￿MJDA
￿A?AIIEJO ￿B JDA HAIE@K=￿?￿￿F￿￿A￿JI￿ 9A?￿￿IE@AH JM￿
IE￿F￿AAN=￿F￿AI￿
￿ ) JDHAA￿FD=IA L￿￿J=CA MEJD = F￿IEJELA IAGKA￿?A
BK￿@=￿A￿J=￿=￿@=F￿IEJELAIAGKA￿?AJDEH@D=H￿￿￿E?￿
1 3
1 3
1 3
( ) 2 (2 50 ) 2 (2 150 )
2 2
( ) 2 (2 50 ) 2 (2 150 )
3 3
2 2
( ) 2 (2 50 ) 2 2 150 )
3 3
a
b
c
v t V sin t V sin t
v t V sin t V sin t
v t V sin t V sin( t
F F
F F
F F
F F
F F
= +
= − + −
= + + +
￿ ) JDHAA￿FD=IA L￿￿J=CA MEJD = F￿IEJELA IAGKA￿?A
BK￿@=￿A￿J=￿=￿@=￿AC=JELAIAGKA￿?AJDEH@D=H￿￿￿E?￿
1 3
1 3
1 3
( ) 2 (2 50 ) 2 (2 150 )
2 2
( ) 2 (2 50 ) 2 (2 150 )
3 3
2 2
( ) 2 (2 50 ) 2 (2 150 )
3 3
a
b
c
v t V sin t V sin t
v t V sin t V sin t
v t V sin t V sin t
F F
F F
F F
F F
F F
= +
= − + +
= + + −
!
1￿JDABEHIJAN=￿F￿AJDA=￿=￿OIEI￿BJDAFHAIA￿JF=FAHOEA￿@I
JDA BK￿@=￿A￿J=￿?￿￿F￿￿A￿J =I JDA CA￿AH=￿E￿A@ F￿IEJELA
IAGKA￿?A?￿￿F￿￿A￿J￿ JDA JDEH@ D=H￿￿￿E? =I JDA HAIE@K=￿
?￿￿F￿￿A￿J￿ =￿@ ￿￿ ￿AC=JELA IAGKA￿?A ￿H ￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿JI￿6DAI=￿A?￿￿?￿KIE￿￿EI￿>J=E￿A@B￿HJDAIA?￿￿@
AN=￿F￿A￿
.￿H>￿JDAN=￿F￿AI,AFA￿>H￿?￿\IJDA￿HOOEA￿@IJDA?￿￿F￿AJA
JDHAA￿FD=IA IEC￿=￿ =I JDA CA￿AH=￿E￿A@ F￿IEJELA IAGKA￿?A
?￿￿F￿￿A￿J￿ =￿@ ￿￿ ￿AC=JELA IAGKA￿?A ￿H ￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿J￿0￿MALAHEJEI?￿A=HJD=JJDAJDHAA￿FD=IAIEC￿=￿I
@￿￿￿JD=LAF￿IEJELAIAGKA￿?AIO￿￿AJHO￿IE￿?AJDAL￿￿J=CAI
E￿JDAFD=IAI>=￿@??=￿￿￿J>A￿>J=E￿A@BH￿￿JDAL￿￿J=CAE￿
FD=IA=>O￿=CCE￿C￿LAHHAIFA?JELA￿O6￿!=￿@ 6￿!￿
￿.7460-4￿*5-48)61￿￿5
1￿JDA￿EJAH=JKHA￿ ￿"￿￿￿A￿BJA￿IAAIJDAIJ=JA￿A￿JJD=JB￿H
JDHAA￿FD=IA FAHE￿@E? @EIJ￿HJA@?KHHA￿JI ￿H L￿￿J=CAI JDA
D=H￿￿￿E?I￿B￿H@AH￿"￿%￿￿￿￿￿HE￿CA￿AH=￿!￿￿￿?￿HHAIF￿￿@
J￿=F￿IEJELAIAGKA￿?A?KHHA￿J￿JDAD=H￿￿￿E?I￿B￿H@AH ￿#￿&￿
￿￿￿￿HE￿CA￿AH=￿!￿`￿?￿HHAIF￿￿@J￿=￿AC=JELAIAGKA￿?A
?KHHA￿J￿ JDA D=H￿￿￿E?I ￿B ￿H@AH!￿ $￿'￿ ￿H E￿ CA￿AH=￿!￿￿
?￿HHAIF￿￿@ J￿ = ￿AH￿￿IAGKA￿?A?KHHA￿J￿ 6DA =￿=￿OIEI ￿B
5A?JE￿￿'ID￿MIJD=JJDEIIJ=JA￿A￿JID￿K￿@>AKIA@?=HABK￿￿O￿
1JEIE￿@AA@￿￿￿OL=￿E@EBJDAJDHAA￿FD=IA?KHHA￿J￿HL￿￿J=CA
E￿FD=IA>EIJDAI=￿A=IE￿FD=IA=￿>KJ￿=CCE￿C￿LAH6￿!￿
=￿@JDA?KHHA￿J￿HL￿￿J=CAE￿FD=IA?EIJDAI=￿A=IE￿FD=IA
=￿>KJ￿=CCE￿C￿LAH 6￿!￿1BJDEIIO￿￿AJHOFH￿FAHJO@￿AI￿￿J
D￿￿@￿JDA￿JDAIJ=JA￿A￿JEI￿￿JL=￿E@￿
￿.1￿)￿4-￿)4￿5
5K￿￿=HE￿E￿CJDAB￿￿￿￿ME￿C?￿￿?￿KIE￿￿IB￿￿￿￿MBH￿￿JDA
=￿=￿OIEIE￿JDEIF=FAH￿
a 6DAJDHAA￿FD=IA?KHHA￿J￿HL￿￿J=CA?=￿￿￿J=￿M=OI>A
@AHELA@BH￿￿CA￿AH=￿E￿A@F￿IEJELAIAGKA￿?A￿CA￿AH=￿E￿A@
￿AC=JELA IAGKA￿?A =￿@ CA￿AH=￿E￿A@ ￿AH￿￿IAGKA￿?A
?￿￿F￿￿A￿JI￿)HAIE@K=￿?￿￿F￿￿A￿J￿=O>AHAGKEHA@￿
a 6￿?￿￿FKJA JDA CA￿AH=￿E￿A@ F￿IEJELA =￿@ ￿AC=JELA
IAGKA￿?A?￿￿F￿￿A￿JI￿ JDA ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI
ID￿K￿@BEHIJ>AIK>JH=?JA@BH￿￿JDAFD=IAGK=￿JEJEAI￿E￿
?￿￿JH=IJ MEJD JDA IE￿KI￿E@=￿?=IA MDAHA JDEI EI ￿￿J
￿A?AII=HO￿
a 1￿JDAIE￿KI￿E@=￿?=IAJDAHAIE@K=￿?￿￿F￿￿A￿JEI=>IA￿J
=￿@ JDA ￿JDAH?￿￿F￿￿A￿JI HA@K?A J￿ JDA?￿=IIE?=￿
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI￿
a 6DACA￿AH=￿E￿A@F￿IEJELAIAGKA￿?A￿￿AC=JELAIAGKA￿?A￿
=￿@ ￿AH￿￿IAGKA￿?A?￿￿F￿￿A￿JI D=LA?￿￿F￿AJA FD=IA
IO￿￿AJHOKFJ￿@EIF￿=?A￿A￿J￿LAH=JDEH@￿B=FAHE￿@￿
6DEIE￿F￿EAIJD=JJDAJDHAA￿FD=IA=￿=￿OIEI?=￿>AHA@K?A@
J￿ = IE￿C￿A￿FD=IA =￿=￿OIEI B￿H = >=￿=￿?A@ JDHAA￿FD=IA
F￿MAHIOIJA￿￿
a 6DA HAIE@K=￿?￿￿F￿￿A￿J @￿AI ￿￿J D=LA JDA I=￿A
IO￿￿AJHO￿=￿@JDA?￿HHAIF￿￿@E￿CJDHAA￿FD=IA=￿=￿OIEI
?=￿￿￿J >A HA@K?A@ J￿ = IE￿C￿A￿FD=IA =￿=￿OIEI￿ 1J
?￿HHAIF￿￿@IJ￿=FAHE￿@E?JE￿ABK￿?JE￿￿E￿A=?D￿BJDA
JDHAAFD=IAIMEJD=FAHE￿@MDE?DEI￿￿AJDEH@￿BJDAFAHE￿@
￿BJDA?KHHA￿JI=￿@L￿￿J=CAI￿JDEI=￿I￿￿=OIE￿F￿EBOJDA
=￿=￿OIEI￿
)22-￿,1:
,AHEL=JE￿￿￿BJDAIAGKA￿?A?￿￿F￿￿A￿JI
6￿ @AHELA JDA ANF￿E?EJ ANFHAIIE￿￿I B￿H JDA CA￿AH=￿E￿A@
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI MA BEHIJ B￿H￿K￿=JA JDA?￿=IIE?=￿
IO￿￿AJHE?=￿?￿￿F￿￿A￿JIE￿=￿E￿A=H=￿CA>H=E?IAJJE￿C￿
) CA￿AH=￿ JDHAA￿FD=IA IE￿KI￿E@=￿ GK=￿JEJO ?KHHA￿J ￿H
L￿￿J=CA?=￿ >A HAFHAIA￿JA@ AEJDAH =I = JDHAA￿@E￿A￿IE￿￿=￿
LA?J￿H￿BFD=I￿HI￿
a
b
c
f
f
f
 
 
=
 
 
 
f
)￿
MDAHA JDA A￿JHEAI =HA?￿￿F￿AN ￿K￿>AHI￿ ￿H =I = JDHAA￿
@E￿A￿IE￿￿=￿LA?J￿H￿BIE￿KI￿E@=￿JE￿ABK￿?JE￿￿I￿
2 ( )
( ) 2 ( )
2 ( )
a a
b b
c c
F cos t
t F cos t
F cos t
M =
M =
M =
 
+
 
= +
 
 
+
 
 
f
)￿ 
6DA @A?￿￿F￿IEJE￿￿ E￿J￿ IO￿￿AJHE?=￿?￿￿F￿￿A￿JI EI
AGKEL=￿A￿J J￿ MHEJE￿C JDA FD=I￿H LA?J￿H =I = IK￿ ￿B JDHAA
LA?J￿HI￿HAIFA?JELA￿OBH￿￿JDA￿E￿A=HIK>IF=?AI@ABE￿A@>O
LA?J￿HI￿BJDAB￿H￿I￿
o
o o
o
f
f
f
 
 
=
 
 
 
f
￿

2 3
2 3
p
j/
p p
j/
p
f
f e
f e
F
F

 
 
 
=
 
 
 
f
￿
2 3
2 3
n
j/
n n
j/
n
f
f e
f e
F
F

 
 
=
 
 
 
f


)￿!
=￿@IE￿E￿=H￿OB￿HJDA?￿HHAIF￿￿@E￿CLA?J￿HI￿BIE￿KI￿E@=￿JE￿A
BK￿?JE￿￿I￿
6DAIO￿￿AJHE?=￿?￿￿F￿￿A￿JI=HA@AHELA@>OANFHAIIE￿C
JD=J JDA IK￿ ￿B JDHAA?￿￿F￿￿A￿J LA?J￿HI￿ ￿￿A BH￿￿ A=?D
IK>IF=?A￿ ID￿K￿@ >A AGK=￿ J￿ JDA CELA￿ JDHAA￿@E￿A￿IE￿￿=￿
LA?J￿H￿)￿￿JDAH=FFH￿=?DEIJ￿ANFHAIIJD=JJDA?￿￿F￿￿A￿J
LA?J￿H BH￿￿ A=?D IK>IF=?A EI JDA FH￿￿A?JE￿￿ ￿B JDA CELA￿
LA?J￿H￿￿J￿JD=JIK>IF=?A￿-GKEL=￿A￿J￿O￿MA?=￿￿>J=E￿JDA
?￿￿F￿￿A￿JI >O?￿￿IE@AHE￿C JDA?￿￿F￿￿A￿J LA?J￿H E￿ JDA
IK>IF=?A=IJDA>AIJ=FFH￿NE￿=JE￿￿￿BJDACELA￿LA?J￿HE￿
JDA ￿A=IJ IGK=HAI IA￿IA￿
6￿ CA￿AH=￿E￿A JDEI?￿￿?AFJ J￿ FAHE￿@E? JDHAA￿FD=IA
GK=￿JEJEAI￿MA?￿￿IE@AH=JDHAA￿@E￿A￿IE￿￿=￿LA?J￿HE￿JDA￿E￿A=H
IF=?A￿
( )
( ) ( )
( )
a
b
c
f t
t f t
f t
 
 
=
 
 
 
f
)￿"
MDAHAB
=
J￿B
>
J=￿@B
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6￿
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BH￿￿ JDA ￿E￿A=H IK>IF=?AI I
￿
￿ I
F
 =￿@ I
￿
 ￿B JDA LA?J￿HI￿
HAIFA?JELA￿O￿BJDAB￿H￿I￿
2￿6A￿JEAJ=￿￿￿/A￿AH=￿E￿A@5O￿￿AJHE?=￿+￿￿F￿￿A￿JIB￿H2AHE￿@E?￿￿￿￿5E￿KI￿E@=￿6DHAA￿2D=IA5EC￿=￿I
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2￿MAH3K=￿EJO=￿@7JE￿E￿=JE￿￿￿￿￿KH￿=￿_8￿￿￿:111￿￿￿￿ %
( )
( ) ( )
( ) ( ) ( ) ( 3) ( ) ( 3)
( ) ( 2 3)
( 2 3)
p
o n
o o p p n n
o n
p
f t
f t f t
t f t,t f t T/,t f t T/
f t f t T/
f t T/
 
   
 
   
= = − = +
 
   
 
   
+

   
 
 
f f f
)￿#
MDAHAB
￿
J￿B
F
J=￿@B
￿
J=HAFAHE￿@E?BK￿?JE￿￿IMEJDFAHE￿@6￿
6DAB￿￿￿￿ME￿CGKAIJE￿￿I=HAAIIA￿JE=￿B￿HJDA=￿=￿OIEI￿B
JDAF￿IIE>E￿EJO=￿@JDAK￿E?EJO￿BJDA@A?￿￿F￿IEJE￿￿￿
a =HAJDAJDHAAIK>IF=?AI￿KJK=￿￿OE￿@AFA￿@A￿J￿
a =HAJDAJDHAAIK>IF=?AI￿KJK=￿￿O￿HJD￿C￿￿=￿￿
a @￿JDAJDHAAIK>IF=?AIIF=￿JDAMD￿￿A￿E￿A=HIF=?A￿
6DA =￿IMAH J￿ JDAIA GKAIJE￿￿I EI =BBEH￿=JELA E￿ JDA
IE￿KI￿E@=￿?=IA￿ 6DEI EI JDA BK￿@=￿A￿J=￿ HA=I￿￿ MDO
IO￿￿AJHE?=￿?￿￿F￿￿A￿JI=HAI￿KIABK￿￿1￿JDACA￿AH=￿FAHE￿@E?
?=IAJDAIEJK=JE￿￿EI￿￿JI￿B=L￿H=>￿A￿
6DA JDHAA IK>IF=?AI =HA E￿@AA@ ￿￿J E￿@AFA￿@A￿J =￿@
DA￿?A ￿￿J ￿KJK=￿￿O ￿HJD￿C￿￿=￿￿ 6DEI?=￿ >A IAA￿ >O
?￿￿IE@AHE￿CJDAAN=￿F￿A￿BJDAB￿￿￿￿ME￿CLA?J￿HMDE?D￿EAIE￿
A=?D￿BJDAJDHAAIK>IF=?AI￿
2 (3 )
( ) 2 (3 )
2 (3 )
A cos t
t A cos t
A cos t
M
M
M
 
 
=
 
 
 
 
f
)￿$
1J?=￿ =￿I￿ >A ID￿M￿ JD=J JDA IK>IF=?AI @￿ ￿￿J IF=￿ JDA
J￿J=￿￿E￿A=HIF=?A￿BFAHE￿@E?LA?J￿HI￿*A￿￿MEI=￿AN=￿F￿A￿B
=FAHE￿@E?LA?J￿HMDE?D?=￿￿￿J>A?￿￿F￿IA@>OLA?J￿HIBH￿￿
JDA JDHAA IK>IF=?AI￿
2 (3 )
( ) 2 (3 2 3)
2 (3 4 3)
A cos t
t A cos t/
A cos t/
M
M F
M F
 
 
= −
 
 

 
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*A￿CE=￿ 5J=JA￿ 2HE￿A ￿E￿EIJAH\I ￿BBE?A B￿H 5?EA￿?A￿ 6A?D￿￿￿￿CO =￿@
+K￿JKHA￿ 6DA I?EA￿JEBE? HAIF￿￿IE>E￿EJO HAIJI MEJD JDA =KJD￿H￿
)@@HAII￿
,AF=HJ￿A￿J --5)￿
/DA￿J 7￿ELAHIEJO￿
6A?D￿￿￿￿CEAF=H￿￿￿ME￿￿==H@A￿'"￿ *￿'# /A￿J￿ *A￿CEK￿￿
A￿￿=E￿￿ ￿=?GKAI￿9E￿￿A￿I(7/A￿J￿>A￿
JA￿￿￿ ￿! ￿'￿ $"#$"&￿ B=N￿ ￿! ￿'￿ $"#&"￿
2=￿￿￿ ￿=JJ=LA￿￿E
HA?AELA@ JDA,H￿ 1￿C￿,ACHAA MEJD D￿￿￿HI =￿@ JDA
2D￿,￿ @ACHAA￿ >￿JD E￿ A￿A?JHE?=￿ A￿CE￿AAHE￿C￿ BH￿￿
JDA 7￿ELAHIEJO ￿B 2=@￿L= 1J=￿O E￿ ''  =￿@ E￿ ''#￿
HAIFA?JELA￿O￿.H￿￿ ''# J￿ ￿ DA M=I = HAIA=H?DAH
=J JDA 7￿ELAHIEJO ￿B 2=@￿L=￿ 1￿  DA ￿￿E￿A@ JDA
,AF=HJ￿A￿J ￿B -￿A?JHE?=￿￿ ￿A?D=￿E?=￿ =￿@
￿=￿=CA￿A￿J -￿CE￿AAHE￿C ,1-/￿ ￿B JDA 7￿ELAHIEJO
￿B 7@E￿A￿ MDAHA DA M=I =￿ )II￿?E=JA 2H￿BAII￿H ￿B
-￿A?JH￿￿E?I BH￿￿   J￿ #￿.H￿￿ ￿ # DA M=I ￿A=@E￿C JDA
2￿MAH -￿A?JH￿￿E?I ￿=>￿H=J￿HO ￿B JDA,1-/￿ =J JDA 7￿ELAHIEJO ￿B
7@E￿A￿ MDE?D DA B￿K￿@A@ E￿ ￿ 5E￿?A # DA D=I >AA￿ MEJD JDA
,6/ =J JDA 7￿ELAHIEJO ￿B 2=@￿L= MEJD JDA I=￿A @KJEAI￿
0EI ￿=￿￿H BEA￿@I ￿B E￿JAHAIJ E￿?￿K@A =￿=￿OIEI￿ ￿￿@A￿E￿C =￿@?￿￿JH￿￿ ￿B
F￿MAH?￿￿LAHJAHI￿ @ECEJ=￿?￿￿JH￿￿ JA?D￿EGKAI B￿H F￿MAH A￿A?JH￿￿E?
?EH?KEJI￿ =￿@ F￿MAH GK=￿EJO EIIKAI￿
2=￿￿￿ ￿=JJ=LA￿￿E EI = ￿A￿>AH ￿B 1--- 2￿MAH -￿A?JH￿￿E?I￿ 1---
1￿@KIJHO )FF￿E?=JE￿￿I￿ 1--- 1￿@KIJHE=￿ -￿A?JH￿￿E?I 5￿?EAJEAI =￿@ JDA
1J=￿E=￿ )II￿?E=JE￿￿ ￿B -￿A?JHE?=￿ =￿@ -￿A?JH￿￿E? -￿CE￿AAHI )-1￿ 0A
=￿I￿ IAHLAI =I =￿ )II￿?E=JA -@EJ￿H B￿H 1--- 6H=￿I=?JE￿￿I ￿￿ 2￿MAH
-￿A?JH￿￿E?I =￿@ ￿A￿>AH￿=J￿￿=HCA ￿B 2-￿5 )@?￿￿ B￿H JAH￿I "￿
$￿
)@@HAII￿
,AF=HJ￿A￿J ￿B 6A?D￿￿￿￿CO =￿@ ￿=￿=CA￿A￿J ￿B 1￿@KIJHE=￿ 5OIJA￿I
,6/￿
7￿ELAHIEJO ￿B 2=@￿L=￿
5JH=@A￿￿= 5￿ ￿E?￿￿=￿!￿ !$ 8E?A￿￿=￿ 1J=￿O￿
A￿￿=E￿￿ F=￿￿￿￿￿=JJ=LA￿￿E(K￿EF@￿EJ￿
JA￿￿￿ ￿!'￿"""￿''&&'￿ B=N￿ ￿!'￿"""￿''&&&&￿
-￿EI=>AJJ= 6A@AI?DE
M=I >￿H￿ E￿ ￿￿LA￿J= 8E?A￿JE￿=￿ 1J=￿O￿ E￿ '&￿ 5DA
HA?AELA@JDA,H￿1￿C￿,ACHAAMEJDD￿￿￿HIE￿A￿A?JHE?=￿
A￿CE￿AAHE￿C BH￿￿ JDA 7￿ELAHIEJO ￿B 2=@￿L=￿ 1J=￿O E￿
#￿ 5DA EI?KHHA￿J￿O FKHIKE￿C JDA 2D, @ACHAA E￿
￿A?D=JH￿￿E?I =￿@ 1￿@KIJHE=￿ 5OIJA￿I =J JDA 7￿ELAHIEJO
￿B 2=@￿L= E￿ 8E?A￿￿=￿
0AH HAIA=H?D E￿JAHAIJI =HA E￿ JDA BEA￿@ ￿B @ECEJ=￿?￿￿JH￿￿
￿B F￿MAH?￿￿LAHJAHI =￿@ E￿ F￿MAH GK=￿EJO EIIKAI￿
)@@HAII￿
,AF=HJ￿A￿J ￿B 6A?D￿￿￿￿CO =￿@ ￿=￿=CA￿A￿J ￿B 1￿@KIJHE=￿ 5OIJA￿I
,6/￿
7￿ELAHIEJO ￿B 2=@￿L=￿
5JH=@A￿￿= 5￿ ￿E?￿￿=￿!￿ !$ 8E?A￿￿=￿ 1J=￿O￿
A￿￿=E￿￿ A￿EI=>AJJ=￿JA@AI?DE(K￿EF@￿EJ￿
JA￿￿￿ ￿!'￿"""￿''&&##￿ B=N￿ ￿!'￿"""￿''&&&&￿
2￿6A￿JEAJ=￿￿￿/A￿AH=￿E￿A@5O￿￿AJHE?=￿+￿￿F￿￿A￿JIB￿H2AHE￿@E?￿￿￿￿5E￿KI￿E@=￿6DHAA￿2D=IA5EC￿=￿I