5A?C1;8:0=A:0O D878G5A:0O
>;8<?8040 (III MB0?)
2006 3>4
"5>@5B8G5A:89 BC@
9 :;0AA.
040G0 1. @0D>?>AB@>8B5;L
040G0 2. 5B0;;K B>65 :8?OB!
040G0 3. -;5:B@>42830B5;L
040G0 4. >475<=0O @5:0
5H5=8O
8
:><<5=B0@88
1
040G0 1. @0D>?>AB@>8B5;L
@8 2K?>;=5=88 40==>3> 7040=8O B@51C5BAO BI0B5;L=>ABL 8 0::C@0B=>ABL. 5H5=85
7040G8 4>AB0B>G=> >G5284=>, ?>MB><C ?@82545< B@51C5<K5 @8AC=:8 157 :><<5=B0@852.
1.1. @8 7040==>< 70:>=5 87<5=5=8O A:>@>AB59 @57C;LB0B>< ?>AB@>5=8O O2;O5BAO :204@0B
4A< 4A< - 8A. 1.
1.2 #:070==K9 A1>9 8A:0605B @8AC=>: A;54CNI8<
>1@07>< 8A. 2.
1.3 7<5=5=85 ?5@8>40 45;05B :204@0B =5C7=0205<K< - 8A. 3.
1.4 -B>B ?>25@=CBK9 :204@0B <>65B 1KBL =0@8A>20= @07;8G=K<8 A?>A>10<8, >48= 87
:>B>@KE ?>:070= =0 @8A. 4
2
040G0 2. 5B0;;K B>65 :8?OB!
1.1 ;O B>3>, GB>1K 684:>ABL 8A?0@O;0AL, 59 =5>1E>48<> A>>1I0BL B5?;>BC MB0 M=5@38O
70B@0G8205BAO =0 ?@5>4>;5=8O A8; <56<>;5:C;O@=>3> ?@8BO65=8O <>;5:C; 2 684:><
A>AB>O=88. '5< A8;L=55 A2O7L <564C <>;5:C;0<8, B5< 1>;LHCN M=5@38N =5>1E>48<>
A>>1I8BL <>;5:C;5. B45;L=0O <>;5:C;0 ?@5>4>;5205B ?@8BO65=85 4@C38E <>;5:C;
1;03>40@O A2>59 :8=5B8G5A:>9 M=5@388 (:>B>@0O ?@>?>@F8>=0;L=0 B5<?5@0BC@5). >MB><C
G5< A8;L=55 A2O7L <564C <>;5:C;0<8, B5< 2KH5 4>;6=0 1KBL B5?;>B0 8A?0@5=8O 8
B5<?5@0BC@0 :8?5=8O. !;54>20B5;L=>, A @>AB>< B5<?5@0BC@K 4>;6=0 2>7@0AB0BL 8 B5?;>B0
8A?0@5=8O.
5B0;; >;O@=0O
B5?;>B0
#45;L=0O B5?;>B0 8A?0@5=8O,
"5<?5@0BC@0
6 :6
:8?5=8O,
B><=0O
8A?0@5=8O L, L ,
t, C
!8<2>; <0AA0 A :3 <>;L
;N<8=89 Al 27,0 2056 10,8 291,4
>;LD@0< W 183,9 5910 4,96 912,0
5;57> Fe 55,85 3200 6,09 340,0
>;>B> Au 197,2 2966 ?
0;89 K 39,1 760 2,05 80,2
03=89 Mg 24,3 1107 5,26 127,8
54L Cu 63,54 2600 4,8 305,0
;>2> Sn 118,7 2270 2,28 270,6
;0B8=0 Pt 195,1 4530 2,41 470,2
BCBL Hg 200,6 356 0,29 58,2
!28=5F Pb 207,2 1725 0,86 178,2
!5@51@> Ag 107,9 2163 2,36 254,6
&8=: Zn 65,4 913 1,76 115,1
1.2 A;8 =0=5AB8 =0 4803@0<<C 7=0G5=8O B5<?5@0BC@K :8?5=8O 8 C45;L=>9 B5?;>BK
8A?0@5=8O, B> =8:0:>9 7028A8<>AB8 =5 ?@>A<0B@8205BAO ( 8A. 1)
3
4=0:> 5A;8 @0AA<0B@820BL B5?;>BC 8A?0@5=8O, ?@8E>4OICNAO =0 >4=C <>;5:C;C
(8;8 GB> @02=>A8;L=> =0 1 <>;L <>;O@=0O B5?;>B0 8A?0@5=8O L = LM , 40==K5
@0AG5B>2 ?@54AB02;5=K 2 B01;8F5), B> =01;N405BAO ?@0:B8G5A:8 ;8=59=0O 7028A8<>ABL
( 8A. 2)..
@0D8: MB>9 ?@8<5@=>9 7028A8<>AB8 <>65B 1KBL >?8A0= ?@>AB>9 D>@<C;>9
L = Ct:8?. , (1)
:6
345 ! H" 0,14 .
3@04
0==0O 7028A8<>ABL <>65B 1KBL 0??@>:A8<8@>20=0 8 1>;55 B>G=K<8 D>@<C;0<8.
1.3 > 3@0D8:C <>6=> ?@81;878B5;L=> >F5=8BL <>;O@=CN B5?;>BC 8A?0@5=8O 7>;>B0
:6
L H" 380 .
<>;L
L 6
">340 >F5=:0 C45;L=>9 B5?;>BK 8A?0@5=8O L = H" 1,9 .
M :3
6
0<5B8<, GB> M:A?5@8<5=B0;L=>5 7=0G5=85 MB> 25;8G8=K @02=> L = 1,74 .
:3
2.1 @0D8: 7028A8<>AB8 C45;L=>9 B5?;>BK 8A?0@5=8O >B B5<?5@0BC@K ?>:070= =0 @8A. 3.
2.2 ! @>AB>< B5<?5@0BC@K =57=0G8B5;L=> 2>7@0AB05B @0AAB>O=85 <564C <>;5:C;0<8, 2
A;54AB285 G53> C<5=LH05BAO M=5@38O 8E 2708<>459AB28O 8 A>>B25BAB25==> C45;L=0O
B5?;>B0 8A?0@5=8O. >AB@>5==CN 7028A8<>ABL ?@81;865==> <>6=> >?8A0BL ;8=59=>9
DC=:F859
L = L0 - at , (2)
6 6
L0 H" 2,5 , a H" 0,003 .
:3 :3 " 3@04
4
040G0 3. -;5:B@>42830B5;L
3.1 5@2>5 C@02=5=85 B@51C5<>9 A8AB5<K ?@82545=> 2 CA;>288 7040G8
I = km . (1)
B>@>5 C@02=5=85 O2;O5BAO C@02=5=85< 70:>=0 A>E@0=5=8O M=5@388 (70?8A0==>5 4;O
<>I=>AB59)
2
IU0 = I R + R0 + mgv . (2)
( )
MB>9 A8AB5<5 425 =58725AB=KE 25;8G8=K, ?>MB><C >15 <>3CB 1KBL =0945=K.
3.2 K@060O 7=0G5=85 A8;K B>:0 87 C@02=5=8O (1) 8 ?>4AB02;OO 53> 2 C@02=5=85 (2),
?>;CG8<
kmU0 = k2m2 R + R0 + mgv , (3)
( )
B:C40 =0E>48< 7028A8<>ABL A:>@>AB8 ?>4J5<0 3@C70 >B ?0@0<5B@>2 A8AB5<K
kU0 - k2m R + R0
( )
v = . (4)
g
3.3 0:A8<0;L=0O <0AA0 3@C70 >?@545;O5BAO 87 D>@<C;K (4), 2 :>B>@>9 A;54C5B CAB@5<8BL
A:>@>ABL ?>4J5<0 : =C;N, A>?@>B82;5=85 @5>AB0B0 B0:65 4>;6=> 1KBL <8=8<0;L=K< (B>
5ABL =C;L):
U0
mmax = . (5)
kR0
3.4 40==>< A;CG05 5ABL >B=>H5=85 <>I=>AB8, 70B@0G8205<>9 =0 ?>4J5< 3@C70 :
<>I=>AB8 8AB>G=8:0:
mgv gv
= = . (6)
IU0 kU0
>4AB02;OO 2K@065=85 4;O A:>@>AB8 ?>4J5<0 87 D>@<C;K (4), ?>;CG8< 7028A8<>ABL
>B <0AAK 3@C70
km R - R0
( )
=1- . (7)
U0
5
040G0 4 >475<=0O @5:0
r
1>7=0G8< ?@>5:F88 8A:><>9 A8;K 402;5=8O F 684:>AB8 =0 3>@87>=B0;L=CN >AL
Fx , =0 25@B8:0;L=CN FC (@8A. 2). ">340 87 A>>B25BAB2CNI53> ?@O<>C3>;L=>3>
B@5C3>;L=8:0 A8; 8<55<
Fy
F = Fx2 + Fy2 , tgą = . (1)
Fx
!8;0 3>@87>=B0;L=>3> 402;5=8O Fx =0 ?>;>28=:C ! :C?>;0 @02=0 ?> <>4C;N (8
?@>B82>?>;>6=0 ?> =0?@02;5=8N) A8;5 402;5=8O
=0 25@B8:0;L=>5 A5G5=85 A8AB5<K. -B>
A;54C5B 87 CA;>28O @02=>25A8O :064>9 87
?>;>28=>: :C?>;0. B<>AD5@=>5 402;5=85 <>6=>
=5 CG8BK20BL, B.:. >=> >:07K205B 459AB285 A >158E
AB>@>= :C?>;0.
>A:>;L:C 384@>AB0B8G5A:>5 402;5=85 A
3;C18=>9 2>7@0AB05B ;8=59=>, 0 M;5<5=B0@=K5
?;>I04:8 8<5NB >48=0:>2CN 4;8=C L , B>
A@54=55 402;5=85 4;O MB>3> A5G5=8O (A<. @8A. 2)
gh g R
@A@ = = .
8A. 2
2 2
#G8BK20O, GB> ?;>I04L 2A53>
?@O<>C3>;L=>3> A5G5=8O S = L R , >:>=G0B5;L=> ?>;CG8<
g R g LR2
Fx = " LR = . (2)
2 2
;O =0E>645=8O FC 2>A?>;L7C5<AO 384@>AB0B8G5A:8< ?0@04>:A><. !8;0
25@B8:0;L=>3> 402;5=8O =0 4=> (A5G5=85 ! ) A >4=>9 AB>@>=K <>65B 1KBL =0945=0 ?>
>?@545;5=8N :0: ?@>872545=85 ?>AB>O==>3> 402;5=8O @ = g R =0 ?;>I04L 4=0 S !
F ! = @ " S ! = g R " LR = g LR2 , (3)
0 A 4@C3>9 :0: AC<<0 A8;K BO65AB8 2>4K =0 @8A. 2 8 FC . 59AB28B5;L=>, ?> III 70:>=C
LNB>=0, 15B>==K9 :C?>; ?@868<05B 2>4C :> 4=C B>65 A A8;>9 FC . "0:8< >1@07><,
Ą R2
g LR2 = Fy + g L (4)
4
7 (4) =0E>48<
4 - Ą
Fy = g LR2 (5)
4
:>=G0B5;L=>, A CG5B>< (1), (2) 8 (5) ?>;CG05<
g LR2
F = 4 + ( 4 - Ą )2 = 2,1"105 = 0,21 (6)
4
4 - Ą
tg ą = = 0,43 ! ą = 23 . (7)
2
0: 2848< 87 (6), A8;0 402;5=8O 2>4K 4>AB0B>G=> 25;8:0 :@5?;5=8O ?>;>28=>:
@CA;0 4>;6=K 1KBL =0456=K<8! @><5 B>3>, 87 (7) A;54C5B, GB> 7=0G5=85 C3;0 ą =0 @8A. 1
?@5C25;8G5=> ?@8<5@=> 2 2 @070, B.5. 2>40 7=0G8B5;L=> 1>;LH5 B>;:05B ?>;>28=:C, G5<
?@8?>4=8<05B.
6
5A?C1;8:0=A:0O D878G5A:0O
>;8<?8040 (III MB0?)
2006 3>4
"5>@5B8G5A:89 BC@
10 :;0AA.
@CB8;L=K5 25AK
C;>=0
5H5=85
8
:><<5=B0@88
7
'0ABL 1. #AB0=>2:0.
1.1 >=5G=> 65, 4;O =>@<0;L=>9 @01>BK CAB0=>2:8 A;54C5B 70@O60BL 1;86=89 :
2=5H=5<C 70@O4C H0@8: :>@><KA;0 3, ?>A:>;L:C >1@0B=>5 ?@>B82>@5G8B CA;>28N 8
74@02><C A<KA;C.
1.2 5 =0@CH8BAO, B.:. <0AA0 M;5:B@>=>2 =8GB>6=> <0;0 ?> A@02=5=8N A
<0AA0<8 @50;L=KE >1J5:B>2. 065 ?@8 >3@><=>< 70@O45 q = 1; AC<<0@=0O <0AA0
M;5:B@>=>2 A>AB028B B = 5,7 "10-12 :3 , GB> 7=0G8B5;L=> <5=LH5 <0AAK ;N1>9 ?K;8=:8,
;5B0NI59 2 2>74CE5.
'0ABL 2. >4C;L :@CG5=8O.
2.1 >:065<, GB> 45D>@<0F8O :@CG5=8O A2>48BAO : 45D>@<0F88 A42830. ;O MB>3>
<KA;5==> @07>1L5< B>=:>AB5==K9 F8;8=4@ (B.5. ?@>2>;>:C) =0 B>=:85 A;>8
?5@?5=48:C;O@=K5 >A8 F8;8=4@0, 0 :064K9 87 MB8E A;>52 =0 <0;K5 A;53:0 87>3=CBK5
?0@0;;5;5?8?54K ( 8A. 30). @8 70:@CG820=88 ?@>2>;>:8 :064K9 87 MB8E 2K45;5==KE
M;5<5=B>2 @07185=8O =0:;>=8BAO =0 <0;K9 C3>; , B> 5ABL 8A?KB05B 45D>@<0F8N A42830.
;O 70:@CG820=8O ?@>2>;>:8 : =59 =5>1E>48<> ?@8;>68BL =5:>B>@K9 <><5=B A8;K M ,
:>B>@K9 2 A>AB>O=88 @02=>25A8O 1C45B C@02=>25H820BLAO @02=K< ?> <>4C;N <><5=B><
A8;K C?@C3>AB8. "0=35=F80;L=0O A8;0 C?@C3>AB8, 2>7=8:0NI0O ?@8 45D>@<0F88
2K45;5==>3> M;5<5=B0 ?> 70:>=C C:0 @02=0
"Fi = G"Si , (1)
345 "Si - ?;>I04L 25@E=59 3@0=8 2K45;5==>3> M;5<5=B0. ><5=B MB>9 A8;K,
>B=>A8B5;L=> >A8 F8;8=4@0 @025=
"M = "Fi R = G"Si R . (2)
i
"0: :0: 4;O 2A5E M;5<5=B>2 2K1@0==>3> :>;LF0 C3;K A42830 >48=0:>2K, B> 2KG8A;5=85
AC<<0@=>3> <><5=B0 A8; C?@C3>AB8 A2>48BAO : AC<<8@>20=8N ?;>I0459 25@E=8E 3@0=59:
8
M = = R = GR = GR " 2ĄR"R = 2ĄR2"RG " .
""M i "G"Si ""Si
i i i
45AL = 2ĄR"R - ?;>I04L 25@E=59 3@0=8 2K45;5==>3> :>;5G:0.
""Si
i
"5?5@L =5>1E>48<> 2K@078BL C3>; A42830 G5@57 C3>; 70:@CG820=8O ?@>2>;>:8 ą . CABL
2 @57C;LB0B5 :@CG5=8O (A<. 8A. 32) =5:>B>@0O B>G:0 =0 B>@F5 ?@>2>;>:8 A<5AB8;0AL 2
?>;>65=85 A , B>340, AG8B0O 2A5 C3;K <0;K<8, 4;8=C 4C38 AA <>6=> 2K@078BL 2 42CE
D>@<0E ą R = L , >B:C40 A;54C5B, GB>
R
= ą ,
L
?>MB><C AC<<0@=K9 <><5=B A8; C?@C3>AB8 (:>B>@K9, >G5284=>, ?>AB>O=5= 4;O ;N1>3>
?>?5@5G=>3> A5G5=8O ?@>2>;>:8) @025=.
2Ą R3 " R
M =F " R = G"ą .
L
!@02=820O A D>@<C;>9 70:>=0 C:0, =0E>48<, GB> <>4C;L :@CG5=8O @025=
2Ą R3"R
k = G . (3)
L
2.2 0AG5B ?> D>@<C;5 (3) 4;O 40==KE 87 CA;>28O 405B
" <
k = 0,13
.
@04
'0ABL 3. 0;K5 C3;K
3.1 CABL :>@><KA;> :@CB8;L=KE 25A>2 ?>25@=C;>AL
=0 <0;K9 C3>; ą (@8A.4). ">340 @0AAB>O=85 <564C
H0@8:0<8 r <>6=> 2K@078BL :0: 4;8=C
A>>B25BAB2CNI59 4C38 r = lą . A;54AB285 <0;>AB8
C3;0 <>6=> AG8B0BL, GB> ?;5G> A8;K C;>=0
?@0:B8G5A:8 @02=> @0AAB>O=8N l . @8 @02=>25A88
<><5=B A8;K >BB0;:820=8O 70@O4>2 (A8;K C;>=0)
1C45B C@02=>25H5= 2>72@0I0NI8< <><5=B><,
2>7=8:H8< 2A;54AB285 70:@CG820=8O A5@51@O=>9
?@>2>;>G:8, ?>MB><C
1 q2 1 q2
" l = " l = k "ą . (3)
2
4Ą 0 r 4Ą 0 (lą)2
7 (3) =0945< 7028A8<>ABL C3;0 70:@CG820=8O =8B8 ?@8 =51>;LH>< >B:;>=5=88
q2
3
ą(q) =
. (4)
4Ą 0 l k
9
3.2 ;O ?>AB@>5=8O 3@0D8:0 ?>4AB028< G8A;5==K5 7=0G5=8O ?>AB>O==KE 25;8G8=
q2
3
ą(q) = =
4Ą 0 l k
,
2
2
(10-9) q2
3
3
= H" 8,84 "10-3 q
4Ą 8,85 "10-12 " 0,10 " 0,13
745AL 70@O4 87<5@O5BAO 2 =0=>:C;>=0E. @0D8: MB>9 7028A8<>AB8 ?>:070= =0 @8A.5.
3.3 K@060O 87 D>@<C;K (4) 70@O4, ?>;CG8<
3
q = 4Ą 0 l ką (5)
>4AB02;OO 2 (5) 7=0G5=85 (?@5420@8B5;L=> =5 701K2 ?5@525AB8 53> 2
@0480=K!)ą = 10 = 0,175@04 =0945< 25;8G8=C 70@O40 q1
q1 = 8,8"10-8; = 88=;
.
3.4 ;O =0E>645=8O <0:A8<0;L=>3> 70@O40 ?@8 <0;KE >B:;>=5=8OE ?>4AB028< 2 D>@<C;C
(5) 7=0G5=85 ?@545;L=>3> C3;0 ą = 20 8 2K@078< >BBC40 25;8G8=C 70@O40 q2
q2 = 4Ą 0l ką3 = 2,5 "10-7 ; = 0,25<:;
. (6)
"0:8< >1@07><, ?@8 70@O40E =0 H0@8:0E q d" q2 = 0,25<:; :@CB8;L=K5 25AK
@01>B0NB 2 @568<5 <0;KE C3;>2.
'0ABL 4. >;LH85 C3;K.
4.1 @8 1>;LH8E C3;0E >B:;>=5=8O A;54C5B CG5ABL,
GB> @0AAB>O=85 <564C 2708<>459AB2CNI8<8
H0@8:0<8 5ABL 4;8=0 E>@4K, 0 =5 4C38 (@8A.6), B.5.
r = 2l sin (ą 2) .
!>>B25BAB25==> ?;5G> A8;K C;>=0 2 MB><
A;CG05 A;54C5B @0AAG8B0BL :0:
d = l cos(ą 2) .
"0:8< >1@07><, ?@028;> <><5=B>2 4;O MB>3>
A;CG0O 70?8H5BAO 2 2845
10
1 q2
"l cos(ą 2) = k "ą . (7)
4Ą 0 (2l sin (ą 2))2
K@065=85 (7) C4>1=> ?@5>1@07>20BL : A;54CNI5<C 284C
1 q2 cos(ą 2)
ą = " "
. (8)
16Ą 0 k l sin2 (ą 2)
-B> C@02=5=85 <>65B 1KBL ?@54AB02;5=> 8 2 4@C38E M:2820;5=B=KE D>@<0E.
4.2 K@060O 87 (8) 25;8G8=C 70@O40 q , ?>;CG8<
Ą 0 k lą
q2 = 4sin (ą / 2) =
. (9)
cos(ą / 2)
~
A;8 B>G=>5 7=0G5=85 70@O40 q2 >?@545;O5BAO D>@<C;>9 (9), 0 ?@81;865==>5 q2 -
D>@<C;>9 (6), B> >B=>A8B5;L=0O ?>3@5H=>ABL ?>A;54=59 @02=0
~ ~
q2 - q2 q2 ą cos(ą / 2)
= = -1 = -1.
q2 q2 2sin (ą / 2)
Ą
@8 C3;5 ą = 20 = @04 @0AG5B ?> MB>9 D>@<C;5 ?@82>48B : 7=0G5=8N H" 2,6 "10-3
9
0==K9 D0:B ?>72>;O5B A45;0BL 2K2>4, GB> 3@0=8FK ?@81;865=8O <0;KE C3;>2 A<5;>
<>6=> C25;8G8BL.
4.3 #@02=5=85 (8) >B=>A8B5;L=> C3;0 70:@CG820=8O =8B8 ą O2;O5BAO B@0=AF5=45=B=K<,
B.5. =52>7<>6=> ?@54AB028BL 7028A8<>ABL ą(q) 2 O2=>< 0=0;8B8G5A:>< 2845. 4=0:> =5
A>AB02;O5B >A>1>3> B@C40 @0AAG8B0BL =0 :0;L:C;OB>@5 =5A:>;L:> 7=0G5=89 >1@0B=>9
DC=:F88 q(ą), GB> ?>72>;O5B ?>AB@>8BL ?@8<5@=> B@51C5<K9 3@0D8:, :>B>@K9 ?>:070= =0
@8A. 7, 345 7=0G5=8O q(ą) ?>4AG8B0=K ?> D>@<C;5 (9) A H03>< 10 .
> 40==><C 3@0D8:C =0E>48<, GB> ?@8 70@O45 q = 1,8<:; C3>; >B:;>=5=8O :>@><KA;0
?@81;878B5;L=> @025= ą H" 73 .
11
0 @8A.8 ?@54AB02;5=K 3@0D8:8 42CE DC=:F89 q(ą), @0AAG8B0==K5 ?>
?@81;865==>9 D>@<C;5 (6) 8 ?> B>G=>9 D>@<C;5 (9), :0: 284=> @07;8G8O <564C =8<8
=0G8=0NB ?@>O2;OBLAO B>;L:> ?@8 C3;0E >B:;>=5=8O ą e" 60 .
12
'0ABL 5. 7<5@5=85 70@O4>2.
5.1 ><5=B A8; M;5:B@8G5A:>3> 2708<>459AB28O ?@>?>@F8>=0;5= ?@>872545=8N 70@O4>2
H0@8:>2 8 A;>6=K< >1@07>< 7028A8B >B C3;>2>3> @0AAB>O=8O <564C =8<8
M = q0qX f (ą0 ). (10)
M;.
"0: :0: 2 40==>< M:A?5@8<5=B5 ?CB5< 4>2>@>B0 :>@><KA;> 2>72@0I05BAO 2 8AE>4=>5
?>;>65=85, B> O2=K9 284 7028A8<>AB8 MB>3> <><5=B0 >B C3;0 f (ą0 ) =5 ACI5AB25=5=.
><5=B A8; C?@C3>AB8 =8B8 ?@>?>@F8>=0;5= C3;C 70:@CG820=8O =8B8,
M = -k(ą0 ą "ą), (11)
2 MB>9 D>@<C;5 A;54C5B 1@0BL 7=0: <8=CA, ?@8 4>2>@>B5 2 =0?@02;5=88 ?5@2>=0G0;L=>3>
:>@><KA;0; 7=0: ?;NA ?@8 4>2>@>B5 2 ?@>B82>?>;>6=>< =0?@02;5=88. ! ?><>ILN MB8E
2K@065=89 ?>;CG05<, GB> >B=>H5=85 70@O4>2 ;53:> >?@545;O5BAO A ?><>ILN ?@>?>@F88
qX ą0 ą "ą
=
. (12)
q0 ą0
"0: ?@8 4>2>@>B5 2 AB>@>=C =0?@02;5=8O ?>2>@>B0 :>@><KA;0
qX ą0 - "ą
= H" 0,5 ,
q0 ą0
A;54>20B5;L=>, 2 MB>< A;CG05 70@O4 C<5=LH8;AO 2 420 @070.
@8 4>2>@>B5 2 ?@>B82>?>;>6=>< =0?@02;5=88
qX ą0 + "ą
= H" 1,5,
q0 ą0
A;54>20B5;L=>, 2 MB>< A;CG05 70@O4 C25;8G8;AO 2 ?>;B>@0 @070.
13
'0ABL 6. O7:>ABL.
6.1 >2>@>B 25@E=53> @KG030-AB@5;:8 A<5I05B ?>;>65=85 @02=>25A8O :>@><KA;0, B5?5@L
?>;>65=8N @02=>25A8O (:>340 ?@>2>;>:0 =5 70:@CG5=0) A>>B25BAB2C5B C3>; ?>2>@>B0
:>@><KA;0, @02=K9 C3;C ?>2>@>B0 AB@5;:8, B> 5ABL 90 . ;O B>G=>3> >?8A0=8O 2@0I5=8O
:>@><KA;0 A;54C5B ?>;L7>20BLAO C@02=5=85< 48=0<8:8 2@0I0B5;L=>3> 42865=8O. 4=0:>,
B0: :0: <0A;> 2O7:>5, B> <>6=> ?>;L7>20BLAO :2078AB0B8G5A:8< ?@81;865=85<, B> 5ABL
AG8B0BL, GB> 2 ;N1>9 <><5=B 2@5<5=8 AC<<0 <><5=B>2 A8;, 459AB2CNI8E =0 :>@><KA;> (2
40==>< A;CG05 MB> <><5=BK A8; C?@C3>AB8 8 2O7:>3> B@5=8O) @02=0 =C;N, GB> ?@82>48B :
C@02=5=8N
"ą
k("ą0 -ą) = = . (13)
"t
7 C@02=5=8O (13) A;54C5B, GB> C3;>20O A:>@>ABL 2@0I5=8O ?@>?>@F8>=0;L=0 C3;C
>B:;>=5=8O :>@><KA;0 >B ?>;>65=8O @02=>25A8O
"ą k
= = ("ą0 -ą). (14)
"t
"0:8< >1@07><, 4;O ?@>25@:8 ?@8<5=8<>AB8 A45;0==KE ?@54?>;>65=89 (<><5=B
A8; 2O7:>3> B@5=8O ?@>?>@F8>=0;5= C3;>2>9 A:>@>AB8, :2078AB0F8>=0@=K9 E0@0:B5@
42865=8O :>@><KA;0) A;54C5B C1548BLAO, GB> <3=>25==0O C3;>20O A:>@>ABL 2@0I5=8O
:>@><KA;0 ?@>?>@F8>=0;L=0O C3;C 53> >B:;>=5=8O >B ?>;>65=8O @02=>25A8O.
> ?@82545==K< 2 "01;8F5 1 40==K< <>6=> ?@81;865==> 2KG8A;8BL 7=0G5=8O
A@54=59 C3;>2>9 A:>@>AB8 2@0I5=8O :>@><KA;0 =0 :064>< 8=B5@20;5 87<5=5=8O C3;0
?>2>@>B0
ąi+1 -ąi
i+0,5 H" , (15)
ti+1 - ti
ąi + ąi+1
:>B>@CN A;54C5B ?@8?8A0BL A5@548=5 C3;>2>3> 8=B5@20;0 ąi+0,5 = .
2
0AG5BK MB>9 M:A?5@8<5=B0;L=>9 7028A8<>AB8 C3;>2>9 A:>@>AB8 >B C3;0 ?>2>@>B0
?@82545=K 2 "01;8F5 2. 0 8A. 9 ?>AB@>5= 3@0D8: 7028A8<>AB8 C3;>2>9 A:>@>AB8 >B
>B:;>=5=8O :>@><KA;0 >B ?>;>65=8O @02=>25A8O ("ą0 -ą).
"01;8F0 2.
ą 5 15 25 35 45 55 65 75
8,33 7,69 6,67 5,26 4,55 3,57 2,38 1,45
, 3@04 " A-1
>;CG5==0O ?@>?>@F8>=0;L=0O 7028A8<>ABL ?>4B25@6405B ?@8<5=8<>ABL
A45;0==KE ?@54?>;>65=89. 7 C@02=5=8O (14) A;54C5B, GB> :>MDD8F85=B
k
?@>?>@F8>=0;L=>AB8 <564C C3;>2>9 A:>@>ABLN 8 C3;>< >B:;>=5=8O @025= , G8A;5==>5
7=0G5=85 MB>3> 65 :>MDD8F85=B0, =0945==>5 87 3@0D8:0, ?@81;878B5;L=> @02=>
k
k
H" = 1,3 " < " A
H" 0,10 c-1 . !;54>20B5;L=>, B@51C5<>5 7=0G5=85 ?0@0<5B@0 .
0,10c-1
6.2 7 C@02=5=8O (13) A;54C5B, GB> E0@0:B5@=>5 2@5<O CAB0=>2;5=8O @02=>25A8O,
>?@545;5==>5 A ?><>ILN >4A:07:8 2, @02=>
T0 = H" 10 c
. (18)
k
14
@>8;;NAB@8@C5< 1>;55 ?>4@>1=> <5B>48:C M:A?5@8<5=B0;L=>9 ?@>25@:8
?@8<5=8<>AB8 C@02=5=8O (13) .
0 8A. 10 ?>:070= 3@0D8: ?@>E>645=8O 48AB0=F88 :>@><KA;><, B> 5ABL
2@5<5=0 ?>2>@>B0 >B =0G0;L=>3> ?>;>65=8O 4> C:070==>3> 7=0G5=8O C3;0 ?>2>@>B0,
?>AB@>5==K9 ?> 40==K< "01;8FK 1.
"0:, =0?@8<5@, CG0AB>: >B 40 4> 50 :>@><KA;> ?@>H;> 70 2@5<O
"t = t50 - t40 = 8,1- 5,9 H" 2,2c . !;54>20B5;L=>, =0 MB>< CG0AB:5 A@54=OO C3;>20O A:>@>ABL
"ą 10 3@04
?@8<5@=> @02=0 A@. = = H" 4,55 . ! =08<5=LH59 ?>3@5H=>ABLN <>6=>
"t 2,2 A
AG8B0BL, GB> A MB>9 A:>@>ABLN :>@><KA;> ?@>H;> A@54=NN B>G:C 8=B5@20;0 ąA@ = 45.
=0;>38G=K5 @0AG5BK ?@>45;0=K 4;O 2A5E 8=B5@20;>2 C3;>2 ?>2>@>B0, 0 8E @57C;LB0BK
?@82545=K 2 "01;8F5 2: 2 ?5@2>9 AB@>:5 A5@548=K 8=B5@20;>2, 2 =86=59
A>>B25BAB2CNI85 C3;>2K5 A:>@>AB8. @8 ?>AB@>5=88 3@0D8:0 ?> >A8 01AF8AA >B;>65=
C3>;, 4> ?>;>65=8O @02=>25A8O, B> 5ABL ("ą0 -ą)= (90 -ą), :>B>@K9 >?@545;O5B
<><5=B A8; C?@C3>AB8. @8 B0:>< ?>AB@>5=88 C3;K, ?@82545==K5 2 B01;8F5 8 =0
3@0D8:5, 84CB 2 >1@0B=>< ?>@O4:5.
15
'0ABL 7. 2865=85 2 <0A;5.
7.1 @8 ?><5I5=88 70@O65==KE H0@8:>2 2 <0A;> A8;0 2708<>459AB28O <564C =8<8
C<5=LH05BAO 2 @07, 2> AB>;L:> 65 @07 C<5=LH05BAO 8 <><5=B A8;K, 459AB2CNI59 =0
:>@><KA;>. >MB><C CA;>285 @02=>25A8O, >?8AK205<>5 C@02=5=85< (3), ?@8>1@5B5B 284
2
1 q0
" l = k "ą . (19)
4Ą 0 (lą)2
>MB><C C3>; @02=>25A8O 2 <0A;5 AB0=5B @02=K<
2
q0 ą0
3
ą = = H" 12
. (20)
3
4Ą 0l k
3
"0:8< >1@07><, C3>; ?>2>@>B0 C<5=LH8BAO 2 H" 1,7 @07.
7.2 0?8H5< C@02=5=85 42865=8O :>@><KA;0 2 :2078AB0F8>=0@=>< ?@81;865=88
1 q2
- ką - = 0 . (21)
2
4Ą0l ą
;O C?@>I5=8O (B.5. M:>=><88 <5AB0) @0745;8< C@02=5=85 =0 <>4C;L :@CG5=8O ?@>2>;>:8
8 8A?>;L7C5< 22545==K5 @0=55 >1>7=0G5=8O:
= T0 - E0@0:B5@=>5 2@5<O CAB0=>2;5=8O @02=>25A8O 2 <0A;5;
k
2
q0
3
= ą , 345 ą - C3>; @02=>25A8O :>@><KA;0 ?@8 70@O45 H0@8:>2 @02=KE q0 ,
4Ą 0l k
>?@545;O5<K9 ?> D>@<C;5 (12);
"ą
= - C3;>20O A:>@>ABL 2@0I5=8O :>@><KA;0.
"t
! CG5B>< MB8E >1>7=0G5=89 C@02=5=85 (21) 70?8AK205BAO 2 2845
3
"ą ą q2
T0 = -ą . (22)
2 2
"t q0 ą
;O >?8A0=8O =0G0;L=>3> MB0?0 42865=8O :>@><KA;0, :>340 <>6=> ?@5=51@5GL <><5=B><
A8; C?@C3>AB8 8 ?@8 =587<5==KE 70@O40E H0@8:>2 q = q0 C@02=5=85 (22) C?@>I05BAO
3
ą
T0 = . (23)
2
ą
;O 53> @5H5=8O 2>A?>;L7C5<AO >4A:07:>9 3, B> 5ABL ?>?KB05<AO =09B8 7028A8<>ABL C3;0
?>2>@>B0 >B 2@5<5=8 2 2845
ą(t) = Ctł , (24)
B>340 (t) = Cłtł -1. >4AB02;OO MB8 2K@065=8O 2 C@02=5=85 (23) ?>;CG8<
3
ą
-2ł
T0Cł tł -1 = t . (25)
2
C
'B>1K MB> 2K@065=85 1K;> A?@0254;82> 2 ;N1>9 <><5=B 2@5<5=8, =5>1E>48<>, GB>1K
?>:070B5;8 AB5?5=59 ?5@5<5==>9 t A>2?040;8. -B> CA;>285 ?>72>;O5B >?@545;8BL
?>:070B5;L AB5?5=8 2 DC=:F88 (24):
1
ł -1 = -2ł ! ł = . (26)
3
16
3
ą
3
"5?5@L 87 @025=AB20 (25) ;53:> >?@545;8BL ?>AB>O==CN C = 3 .
T0
:>=G0B5;L=>, 70:>= 42865=8O :>@><KA;0 (24) 2 @0<:0E >3>2>@5==KE ?@81;865=89 8<55B
>G0@>20B5;L=K9 284
t
3
ą(t) = ą " 3
. (27)
T0
7.3 :@C68< >48= 87 H0@8:>2 70<:=CB>9 ?>25@E=>ABLN &! ( 8A. 11). !8;0
M;5:B@8G5A:>3> B>:0 ?5@5A5:0NI53> G5@57 MBC ?>25@E=>ABL,
@02=0 A:>@>AB8 C<5=LH5=8O 70@O40 2=CB@8 ?>25@E=>AB8. -B0
A8;0 B>:0 @02=0 ?>B>:C 25:B>@0 ?;>B=>AB8 B>:0 G5@57
r
?>25@E=>ABL I = Ś . "0: :0: ?> 70:>=C <0 ?;>B=>ABL B>:0
j
?@>?>@F8>=0;L=0 =0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O
r
r
1
j = E , B> ?>B>:8 MB8E 25:B>@>2 A2O70=K 0=0;>38G=K<
1
r r
A>>B=>H5=85< Ś = ŚE . >B>: 25:B>@0 =0?@O65==>AB8
j
q
r
M;5:B@8G5A:>3> ?>;O >?@545;O5BAO B5>@5<>9 0CAA0 ŚE = . "0:8< >1@07><, 25;8G8=0
0
70@O40 2=CB@8 @0AA<0B@8205<>9 >1;0AB8 ?>4G8=O5BAO C@02=5=8N
"q 1 q
r r
= -I = -Ś = - ŚE = - . (28)
j
"t 0
7 :>B>@>3> =5?>A@54AB25==> A;54C5B, GB> E0@0:B5@=>5 2@5<O AB5:0=8O 70@O4>2 A H0@8:0
@02=>
T1 = 0 H" 2,2 c . (29)
0<5B8<, GB> 40==0O >F5=:0 <>65B 1KBL ?>;CG5=0 8 8=K<8 A?>A>10<8, =0?@8<5@,
@0AA<0B@820O O2=K5 2K@065=8O 4;O =0?@O65==>AB8 ?>;O 2>:@C3 C548=5==>3> H0@8:0.
7.4 40==>< A;CG05 2 ?@>F5AA5 42865=8O :>@><KA;0 ?@>8AE>48B CB5G:0 70@O4>2 A >1>8E
H0@8:>2, ?>MB><C <><5=B A8;, 459AB2CNI8E =0 :>@><KA;>, 87<5=O5BAO, =5 B>;L:> 87-70
87<5=5=8O 53> C3;0 ?>2>@>B0, => 8 87-70 87<5=5=8O 70@O4>2 2708<>459AB2CNI8E H0@8:>2.
>MB><C C3;>20O A:>@>ABL 2@0I5=8O :>@><KA;0, >?@545;O5<0O C@02=5=85< (22) O2=>
7028A8B :0: >B 2@5<5=8, B0: 8 >B C3;0 ?>2>@>B0. 5H0BL B>G=> B0:85 C@02=5=8O 4>AB0B>G=>
A;>6=>. 4=0:>, 2 40==>< A;CG05 =0< =5 B@51C5BAO =0E>48BL 70:>= 42865=8O :>@><KA;0
=0< B@51C5BAO =09B8 B>;L:> <0:A8<0;L=K9 C3>; >B:;>=5=8O! G5284=>, GB> 2 <><5=B
<0:A8<0;L=>3> >B:;>=5=8O :>@><KA;0 tm 53> C3;>20O A:>@>ABL 2@0I5=8O @02=0 =C;N,
A;54>20B5;L=>, 2 MB>B <><5=B A?@0254;82> A>>B=>H5=85
2
3
2
3
q(tm )ź# = ś#
ą q(tm ) # ś#
#ąmax(tm )ź# . (30)
ś#
ś#
-ąmax = 0 , 8;8
2 2
ś# ź#
q0 ąmax q0 # ą
#
# #
'B>1K B>G=> @5H8BL MB> C@02=5=85 (B> 5ABL =09B8 <><5=B >AB0=>2:8 :>@><KA;0 tm )
=5>1E>48<> 7=0BL O2=K9 284 7028A8<>AB59 C3;0 ?>2>@>B0 8 70@O4>2 H0@8:>2 >B 2@5<5=8
(B> 5ABL >?OBL 65 @5H8BL C@02=5=85 (22)). ;O >F5=:8 2@5<5=8 42865=8O 4> >AB0=>2:8
<>6=> 2>A?>;L7>20BLAO B@048F8>==>9 <5B>48:>9, >?8A0==>9 2 >4A:07:5 2 .
;O MB>3> 2 C@02=5=85 (30) A;54C5B ?>4AB028BL 2@5<5==K5 7028A8<>AB8, A>>B25BAB2CNI85
=0G0;L=><C MB0?C 42865=8O! "0:0O 7028A8<>ABL 4;O C3;0 ?>2>@>B0 ?>;CG5=0 8 70405BAO
17
DC=:F859 (27). 7 C@02=5=8O (28) A;54C5B, GB> =0G0;L=K9 MB0? @07@O4:8 >?8AK205BAO
;8=59=K< C@02=5=85<
# ś# # ś#
t t
ś#
q(t) = q0 ś#1- ź# ś# ź#
= q0 . (31)
0 ź# ś#1- T1 ź#
# # # #
B>< 65 ;8=59=>< ?@81;865=88 :204@0B 70@O40 >?8AK205BAO 2K@065=85<
2
# ś# # ś#
t t
2 2
ś#
q2 = q0 ś#1- ź# ś# ź#
H" q0 2 . (32)
0 ź# ś#1- T1 ź#
# # # #
>4AB0=>2:0 MB8E DC=:F89 2 C@02=5=85 (30) ?@82>48B : ?@>AB><C ;8=59=><C C@02=5=8N
tm tm
1- 2 = 3 , (33)
T1 T0
87 :>B>@>3> A;54C5B, GB> >AB0=>2:0 ?@>87>945B 2 <><5=B 2@5<5=8
T0T1
tm = . (35)
2T0 + 3T1
0:A8<0;L=K9 C3>; ?>2>@>B0 1C45B 4>AB83=CB 2 MB>B <><5=B 2@5<5=8, 53> =0E>48< A
?><>ILN DC=:F88 (7)
1
-
ś#
tm 3T1 # 2T0 3
3 H" 0,63ą H" 7,5
ąmax = ą " 3 = ą " 3 = ą ś# ź#
. (36)
T0 2T0 + 3T1 ś#1+ 3T1 ź#
# #
18
>?>;=5=85, =5 2>H54H55 2 >A=>2=>5 A>1@0=85 A>G8=5=89
;O >1>A=>20=8O ?@54;>65==>9 <5B>48:8 >F5=:8 <0:A8<0;L=>3> C3;0 ?@82545<
B>G=>5 @5H5=85 40==>9 7040G8.
0G=5< A> AB@>3> @5H5=8O C@02=5=8O (22)
3
"ą ą q2
T0 = -ą . (41)
2 2
"t q0 ą
ą t
2545< 157@07<5@=K5 ?0@0<5B@K x = , = , 0 B0:65 ?>4AB028< O2=>5 2K@065=85
ą T0
t
4;O 70@O40 H0@8:>2 q = q0 exp(- ) . MB8E ?5@5<5==KE C@02=5=85 8<55B 284
T1
dx exp(-b )
= - x , (42)
d x2
2T0
345 >1>7=0G5=> b = .
T1
0<5=0 x( ) = z( )" exp(- ) ?@82>48B : C@02=5=8N A @0745;ONI8<8AO ?5@5<5==K<8
dz exp(- b )
exp(- )- z exp(- ) = - z exp(- ) !
d z2 exp(- 2 )
,
dz exp(- (b - 3) )
=
d z2
:>B>@>5 ;53:> 8=B53@8@C5BAO (A CG5B>< =0G0;L=KE CA;>289 ?@8 = 0 z = 0 ):
z
z2dz =
+" +"exp(- (b - 3) ) d
0 0
z3 1- exp(- (b - 3) )
= .
3 (b - 3)
1- exp(- (b - 3) )
3
z( ) = 3
(b - 3)
"5?5@L =0E>48<
exp(- 3 )- exp(- b )
3
x( ) = z( )" exp(- ) = 3 . (43)
(b - 3)
>72@0I0OAL : ?@56=8< ?5@5<5==K<, ?>;CG8<
ś#
3T1 # #
ś#exp# 3 t ś# expś#- 2 t ś#ź#
ą(t) = ą 3 ś#- ź# - ś# ź#ź#
. (430)
2T0 - 3T1 ś# ś# T0 ź# # T1 ź# #
# # #
#
@8 <0;KE 70:>= 87<5=5=8O C3;0 8<55B 284
exp(- 3 )- exp(- b )
3
3
x( ) = 3 H" 3 , (44)
(b - 3)
8;8
t
ą = ą 3 , (440)
3
T0
GB> A>2?0405B A (27).
19
0 8A.1 ?>:070= 3@0D8: 7028A8<>AB8 #3;0 ?>2>@>B0 :>@><KA;0 >B 2@5<5=8,
@0AAG8B0==K9 ?> D>@<C;5 (430): 1KAB@>5 >B:;>=5=85 : <0:A8<0;L=><C 7=0G5=8N 8
?>A;54CNI55 <54;5==>5 2>72@0I5=85 : 8AE>4=><C ?>;>65=8N.
0 @8A. 2 ?>:070= CG0AB>: MB>9 65 DC=:F88 =0 <5=LH5< 2@5<5==>< 8=B5@20;5,
745AL 65 4;O A@02=5=8O ?@82545= 4;O A@02=5=8O (B>=:0O ;8=8O) 3@0D8: ?@81;865==>3>
@5H5=8O, @0AAG8B0==K9 ?> D>@<C;5 (27).
@>8;;NAB@8@C5< 3@0D8G5A:8 @5H5=85 C@02=5=8O (30) CA;>28O @025=AB20 <><5=B>2 A8;
C?@C3>AB8 8 M;5:B@8G5A:>3> 2708<>459AB28O 2 <><5=B >AB0=>2:8 :>@><KA;0. 0 8A. 3
?>AB@>5=K 3@0D8:8 DC=:F89, D83C@8@CNI8E 2 MB>< C@02=5=88: B>;ABK5 ;8=88
A>>B25BAB2CNB B>G=><C @5H5=8N, B>=:85 8A?>;L7>20==><C 2 >D8F80;L=>< @5H5=88
?@81;865=8N.
20
84=>, GB> 7=0G5=85 <><5=B0 2@5<5=8 tm >?@545;O5BAO A> 7=0G8B5;L=>9
?>3@5H=>ABLN, >4=0:> B@51C5<>5 7=0G5=85 <0:A8<0;L=>3> C3;0 >B:;>=5=8O
@0AAG8BK205BAO 4>AB0B>G=> :>@@5:B=>: 8<55B <5AB> A2>5>1@07=0O :><?5=A0F8O
?>3@5H=>AB59 0??@>:A8<0F88 DC=:F89. @><5 B>3>, 21;878 M:AB@5<C<0 ;N10O DC=:F8O
87<5=O5BAO <54;5==>, ?>MB><C ?>3@5H=>ABL 0@3C<5=B0 A;01> 2;8O5B =0 ?>3@5H=>ABL
>?@545;5=8O <0:A8<C<0 DC=:F88.
70:;NG5=85 C:065< B>G=>5 7=0G5=85 <0:A8<0;L=>3> >B:;>=5=8O :>@><KA;0,
=0945==K9 B@048F8>==K< <5B>4>< <0:A8<C< DC=:F88 (430):
2
-
3(2-3)
2
# ś#
ą = ą ś# ź#
, (45)
max
3
# #
T0
345 >1>7=0G5=> = - >B=>H5=85 2@5<5=8 CAB0=>2;5=8O @02=>25A8O :> 2@5<5=8
T1
@07@O4:8. 0=55 1K;> ?>;CG5=> ?@81;865==>5 7=0G5=85 MB>9 65 DC=:F88 (36), :>B>@0O
2K@0605BAO G5@57 >B=>H5=85 2@5<5= 2 2845
1
-
3
2
#1+ ś#
ąmax = ą ś# ź#
. (450)
3
# #
@0D8:8 MB8E DC=:F89 ?>:070=K =0 8A. 4 (B>;AB0O ;8=8O B>G=>5 @5H5=85, B>=:0O
?@81;865==>5): ?>3@5H=>ABL @5H5=8O =525;8:0 8 2?>;=5 4>?CAB8<0 4;O 40==>9
>;8<?804=>9 7040G8.
21
5A?C1;8:0=A:0O D878G5A:0O
>;8<?8040 (III MB0?)
2006 3>4
"5>@5B8G5A:89 BC@
11 :;0AA.
!25@E?@>2>4OI0O
@578=0
5H5=85
8 :><<5=B0@88
22
5H5=85.
'0ABL 1. ;>B=>ABL.
1. ;>I04L ?>25@E=>AB8 >B@57:0 B@C1:8 @02=0 S = 2Ąr0l0 . > >?@545;5=8N
m m 1,5 "10-3 :3 :3
= = = H" 0,119
(1)
S 2Ąr0l0 2Ą "1,0 "10-2 < " 20 "10-2 < <2 .
'0ABL 2. #?@C3>ABL, ?@>G=>ABL 8 45D>@<0F88.
2.1 40==>< M:A?5@8<5=B5 A8;K C?@C3>AB8 =0?@02;5=K 24>;L >A8 B@C1:8, ?>MB><C
;8=59=>5 =0?@O65=85 A8; C?@C3>AB8, 2>7=8:0NI59 2 B@C1:5 @02=>
mg
f = , (2)
2Ąr0
>B=>A8B5;L=>5 C4;8=5=85 <0B5@80;0 B@C1:8 5ABL
"l
= . (3)
l0
> MB8< D>@<C;0< =5>1E>48<> @0AAG8B0BL A>>B25BAB2CNI85 7=0G5=8O ?> 40==K<
"01;8FK 1 CA;>28O 7040G8. 57C;LB0BK @0AG5B>2 ?@82545=K 2 B01;8F5 1, =0 >A=>20=88
:>B>@>9 ?>AB@>5= 3@0D8: B@51C5<>9 7028A8<>AB8 ( 8A. 1). 0 MB>< 65 3@0D8:5 =0=5A5=0
B>G:0 @07@K20.
@820O =0 MB>< 3@0D8:5 ?@>2545=0 =0 3;07.
"01;8F0 1. 028A8<>ABL >B=>A8B5;L=>3> C4;8=5=8O >B ;8=59=>3> =0?@O65=8O.
78,1 15,6 23,4 31,2 39,0 46,8 54,6 62,4 70,2 78,1 85,9
f ,
<
0,048 0,100 0,148 0,202 0,247 0,285 0,322 0,355 0,368 0,385 0,392
23
2.2 > 3@0D8:C 284=>, GB> ?5@2K5 B>G:8 A E>@>H59 B>G=>ABLN ;560B =0 ?@O<>9, GB>
?>4B25@6405B A?@0254;82>ABL 70:>=0 C:0 2 MB>< 480?07>=5.
'5@57 MB8 B>G:8 8 =0G0;> :>>@48=0B ?@>2545< ?@O<CN 8 >?@545;8< :>MDD8F85=B
"
55 =0:;>=0 . -B>B :>MDD8F85=B <>6=> >?@545;8BL:
"f
- ?> >4=>9 B>G:5 (B>;L:> :0:CN 2K1@0BL?);
- ?> =5A:>;L:8< =0G0;L=K< B>G:0< (?> 3-5), A ?>A;54CNI8< CA@54=5=85<;
- ?> 3@0D8:C ;8=59=>9 7028A8<>AB8;
- ?> <5B>4C =08<5=LH8E :204@0B>2 ( ) ?> ?5@2K< 5 B>G:0< (MB>B <5B>4
=081>;55 ?@54?>GB8B5;L=K9).
" <
0AG5B ?> ?> ?@82>48B : @57C;LB0BC H" 6,405 "10-3 . !;54>20B5;L=>, 7=0G5=85
"f
<>4C;O @0ABO65=8O @02=>
-1
# ś#
"
ł = ś# ź# H" 1,56 "102 H" 1,6 "102 .
ś# ź# (4)
"f < <
# #
5 A<>B@O =0 B>, GB> 2 CA;>288 ?@82545=K 40==K5 A B@5<O 7=0G0I8<8 F8D@0<8,
8<5NI89AO @071@>A M:A?5@8<5=B0;L=KE 40==KE =5 ?>72>;O5B ?>;CG8BL 7=0G5=85 ?0@0<5B@0 ł A
B>G=>ABLN 1>;LH59 42CE 7=0G0I8E F8D@.
! ?><>ILN 3@0D8:0 >?@545;O5<, GB> ?5@2K9 ?OBL B>G5: ;560B 1;87:> : ;8=59=>9
7028A8<>AB8 (H5AB0O 8 ?>A;54CNI85 B>G:8 70:>=><5@=> >B:;>=ONBAO >B =55), ?>MB><C
<>6=> >F5=8BL 3@0=8FC ?@8<5=8<>AB8 70:>=0 C:0 :0:
< 0,25
(5)
'0ABL 3. 02;5=85.
@8 =0;8G88 871KB>G=>3> 402;5=8O 3070 2=CB@8 B@C1:0
?>A;54=OO @0ABO38205BAO, C25;8G8205BAO 55 @048CA
r8,
A;54>20B5;L=>, 4;8=0 55 >:@C6=>AB8. !8;0 402;5=8O 3070 F
( 8A. 2), 459AB2CNI0O =0 ?>25@E=>ABL, =0?@02;5=0 ?> =>@<0;8 :
r
?>25@E=>AB8, 0 2>7=8:0NI85 A8;K C?@C3>AB8 FC?@. =0?@02;5=K
?> :0A0B5;L=>9 : ?>25@E=>AB8 B@C1:8 ?5@?5=48:C;O@=> 55 >A8
(0 =5 24>;L >A8, :0: 2 ?@54K4CI59 G0AB8 7040G8).
B=>A8B5;L=>5 C4;8=5=85 ?5@8<5B@0 B@C1:8 l = 2Ąr 2 MB>< A;CG05 @02=> >B=>A8B5;L=><C
C25;8G5=8N 55 @048CA0:
"l "r
= = . (6)
l0 r0
3.1 KA;5==> 2K45;8< =0 45D>@<8@>20==>9 B@C1:5 =51>;LH>5 :>;LF>, 2KA>B>9 "z
( 8A. 3), 0 =0 =5< <0;CN ?;>I04:C "S , 2848<CN 87 F5=B@0 ?>4 <0;K< C3;>< ą . ;>I04L
MB>9 ?;>I04:8 @02=0 "S = rą"z . "0: :0: ?;>I04:0 =0E>48BAO 2 @02=>25A88, B> AC<<0 A8;,
459AB2CNI8E =0 =55, @02=0 =C;N. !8;0 402;5=8O 3070 =0?@02;5=0 ?5@?5=48:C;O@=>
2K45;5==>9 ?;>I04:5 8 @02=0 F = "P"S = rą"z"P . !8;K C?@C3>AB8, 459AB2CNI85 =0
1>:>2K5 AB>@>=K 2K45;5==>9 ?;>I04:8, =0?@02;5=K ?> :0A0B5;L=>9 : 1>:>2>9
?>25@E=>AB8 8 @02=K ?> <>4C;N FC?@. = f"z , 345 f - ;8=59=>5 =0?@O65=85 AB5=>:.
24
0?8H5< CA;>285 @02=>25A8O 2K45;5==>9 ?;>I04:8 2 ?@>5:F88 =0 =>@<0;L
ą
F = 2FC?@. sin .
2
>4AB028< 2K@065=8O 4;O A8; 402;5=8O 8 C?@C3>AB8 8 AG8B0O C3>; ą <0;K< (?@8 MB><
ą ą
sin H" ) 2 MB> C@02=5=85
2 2
ą
rą"z"P = 2 f"z ,
2
8 ?>A;5 A>:@0I5=8O ?>;CG05< B@51C5<>5 C@02=5=85 (:>B>@>5 O2;O5BAO >A=>2>9
40;L=59H53> @5H5=8O):
r"P = f (7)
>7<>6=K 8 4@C385 20@80=BK 2K2>40 MB>3>
C@02=5=8O. 0?@8<5@, <>6=> @0AA<>B@5BL CA;>28O
@02=>25A8O ?>;>28=K :>;LF0 ( 8A. 4). ">340 AC<<0 A8;
402;5=8O "P"z " 2r C@02=>25H8205BAO 42C<O A8;0<8
C?@C3>AB8 FC?@. = f"z , GB> B0:65 ?@82>48B :
C@02=5=8N (7).
0<5B8<, GB> 2 40==>< A;CG05 <>6=>
3>2>@8BL > 402;5=88 A8; C?@C3>AB8. ;O MB>3>
A;54C5B 2K45;8BL =0 1>:>2>9 ?>25@E=>AB8
F8;8=4@0 =51>;LHCN ?;>I04:C, =09B8 25:B>@=CN AC<<C B0=35=F80;L=KE A8; C?@C3>AB8,
(:>B>@0O =0?@02;5=0 ?5@?5=48:C;O@=> : ?>25@E=>AB8) 8 @0745;8BL 55 =0 ?;>I04L
2K45;5==>9 ?;>I04:8. 0: A;54C5B 87 ?@82545==>3> @5H5=8O MB> 402;5=85 2 40==><
f
A;CG05 @02=> PC?@. = .
r
3.2 7 40==KE '0AB8 2. >?@545;O5<, GB> <0:A8<0;L=>5 >B=>A8B5;L=>5 C4;8=5=85 ?;5=:8
@02=>
"lmax 8,00
max = = = 0,400 , (8)
l0 20,0
0 <0:A8<0;L=>5 ;8=59=>5 =0?@O65=85
<
600 "10-3 :3 " 9,81
mmax g
A2 93,7 .
fmax = = H" (9)
2Ąr0 2Ą "1,00 "10-2 < <
>4AB02;OO MB8 40==K5 2 C@02=5=85 (7), ?>;CG8< 7=0G5=85 <0:A8<0;L=>3> 402;5=8O,
:>B>@>5 <>65B 2K45@60BL B@C1:0
fmax mmax g
"Pmax = = H" 6,69 "103 0 (10)
rmax 2Ąr02(1+ )
max
25
3.3 ;O ?>AB@>5=8O B@51C5<>9 7028A8<>AB8 A;54C5B 2>A?>;L7>20BLAO C@02=5=85< (7), 2
:>B>@>< ;8=59=>5 =0?@O65=85 O2;O5BAO 8725AB=>9 DC=:F859 >B >B=>A8B5;L=>9
45D>@<0F88. @54AB028< B0:65 @048CA 45D>@<8@>20==>9 B@C1:8 G5@57 55 >B=>A8B5;L=CN
45D>@<0F8N r = r0(1+ ). @57C;LB0B5 ?>;CG8< C@02=5=85
f () = "P r0(1+ ). (11)
MB>< C@02=5=88 DC=:F8O f () 7040=0 2 2845 B01;8FK 1, 8;8 3@0D8:0 ( 8A.1),
0=0;8B8G5A:89 284 MB>9 7028A8<>AB8 =58725AB5=. >MB><C C@02=5=85 (11) =5 <>65B 1KBL
@5H5=> >B=>A8B5;L=> 0=0;8B8G5A:8. A;8 =5;L7O =09B8 7028A8<>ABL ("P), B> A;54C5B
?>?KB0BLAO @0AAG8B0BL >1@0B=CN 7028A8<>ABL "P(), GB> <>6=> A45;0BL M;5<5=B0@=> 87
C@02=5=8O (11)
f ()
"P = (12)
r0(1+ ).
A ?><>ILN 40==KE B01;8FK 1. A?>;L7CO MB8 40==K5, A;54C5B @0AAG8B0BL @048CA B@C1:8
r = r0(1+ ) 8 ?> D>@<C;5 (12) A>>B25BAB2CNI55 402;5=85. 57C;LB0BK @0AG5B>2
?@54AB02;5=K 2 "01;8F5 2, ?> :>B>@K< ?>AB@>5= 3@0D8: ( 8A. 5)
"01;8F0 2. 028A8<>ABL @048CA0 B@C1:8 >B 871KB>G=>3> 402;5=8O 2=CB@8 =55.
78,1 15,6 23,4 31,2 39,0 46,8 54,6 62,4 70,2 78,1 85,9
f ,
<
0,048 0,100 0,148 0,202 0,247 0,285 0,322 0,355 0,368 0,385 0,392
"P, : 0 0,74 1.412 2,04 2,59 3,13 3,64 4,13 4,61 5,13 5,63 6,16
r A<
1,05 1,10 1,15 1,20 1,25 1.29 1,32 1,36 1,37 1,39 1,39
26
#@02=5=85 (12) B0:65 <>65B 1KBL @5H5=> 3@0D8G5A:8< <5B>4><. ;O MB>3> ?@54AB028< 53> 2
2845
f () = "P r0(1+ ). (120)
@0D8: DC=:F88 f () ?>AB@>5= @0=55, B5?5@L =0 =5< A;54C5B ?@>25AB8 A5<59AB2>
?@O<KE, >?8AK205<KE DC=:F8O<8 y = "P r0(1+ ). @48=0B0 B>G:8 ?5@5A5G5=8O ?@O<>9
y = "P r0(1+ ) A 3@0D8:>< DC=:F88 f () 40AB @5H5=85 C@02=5=8O (120) ?@8 7040==>< 7=0G5=88
"P . >AB@>5=85 C:070==KE ?@O<KE >G5=L ?@>AB>: 2A5 >=8 ?5@5A5:0N 25@B8:0;L=CN >AL ?@8
= -1, 0 3>@87>=B0;L=CN ?@8 f = r0"P . #:070==>5 ?>AB@>5=85 ?@82545=> =0 8A. 6.
AB5AB25==>, GB> ?>;CG5==>5 B0:8< A?>A>1>< @5H5=85 A>2?0405B A ?>AB@>5==K< @0=55.
27
3.4.1 ?@81;865=88 70:>=0 C:0, 2 C@02=5=85 (7) =5>1E>48<> ?>4AB028BL O2=>5
r - r0
2K@065=85 4;O ;8=59=>3> =0?@O65=8O f = ł = ł :
r0
r - r0
ł = "P " r .
r0
7 MB>3> C@02=5=8O >?@545;O5< B@51C5<CN 7028A8<>ABL
r0
r = (13)
r0
1- "P
ł
@0D8: MB>9 DC=:F88 ?>:070= =0 8A. 5.
3.4.2 0=55 <K CAB0=>28;8, GB> 70:>= C:0 A?@0254;82 ?@8 >B=>A8B5;L=KE 45D>@<0F8OE
<5=LH8E < 0,25 . -B0 45D>@<0F8O 2>7=8:05B ?@8 871KB>G=>< 402;5=88 "P H" 3"104 0 ,
A;54>20B5;L=>, ?@81;865=85< 70:>=0 C:0 8 A;54CNI59 87 =53> D>@<C;>9 (13) <>6=>
?>;L7>20BLAO ?@8
"P d" 3"103 0 (14)
7 3@0D8:0 DC=:F88 (13) =0 @8A. 5, B0:65 A;54C5B, GB> MB> ?@81;865=85 4>AB0B>G=>
B>G=> >?8AK205B M:A?5@8<5=B0;L=K5 40==K5 ?@8 r d" 1,25, B> 5ABL ?@8 B>9 65 >B=>A8B5;L=>9
45D>@<0F88.
3.4.3 =0G5=85 @048CA0 B@C1:8, @0AAG8B0==>5 ?> D>@<C;5 (13), AB@5<8BAO : 15A:>=5G=>AB8,
5A;8 7=0<5=0B5;L CAB@5<;O5BAO : =C;N, GB> 4>AB8305BAO ?@8 <0:A8<0;L=> 2>7<>6=><
402;5=88, @02=><
1,56 "102
ł
<
("P) = H" H" 1,6 "104 0
max
(15)
r0 1,00 "10-2 <
0<5B8<, GB> MB> 7=0G5=85 1>;LH5, @0AAG8B0==>3> @0=55 ?> B01;8G=K< 40==K<. @><5
B>3>, 2 MB>< @5H5=88 ?@8G8=0 ACI5AB2>20=8O <0:A8<0;L=>3> 402;5=8O 8=0O ?@8 2>7@0AB0=88
@048CA0 B@C1:8 2>7@0AB0NI85 A8;K C?@C3>AB8 =5 2 A>AB>O=88 C45@60BL 1KAB@55 2>7@0AB0NI85
A8;K 402;5=8O 3070.
3.5.1 !G8B0O 871KB>G=>5 402;5=85 <0;K<, 8A?>;L7C5< ?@81;865==CN D>@<C;C 4;O
C?@>I5=8O DC=:F88 (13)
2
# ś#
r0 r0 # r0 ś#
ź#
r = H" r0 ś#1+ "P + ś# "Pź# .
ś# ź#
r0 ś# ź#
ł ł
# #
# #
1- "P
ł
7 MB>3> 2K@065=8O ?>;CG8< D>@<C;C 4;O >?@545;5=8O 87<5=5=8O @048CA0 B@C1:8
2
#
r0 # ś# # r0 ś#
r0 ś# r02
ś# ź# ś# ź#
"r = r0ś# "P + "Pź# ź# = "Pś#1+ "Pź# . (15)
ś#
ś# ź#
ł ł ł ł
# # # #
# #
@5=51@530O 2B>@K< A;0305<K<, ?>;CG05< ;8=59=CN 7028A8<>ABL <564C 87<5=5=85<
@048CA0 8 871KB>G=K< 402;5=85<
r02
"r = "P (16)
ł
28
3.5.2 =0G5=85 :>MDD8F85=B0 ?@>?>@F8>=0;L=>AB8 2 MB>9 D>@<C;5 @02=>
2
r02 (1,00 "10-2 <)
C = H" H" 6,4 "10-7 < " 0-1 (17)
ł
1,56 "102
<
@0D8: ;8=59=>9 DC=:F88 (16) A =0945==K< :>MDD8F85=B>< B0:65 ?>:070= =0 @8A. 5.
3.5.3 B>@>5 A;0305<>5 2 D>@<C;5 (15), :>B>@>5 1K;> >?CI5=>, <>65B A;C68BL >F5=:>9
?>3@5H=>AB8 D>@<C;K (16). "0:8< >1@07><, >B=>A8B5;L=0O ?>3@5H=>ABL ;8=59=>3>
?@81;865=8O @02=0
r0
= "P .
ł
!;54>20B5;L=>, ?>3@5H=>ABL D>@<C;K (16) =5 1C45B ?@52KH0BL CAB0=>2;5==K9 ?@545;
?@8
1,6 "102
ł
<
"P d" = 0,05 H" 8 "102 0 (18)
r0 1,0 "10-2 <
'0ABL 4. -;5:B@8G5A:85 A2>9AB20 ?;5=:8.
4.1 A?>;L7CO ?@82545==CN D>@<C;C 4;O A>?@>B82;5=8O 8 70:>= <0
1 l0
U0 = I0R = I0 ,
2Ąr0
?>;CG8< 7=0G5=85 C45;L=>3> A>?@>B82;5=8O ?;5=:8
I0l0 2,00 " 0,200<
= = H" 2,55 "103 <-1 (19)
2Ąr0U0 2Ą "1,00 "10-2 < " 2,5 "10-3
4.2 CABL ?>25@E=>AB=0O ?;>B=>ABL 70@O4>2 =0 >1:;04:0E :>=45=A0B>@0 @02=0 . ">340
=0?@O65==>ABL M;5:B@8G5A:>3> ?>;O 2 :>=45=A0B>@5 @02=0 E = , ?@8G5< =0?@O65==>ABL
0
2
?>;O, A>740205<>3> B>;L:> >4=>9 >1:;04:>9, 2 420 @070 <5=LH5 E = . !;54>20B5;L=>,
20
A8;0, 459AB2CNI0O =0 548=8FC ?;>I048 >1:;04:8 (B.5. 402;5=85 M;5:B@8G5A:>3> ?>;O)
@02=0
2 2
0E
2
PM;. = E = = , (20)
20 2
8 A>2?0405B A >1J5<=>9 ?;>B=>ABLN M=5@388 ?>;O.
4.2.1 ;O 40;L=59H53> @5H5=8O 7040G8 <>6=> 2>A?>;L7>20BLAO C@02=5=85< (7), 2 :>B>@><
@>;L 871KB>G=>3> 402;5=8O 1C45B 83@0BL A8;0 402;5=8O M;5:B@8G5A:>3> ?>;O.
0<5B8<, GB> M;5:B@8G5A:85 A8;K, 459AB2CN A=0@C68 >B B@C1:8, => AB@5<OBAO 55
@0ABO=CBL, :0: 871KB>G=>5 402;5=85 3070 2=CB@8 =55, ?>MB><C C:070==0O 70<5=0 4>?CAB8<0.
0;55, >1@0B8< 2=8<0=85, GB> <0:A8<0;L=> 2>7<>6=0O 45D>@<0F8O B@C1:8
(>3@0=8G5==0O 2=5H=8< <5B0;;8G5A:8< F8;8=4@><) A>AB02;O5B 10%, ?>MB><C 4;O
2KG8A;5=8O ;8=59=>3> <5E0=8G5A:>3> =0?@O65=8O AB5=>: B@C1:8 <>6=> ?>;L7>20BLAO
70:>=>< C:0.
29
"0: :0: :>=45=A0B>@ ?>4:;NG5= : 8AB>G=8:C, B> =0?@O65=85 =0 >1:;04:0E >AB05BAO
?>AB>O==K< ?@8 87<5=5=88 @0AAB>O=8O <564C B@C1:0<8. K@078< =0?@O65==>ABL
M;5:B@8G5A:>3> ?>;O 2=CB@8 :>=45=A0B>@0 G5@57 =0?@O65=85 8AB>G=8:0
U0 U0 U0
E = = = , (21)
R - r R - (r0 + "r) h - "r
(745AL R - 2=CB@5==89 @048CA <5B0;;8G5A:>9 B@C1:8). "0:8< >1@07><, 402;5=85 ?>;O
2K@0605BAO D>@<C;>9
2
2
0E 0 U0
# ś#
PM;. = = . (22)
ś# ź#
2 2 h - "r
# #
0:>=5F, 70?8H5< C@02=5=85, A2O7K20NI55 C25;8G5=85 @048CA0 B@C1:8 A ?@8;>65==K<
=0?@O65=85< (:>B>@>5 1C45B O2;OBLAO >A=>2>9 40;L=59H53> @5H5=8O):
2
0 U0 "r 1
# ś#
= ł . (22)
ś# ź#
2 h - "r r0 r0 + "r
# #
MB>< C@02=5=88 A;520 AB>8B 402;5=85 M;5:B@8G5A:>3> ?>;O, A?@020
C@02=>25H820NI55 53> 402;5=85 A8; C?@C3>AB8. #@02=5=85 (22) <>6=> ?@825AB8 :
0;351@08G5A:><C C@02=5=8N B@5BL59 AB5?5=8, ?>MB><C A=0G0;0 =5>1E>48<>
?@>0=0;878@>20BL 53> @5H5=85, 0 70B5< ?@>25AB8 2>7<>6=K5 C?@>I5=8O (>A=>20==K5 =0
@07C<=KE 8 >1>A=>20==KE ?@81;865=8OE). ;O MB>3> 2K@078< C25;8G5=85 @048CA0 B@C1:8
G5@57 >B=>A8B5;L=CN 45D>@<0F8N "r = r0 x 8 ?>4AB028< 2 C@02=5=85 (22)
2
2
#
0 U0 ś# r0 x 1 0U0 r0 1 x
ś# ź#
= ł ! =
2
ś# ź#
2 h - r0 x r0 r0 + r0 x 2h2 ł
# # #1- r0 1+ x
xś#
ś# ź#
h
# #
1>7=0G8< 157@07<5@=K5 25;8G8=K, @0AAG8B05< 8E G8A;5==K5 7=0G5=8O (2A5 25;8G8=K
?@54AB02;5=K 2 A8AB5<5 !)
2 2
2
0U0 r0 8,85 "10-12 "(1,0 "103) (1,0 "10-2)
A0 = = H" 2,84 "10-4 ,
2
2h2 ł 1,56 "102
2 "(1,0 "10-3)
r0
= H" 10 ,
h
8 ?>;CG8< 157@07<5@=>5 C@02=5=85
A0 x
= . (23)
2
1+ x
(1- x)
MB>< C@02=5=88 A0 O2;O5BAO <0;>9 25;8G8=>9, ?>MB><C 8 53> @5H5=85 x B0:65 <0;>,
?>MB><C ?>?KB05<AO ?@5=51@5GL <0;K<8 25;8G8=0<8 x 8 x 2 7=0<5=0B5;OE MB>3>
C@02=5=8O. ">340 @5H5=85 MB>3> C@02=5=8O
2
0U0 r0
x H" A0 = H" 2,84 "10-4 (24)
2h2 ł
0;>ABL ?>;CG5==>3> 7=0G5=8O ?>;=>ABLN >?@024K205B A45;0==>5 ?@81;865=85.
MB><C 65 @5H5=8N <>6=> ?>4>9B8 4@C38<8 A?>A>10<8. "0:, A@07C @0AAG8B02 402;5=85
M;5:B@8G5A:>3> ?>;O ?@8 >BACBAB288 45D>@<0F88,
2 2
2
0E 0 U0 8,85 "10-12 "(1,0 "103)
# ś#
PM;. = = = H" 4,4 0 ,
ś# ź#
2
2 2 h
# #
2 "(1,0 "10-3)
2848<, GB> >=> :@09=5 <0;> (@0=55 1K;8 BKAOG8 ?0A:0;59), ?>MB><C 8 45D>@<0F8O 1C45B <0;>9.
30
!;54>20B5;L=>, ?@8 @0AG5B5 M;5:B@8G5A:>3> 2708<>459AB28O 8 A8; C?@C3>AB8 8< <>6=>
?@5=51@5GL 8 70?8A0BL C@02=5=85 (22) 2 2845
2
0 U0 "r 1
# ś#
= ł " , (220)
ś# ź#
2 h r0 r0
# #
87 :>B>@>3> A@07C A;54C5B (24).
4.2.2 @8G8=>9 2>7<>6=>3> 2KE>40 :>=45=A0B>@0 87 AB@>O <>65B A;C68BL :>@>B:>5
70<K:0=85, :>B>@>9 ?@>87>945B, 5A;8 @0ABO3820NI0OAO B@C1:0 ?@8:>A=5BAO : 2=5H=5<C
h
F8;8=4@C, B> 5ABL :>340 >B=>A8B5;L=0O 45D>@<0F8O AB0=5B @02=>9 x = H" 0,10 .
r0
5?>A@54AB25==0O ?>4AB0=>2:0 MB>3> 7=0G5=8O 2 C@02=5=85 (23) =52>7<>6=>, B0:
?@8 MB>< 7=0<5=0B5;L >1@0I05BAO 2 =C;L. ;O 0=0;870 ?>2545=8O ?>;8<5@=>9 B@C1:8 ?@8
C25;8G5=88 =0?@O65=8O ?>AB@>8< ?@8<5@=K5 3@0D8:8 7028A8<>AB59 M;5:B@8G5A:>3> 8
C?@C3>3> 402;5=89 >B >B=>A8B5;L=>9 45D>@<0F88 2 8=B5@5ACNI5< 480?07>=5
1
0 d" x d" = 0,10 . -B> ?@>I5 A45;0BL 4;O 157@07<5@=KE DC=:F89 ( 8A. 7):
A x
pM;. = ; pC?@. = . (25)
2
1+ x
(1- x)
2
0U r0
2545==K9 745AL ?0@0<5B@ A = 7028A8B >B ?@8;>65==>3> : :>=45=A0B>@C
2h2 ł
=0?@O65=8O U .
@8 <0;KE 7=0G5=8OE ?0@0<5B@0 A 2 8=B5@5ACNI5< =0A 8=B5@20;5 0 < x < 0,1 MB8
3@0D8:8 8<5NB 425 B>G:8 ?5@5A5G5=8O, A;54>20B5;L=>, C@02=5=85 (23) 8<55B 420 :>@=O!
53:> ?>:070BL, GB> ?5@2K9 (=0945==K9 =0<8) A>>B25BAB2C5B CAB>9G82><C ?>;>65=8N
B@C1:8, 2B>@>9 =5CAB>9G82><C. /A=>, GB> ?@8 C25;8G5=88 =0?@O65=8O >B =C;O 1C45B
@50;87>2K20BLAO 8<5==> ?5@2>5, CAB>9G82>5 ?>;>65=85 @02=>25A8O (B0: >=> 1;865 :
=C;N). ! @>AB>< =0?@O65=8O 8AB>G=8:0 (8 A>>B25BAB2CNI53> 2>7@0AB0=8O ?0@0<5B@0 A )
:@820O 7028A8<>AB8 M;5:B@8G5A:>3> 402;5=8O 1C45B ?@8?>4=8<0BLAO 8, =0:>=5F, ?@8
=5:>B>@>< :@8B8G5A:>< 7=0G5=88 A* MB8 :@82K5 1C4CB 8<5BL 548=AB25==CN >1ICN
B>G:C, B>G:C :0A0=8O. <5==> ?@8 MB>< CA;>288 :>=45=A0B>@ ?>B5@O5B CAB>9G82>ABL.
@8B8G5A:>5 7=0G5=85 A* <>65B 1KBL =0945=> @07;8G=K<8 A?>A>10<8. 0AA<>B@8<
A0<K9 ?@>AB>9 (?> =0H5<C <=5=8N) 87 =8E. 0<5B8<, GB> 2 @0AA<0B@8205< 480?07>=5
:@82CN C?@C3>3> 402;5=8O A 2KA>:>9 B>G=>ABLN <>6=> 70<5=8BL =0 ?@O<CN
x
pC?@. = H" x .
1+ x
31
! CG5B>< MB>3>, ?5@5?8H5< C@02=5=85 (23) 2 2845
2
A = x(1- x) . (26)
>=OB=>, GB> MB> @02=>A8;L=>5 (23) C@02=5=85 =5 1C45B 8<5BL :>@=59, 5A;8 7=0G5=85
2
?0@0<5B@0 A ?@52KH05B <0:A8<C< DC=:F88 y = x(1- x) . @>872>4=0O MB>9 DC=:F88
2
2
y = (1- x) - 2x(1- x) = (1- x)(1- 3x)
1
>1@0I05BAO 2 =C;L ?@8 x* = , (7=0G5=8N x* = 1 A>>B25BAB2C5B B>G:0 @07@K20
3
DC=:F88). "0:8< >1@07><, <0:A8<0;L=>5 7=0G5=85 MB>9 DC=:F88 8 :@8B8G5A:>5 7=0G5=85
?0@0<5B@0 =0945<, ?>4AB0282 MB> 7=0G5=85 2 D>@<C;C (26)
2
1 1 4h
#1- ś#
A* = = .
ś# ź#
3 3 # 27r0
#
2
0U r0 4h
A?>;L7CO 2K@065=85 4;O 22545==>3> ?0@0<5B@0, ?>;CG05< = ,
2h2 ł 27r0
>B:C40 A;54C5B, GB> :@8B8G5A:>5 =0?@O65=85 @02=>
3
8h3ł 8(1,0 "10-3) "1,56 "102
U = = H" 7,2 "103 (26)
max
27r020 27(1,0 "10-2)2
"8,85 "10-12
1@0B8B5 2=8<0=85, ?@8 =0?@O65=88 2 1 : 45D>@<0F8O B@C1:8 D0:B8G5A:8
?@5=51@568<> <0;0, 0 ?@8 =0?@O65=88 7 : 2 :>=45=A0B>@ ?@>8AE>48B :>@>B:>5 70<K:0=85.
04> ?@87=0BL =5 >G5=L =0456=>5 CAB@>9AB2> 87>1@5B5=>.
;O >:>=G0B5;L=>3> ?>4B25@645=8O B>G=>AB8 ?>;CG5==>3> @5H5=8O ?>AB@>8< 3@0D8: DC=:F88
x
2
(26) 8A.8 A?;>H=0O ;8=8O. C=:B8@>< ?>AB@>5=0 B>G=0O DC=:F8O A = (1- x) (157
1+ x
A45;0==>3> ?@81;865=8O >1 C?@C3>< 402;5=88).
84=>, GB> MB8 DC=:F88 @07;8G0NBAO =57=0G8B5;L=>. "0:, 2 @09>=5 =0945==>3> <0:A8<C<0
@07;8G8O 2 MB8E DC=:F8OE >F5=8< 87 @07;>65=8O
x
2 2
A = (1- x) H" x(1- x) "(1- x),
1+ x
87 :>B>@>3> A;54C5B, GB> >B=>A8B5;L=0O >H81:0 =0H53> ?@81;865=8O A>AB02;O5B 25;8G8=C
A* 4
H" x = H" 0,015 (2A53> ?>;B>@0 ?@>F5=B0), GB> ?@82>48B : >H81:5 2 >?@545;5=88
A* 27
32
<0:A8<0;L=> =0?@O65=8O (A CG5B>< U ~ A* ) 2 420 @070 <5=LH59, B> 5ABL ?>@O4:0 0,8%. "0:0O
B>G=>ABL =5 >15A?5G8205BAO B>G=>ABLN 8AE>4=KE 40==KE, ?>MB><C O2;O5BAO 87;8H=59.
1@0B8B5 2=8<0=85, GB> ?@8 <0;KE x 3@0D8: MB>9 DC=:F88 ;8=55=, GB> ?>4B25@6405B
4>?CAB8<>ABL A45;0==KE ?@81;865=89 ?@8 >?@545;5=88 45D>@<0F88 ?> D>@<C;5 (24).
B<5B8<, GB> A ?><>ILN 3@0D8:0 =0 8A.8 <>6=> @5H0BL 8 8AE>4=>5 C@02=5=85 (23).
3> :>@=O< A>>B25BAB2CNB B>G:8 ?5@5A5G5=8O 3@0D8:0 MB>9 DC=:F88 A 3>@87>=B0;L=>9 ?@O<>9
(4;O =03;O4=>AB8 7=0G5=85 ?0@0<5B@0 A 702KH5=> ?> A@02=5=8N A @5H5=85< 2 ?@54K4CI5<
@0745;5 7040G8). @><5 B>3>, 284=>, GB> MB> C@02=5=85 8<55B F5;KE B@8 :>@=O, :0: 8 ?>;>65=>
C@02=5=8N B@5BL59 AB5?5=8! @8G8=K ?>O2;5=8O MB>3> :>@=O CAB0=>28B5 A0<>AB>OB5;L=>, =0A
65 >= =5 8=B5@5AC5B, B0: :0: 2A5340 2KE>48B 70 ?@545;K 8=B5@5ACNI53> =0A 8=B5@20;0.
'0ABL 5. 03=8B=>5 2>7459AB285.
5.1 @8 ?@>B5:0=88 ?5@5<5==>3> M;5:B@8G5A:>3> B>:0 ?> >1<>B:5 A>;5=>840, 2=CB@8
=53> 2>7=8:05B ?5@5<5==>5 <03=8B=>5 ?>;5, :>B>@>5 8=4CF8@C5B -! :0: 2
>1<>B:5 A>;5=>840 (A0<>8=4C:F8O), B0: 8 2 ?@>2>4OI59 ?>;8<5@=>9
>1>;>G:5. =4CF8@>20==K9 2 >1>;>G:5 M;5:B@8G5A:89 B>: :@C3>2>9 8
?@>B5:05B ?> 55 ?>25@E=>AB8 ?5@?5=48:C;O@=> >A8 ( 8A. 10). -B>B B>: 1C45B
>:07K20BL 0=0;>38G=>5 >1@0B=>5 2;8O=85 =0 B>: 2 >1<>B:5. "0: :0: 2=CB@8
A8AB5<K 8<55BAO A5@45G=8: A 1>;LH>9 <03=8B=>9 ?@>=8F05<>ABLN, B>
<03=8B=K< ?>B>:>< 2 C7:>< ?@>AB@0=AB25 <564C >1<>B:>9 8 >1>;>G:>9
<>6=> ?@5=51@5GL.
=4C:F8O <03=8B=>3> ?>;O 2 A5@45G=8:5 B1 ,
A>740205<>3> B>:>< 2 >1<>B:5 I1 , @0AAG8BK205BAO ?>
8725AB=>9 D>@<C;5
B1 = 0nI1. (27)
0<5B8<, GB> ?@>872545=85 nI = i ?@54AB02;O5B A>1>9
?>25@E=>AB=CN ?;>B=>ABL B>:0 2 >1<>B:5, ?>MB><C MB0 65
D>@<C;0 A?@0254;820 8 4;O @0AG5B0 8=4C:F88 ?>;O,
A>740205<>3> B>:>< 2 >1>;>G:5
B2 = 0i . (28)
>;>68B5;L=K< =0?@02;5=85< B>:>2 1C45< AG8B0BL
=0?@02;5=85 ?@>B82 G0A>2>9 AB@5;:8 ( 8A. 11) , B>340 ?>;>68B5;L=>5 =0?@02;5=85
25:B>@0 8=4C:F88 >?@545;O5BAO ?> ?@028;C 1C@02G8:0.
0 @8AC=:5 C:070=K ?>;>68B5;L=K5 =0?@02;5=8O B>:>2 8 25:B>@>2 8=4C:F88, E>BO
@50;L=> =0?@02;5=8O B>:>2 2 >1<>B:5 8 >1>;>G:5 ?@>B82>?>;>6=K, 8E ?@028;L=K5
=0?@02;5=8O 02B><0B8G5A:8 ?>O2OBAO ?@8 ?@028;L=>9 70?8A8 C@02=5=89, >?8AK20NI8E MB8 B>:8.
03=8B=K9 ?>B>: G5@57 ?>?5@5G=>5 A5G5=85 @0AA<0B@8205<>9 A8AB5<K 70?8H5< 2
2845
Ś = Ąa2(B1 + B2 ) = Ąa20(nI1 + i). (29)
@5>1@07C5< MB> 2K@065=85, 4;O G53> 70<5B8<, GB> 25;8G8=0
Ąa20nN = L (30)
(345 N = nl - G8A;> 28B:>2 >1<>B:8) O2;O5BAO 8=4C:B82=>ABLN A>;5=>840 (2 >BACBAB288
NŚ1
>1>;>G:8), B0: :0: >=0 <>65B 1KBL ?@54AB02;5=0 2 2845 L = , Ś1 - ?>B>: G5@57 >48=
I1
28B>: <03=8B=>3> ?>;O, A>740205<>3> B>:>< I1 2 >1<>B:5. -;5:B@8G5A:89 B>: 2
?>;8<5@=>9 >1>;>G:5 @02=><5@=> @0A?@545;5= ?> 55 ?>25@E=>AB8. >MB><C <>6=>
33
70?8A0BL 2K@065=85 4;O AC<<0@=>3> B>:0 I2 = il , B5:CI53> ?> >1>;>G:5. B>@>5
A;0305<>5 2 MB>< A;CG05 8<55B 284
I2 nN L
Ąa20i = Ąa20 = Ąa20 I2 = I2 (31)
2
l nNl N
:>=G0B5;L=>, <03=8B=K9 ?>B>: 70?8AK205BAO 2 2845
L L
Ś = I1 + I2 . (32)
2
N N
>7=8:0NI0O ?@8 87<5=5=88 <03=8B=>3> ?>B>:0 -! 8=4C:F88 >48=0:>20 2 >4=>< 28B:5
>1<>B:8 8 >1>;>G:5 8 @02=0
2
ind = -Ś (33)
"0: :0: 0:B82=K< A>?@>B82;5=85< :>=BC@0, A>AB>OI8< 87 8AB>G=8:0 8 >1<>B:>9
A>;5=>840 <>6=> ?@5=51@5GL, B> 4;O =53> AC<<0 -! 8AB>G=8:0 8 8=4C:F88 2> 2A5E
28B:0E A>;5=>840< @02=0 =C;N, ?>MB><C
2
8AB. - NŚ = 0 . (34)
0?8H5< C@02=5=85 AC<<0@=>3> B>:0 I2 = il , B5:CI53> ?> >1>;>G:5. 1I55
A>?@>B82;5=85 >1>;>G:8 ?@8 C:070==>< ?@>B5:0=88 B>:0 @02=> (?> A@02=5=8N A
?@54K4CI8< ?@>4>;L=K< ?@>B5:0=85<, 4;8=0 8 H8@8=0 ?><5=O;8AL <5AB0<8):
1 2Ąr0
R = . (35)
l
@54?>;>68< ?>:0, GB> A>?@>B82;5=85 :>;LF0 =5 7028A8B >B 53> 45D>@<0F88.
;O AC<<0@=>3> B>:0 A?@0254;82 70:>= <0 4;O ?>;=>9 F5?8 I2R = ind , 8;8
2
I2R = -Ś (36)
#@02=5=8O (34) 8 (36) >1@07CNB A8AB5<C, >?8AK20NICN 87<5=5=8O B>:>2 2 >1<>B:5 8
>1>;>G:5. >4AB0282 2 =55 2K@065=85 4;O ?>B>:0, ?>;CG8< MBC A8AB5<C 2 O2=>< 2845
2
ż#
#8AB. - L# I1 + 1 I2 ś# 0
=
ś# ź#
N
#
# #
(37)
#
2
#
L 1
#
I1 + I2 ś#
#I R = - ś# ź#
2
N N
# #
#
1. 57 22545=8O >1>7=0G5=89 4;O 8=4C:B82=>AB8 8 A>?@>B82;5=8O >1>;>G:8 MB0 65 A8AB5<0 1C45B
2K3;O45BL 1>;55 3@><>74:>.
K45;8< B>=:>5 :>;LF> B>;I8=>9 "z ( 8A. 12). !8;0
B>:0, ?@>B5:0NI0O ?> MB><C :>;LFC @02=0 I2 = i"z , 0 53>
1 2Ąr
A>?@>B82;5=85 @02=> R = . 0:>= <0 4;O 2K45;5==>3>
"z
:>;LF0 8<55B 284
2Ąr0
ind = I2R = i .
"0:8< >1@07><, ?>;CG05< A8AB5<C, >?8AK20NI8E 87<5=5=8O B>:>2:
ż# - NĄa20(nI1 + i)2
= 0
8AB.
#
.
#2Ąr0
i = -Ąa20(nI1 + i)2
#
#
2. $0:B8G5A:8 @0AA<0B@8205<>5 CAB@>9AB2> O2;O5BAO B@0=AD>@<0B>@><: ?5@28G=0O >1<>B:0
A>45@68B N 28B:>2 8 ?@5=51@568<> <0;>5 A>?@>B82;5=85, 55 8=4C:B82=>ABL @02=0 L ;
2B>@8G=0O >1<>B:0 A>AB>8B 87 >4=>3> 28B:0, A>?@>B82;5=85 2B>@8G=>9 F5?8 5ABL
34
I2
Ś2 0 l Ąa2 0Ąa2
A>?@>B82;5=85 >1>;>G:8 8 @02=> R , 55 8=4C:B82=>ABL L2 = = =
I2 I2 l
0Ąa2 0Ąa2 nN L
A2O70=0 A 8=4C:B82=>ABLN >1<>B:8 A>>B=>H5=85< L2 = = " = .
2
l l nN N
L
0:>=5F :>MDD8F85=B 2708<=>9 8=4C:F88 L12 = L21 = . >;CG5==0O A8AB5<0 C@02=5=89 5ABL
N
A8AB5<0 C@02=5=89, >?8AK20NI8E B@0=AD>@<0B>@.
5.2 5H8< A8AB5<C C@02=5=89 (37). 0?8H5< 7028A8<>ABL -! 8AB>G=8:0 >B 2@5<5=8 2
2845 8AB. = -U0 cost . K@060O ?@>872>4=CN 87 2B>@>3> C@02=5=8O 8 ?>4AB02;OO 55 2
?5@2>5, ?>;CG8<
I2RN = -U0 cost .
"0:8< >1@07><, A8;0 B>:0 2 >1<>B:5 >?8AK205BAO DC=:F859
U0
I2 = - cost . (38)
RN
7 ?5@2>3> C@02=5=8O A8AB5<C 2K@078<
L
2 2
LI1 = U0 cost - I2 .
N
@54AB028< A8;C B>:0 2 >1<>B:5 :0:
I1(t) = Acost + B sint ,
8 53> ?>4AB028< 2 C@02=5=85 2<5AB5 A 2K@065=85< (38) 4;O B>:0 2 >1>;>G:5:
L U0
- AL sint + BL cost = U0 cost - sint .
N NR
7 CA;>28O @025=AB20 :>MDD8F85=B>2 ?@8 B@83>=><5B@8G5A:8E DC=:F89 =0E>48<:
U0 U0
A = , B = .
2
N R L
:>=G0B5;L=> 7028A8<>ABL A8;K B>:0 2 >1<>B:5 >B 2@5<5=8 8<55B 284
U0 U0
I2 = cost + sint . (39)
2
N R L
<?;8BC4=>5 7=0G5=85 A8;K MB>3> B>:0 @02=>
1 1
I20 = A2 + B2 = U0 2 2 + . (40)
2
(L)
(N R)
@>2545< G8A;5==K5 @0AG5BK ?0@0<5B@>2 87CG05<>3> CAB@>9AB20:
1 2Ąr0 2Ą "1,0 "10-2
A>?@>B82;5=85 >1>;>G:8: R = = H" 1,23"10-4 < ;
l 2,55 "103 " 20 "10-2
G8A;> 28B:>2 >1<>B:8: N = nl = 200 ;
8=4C:B82=>ABL >1>;>G:8:
2
L = Ąa20nN = Ą(0,90 "10-2) "1,0 "103 " 4Ą "10-7 "1,0 "103 " 200 H" 63,9 "10-3 = ;
:@C3>20O G0AB>B0 B>:0: = 2Ą H" 314 A-1.
"5?5@L 2KG8A;8< E0@0:B5@8AB8:8 B>:>2:
- 0<?;8BC4=>5 7=0G5=85 A8;K B>:0 2 >1>;>G:5
U0 36
I20 = = H" 1,46 "103 A .
RN 1,23"10-4 " 200
35
- B@51C5<>5 7=0G5=85 0<?;8BC4K ?>25@E=>AB=>9 ?;>B=>AB8 B>:0 2 >1>;>G:5
I2 0 U0
i0 = = H" 7,3"103 A " <-1 (41)
l RNl
- 0<?;8BC4=>5 7=0G5=85 A8;K B>:0 2 A>;5=>845
1 1 1 1
I20 = U0 2 2 + = 36 + H"
2 2 2
(L)
(N R) (2002 "1,23"10-4) (63,9 "10-3 "314)
(42)
1 1
H" 36 + H" 7,5A
2 2
(4,92) (20,1)
KA>:85 7=0G5=8O ?>;CG5==KE 7=0G5=89 B>:>2 O2;ONBAO A;54AB285< <0;>3> A>?@>B82;5=8O
>1>;>G:8, :>B>@0O, :0: 1K;> >B<5G5=> 2KH5, O2;O5BAO 2B>@8G=K< :>=BC@>< B@0=AD>@<0B>@0.
5.3 >;CG5==K5 7028A8<>AB8 B>:>2 >B 2@5<5=8 ?>72>;ONB ;53:> =09B8 M=5@35B8G5A:85
E0@0:B5@8AB8:8 @0AA<0B@8205<KE ?@>F5AA>2. !@54=OO <>I=>ABL1 B5?;>BK, 2K45;ONI59AO
2 >1>;>G:5, @0AAG8BK205BAO ?> 70:>=C 6>C;O-5=F0 (A CA@54=5=85< ?> 2@5<5=8, :>B>@>5
>1>7=0G05BAO C3;>2K<8 A:>1:0<8)
2
2
U0 U0
# ś#
2
PB5?;. = I2 R = cost R = cos2 t
ś# ź#
2
RN RN
# #
1
"0: :0: cos2 t = , B> >:>=G0B5;L=K9 @57C;LB0B:
2
2
U0
PB5?;. = H" 1,3"102 B (43)
2
2RN
>I=>ABL, ?>B@51;O5<0O >B 8AB>G=8:0, B0:65 ;53:> @0AAG8BK205BAO
2
U0 U0 U0
# ś#
P8AB. = 8AB.I1 = U0 cost cost + sint = , (44)
ś# ź#
2 2
N R L N R
# #
8, :>=5G=>, A>2?0405B A <>I=>ABLN B5?;>BK, 2K45;ONI59AO 2 >1>;>G:5. @8 2K2>45
D>@<C;K (44) CGB5=>, GB> cost sint = 0 .
5.4. -;5:B@8G5A:89 B>: ?>25@E=>AB=>9 ?;>B=>AB8 i , ?@>B5:0NI89 ?> >1>;>G:5,
2708<>459AB2C5B A <03=8B=K< ?>;5<, 2 @57C;LB0B5 G53>
?>O2;ONBAO A8;K, @0ABO3820NI85 >1>;>G:C. @8 @0AG5B5
MB>9 A8;K =5>1E>48<> CG5ABL, GB> =0 >1>;>G:C 459AB2C5B
B>;L:> <03=8B=>5 ?>;5, A>740205<>5 B>:>< 2 A0<>9 >1>;>G:5
2 ?@>AB@0=AB25 <564C >1<>B:>9 8 >1>;>G:>9 (345 =5B
A5@45G=8:0), ?>MB><C 25;8G8=0 8=4C:F88 MB>3> ?>;O @02=0
B0 = 0i . 0AG5B 402;5=8O <03=8B=>3> ?>;O ?>;=>ABLN
0=0;>38G5= @0AG5BC 402;5=8O M;5:B@8G5A:>3> ?>;O. 'B>1K
=09B8 402;5=85 <03=8B=>3> ?>;O =0 ?>25@E=>ABL >1>;>G:8,
2K45;8< =0 55 ?>25@E=>AB8 <0;CN ?;>I04:C ( 8A. 13) A>
AB>@>=0<8 "z (?5@?5=48:C;O@=CN =0?@02;5=8N B>:0) 8 "l (?> =0?@02;5=8N B>:0).
!8;0 B>:0, ?5@5A5:0NI0O ?;>I04:C, @02=0 I = i"z . !8;0 <?5@0, 459AB2CNI0O =0 MBC
?;>I04:C, @02=0
1
MB>< @0745;5 P - >7=0G05B <>I=>ABL, 0 =5 402;5=85! >-?5@2KE, MB> B@048F8>==>5 >1>7=0G5=85, 2>-
2B>@KE, 745AL =5 @0AA<0B@8205BAO 402;5=85, 2-B@5BL8E, C65 4@C38E 1C:2 =5 E20B05B.
36
"F = IB1"l = iB1l"z = iB1"S , (45)
(345 "S - ?;>I04L 2K45;5==>9 ?;>I04:8) 8 =0?@02;5=0 ?5@?5=48:C;O@=> 59. 0;55
=5>1E>48<> CG5ABL, GB> 2 MB>9 D>@<C;5 B1 - 8=4C:F8O ?>;O, A>740205<>3> 2A5<8 B>:0<8,
:@><5 B>:>2 A0<>9 ?;>I04:8. =5?>A@54AB25==>9 1;87>AB8 >B ?;>I04:8 A 2=CB@5==59
AB>@>=K AC<<0@=>5 ?>;5 8=4C:F88 B0 = 0i O2;O5BAO AC<<>9 ?>;59 A>740205<>9 B>:0<8
r r
=0 ?;>I04:5 B' 8 8=B5@5ACNI53> =0A ?>;O 2A5E >AB0;L=KE B>:>2 B1 , ?>MB><C B1 + B'= B0 .
r r
! =0@C6=>9 AB>@>=K ?;>I04:8 25:B>@K B' 8 B1 ?@>B82>?>;>6=K 8 AC<<0@=>5 ?>;5 2
MB>9 >1;0AB8 >BACBAB2C5B, ?>MB><C 745AL B1 - B'= 0 . "0:8< >1@07><, :0: 8 2
M;5:B@>AB0B8:5, ?>;5, 459AB2CNI55 =0 ?;>I04:C, 2 420 @070 <5=LH5 AC<<0@=>3> ?>;O
1
B1 = B0 . "5?5@L A ?><>ILN D>@<C;K (45) 8 ?>;CG5==>3> @57C;LB0B0, >?@545;O5<
2
402;5=85 <03=8B=>3> ?>;O
F 1 0i2
P<03. = = iB1 = iB0 = . (46)
"S 2 2
"0:8< >1@07><, 402;5=85 <03=8B=>3> ?>;O =0 >1>;>G:C ?@>?>@F8>=0;L=> :204@0BC
?>25@E=>AB=>9 ?;>B=>AB8 B>:0. B<5B8<, GB> ?@8 ?5@5<5==>9 A8;5 (:0: 2 40==><
A;CG05) AB@>389 @0AG5B A<5I5=8O B@51C5B @5H5=8O 48=0<8G5A:8E C@02=5=89, B>
C@02=5=89 2B>@>3> 70:>=0 LNB>=0 ma = F (345 F - AC<<0 A8; 459AB2CNI8E =0 B5;>).
A;8 65 <0AA0 B5;0 ?@5=51@568<> <0;0, B> 48=0<8G5A:>5 C@02=5=85 ?5@5E>48B 2
C@02=5=85 AB0B8G5A:>3> @02=>25A8O. "0:8< >1@07><, 2 40==>< A;CG05 4;O @0AG5B0
45D>@<0F88 <>6=> ?>;L7>20BLAO ?>;CG5==K<8 @0=55, D>@<C;0<8 4;O AB0B8G5A:>3>
A;CG0O.
A?>;L7CO @0AAG8B0==>5 @0=55 7=0G5=85 0<?;8BC4K ?>25@E=>AB=>9 ?;>B=>AB8 B>:0,
@0AAG8B05< <0:A8<0;L=>5 402;5=85 ?>;O =0 >1>;>G:C
2
2
0i0 4Ą "10-7 "(7,3"103)
Pmax = = H" 33 0 . (47)
2 2
-B> 402;5=85 <0;> =0 AB>;L:>, GB> 4;O @0AG5B0 C25;8G5=8O @048CA0 <>6=>
?>;L7>20BLAO ;8=59=K< ?@81;865=85<, B> 5ABL ?> D>@<C;5 (16) 87 :>B>@>9 A;54C5B, GB>
<0:A8<0;L=>5 C25;8G5=85 @048CA0 B@C1:8 2 40==>< A;CG05 @02=>
2
r02 (1,0 "10-2)
"rmax = Pmax = " 33,5 H" 2,1"10-5 < (48)
ł 1,56 "102
5.5 @8G8=0 2>7=8:=>25=8O 72C:0 2 >?8A0==>9 A8BC0F88 ?>=OB=0: ?>4 459AB285<
?5@5<5==>9 A8;K A> AB>@>=K ?5@5<5==>3> <03=8B=>3> ?>;O >1>;>G:0 =0G8=05B
281@8@>20BL 8 874020BL 72C:. G5284=>, GB> 3@><:>ABL 72C:0 1C45B <0:A8<0;L=0 ?@8
<0:A8<0;L=>9 0<?;8BC45 :>;510=89 >1>;>G:8. ?@>2545==>< 2KH5 @0AG5B5 402;5=8O
>BACBAB2C5B 7028A8<>ABL <0:A8<0;L=>3> 402;5=8O >B G0AB>BK B>:0 (A<. D>@<C;K (46) 8
(41)). 4=0:> >1>;>G:0 >1;0405B <0AA>9 (8 C?@C3>ABLN),
?>MB><C >=0 A?>A>1=0 A>25@H0BL A2>1>4=K5 :>;510=8O A
=5:>B>@>9 A>1AB25==>9 G0AB>B>9. >MB><C A59G0A <K 4>;6=K
@0AA<>B@5BL 42865=85 >1>;>G:8 :0: 2K=C645==K5 :>;510=8O.
%>@>H> 8725AB=>, GB> <0:A8<0;L=0O 0<?;8BC40 :>;510=89
4>AB8305BAO 2 CA;>28OE @57>=0=A0, :>340 G0AB>B0
2K=C640NI59 A8;K A>2?0405B A A>1AB25==>9 G0AB>B>9
:>;510=89 A8AB5<K. "0:8< >1@07><, A59G0A =0< =5>1E>48<>
=09B8 A>1AB25==CN G0AB>BC :>;510=89 A8AB5<K. G5@54=>9
@07 2K45;8< =0 ?>25@E=>AB8 B@C1:8 <0;CN ?;>I04:C
?;>I04LN "S ( 8A. 14), <0AA0 :>B>@>9 @02=0 "m = "S . @8
37
A<5I5=88 MB>9 ?;>I04:8 =0 25;8G8=C "r(t) (:>B>@0O A59G0A @0AA<0B@8205BAO :0:
DC=:F8O 2@5<5=8) =0 =55 1C45B 459AB2>20BL 2>72@0I0NI0O A8;0 C?@C3>AB8. 5;8G8=C
MB>9 A8;K <K <>65< 2K@078BL 87 ?@81;865==>9 D>@<C;K (16): MB0 D>@<C;0 ?>;CG5=0 87
ł
CA;>28O @025=AB20 A8;0 402;5=8O 8 A8;C C?@C3>AB8, ?>MB><C 2 MB>9 D>@<C;5 "P = "r
r02
@02=> 402;5=8N A8; C?@C3>AB8. "0:8< >1@07><, 2>72@0I0NI0O A8;0 <>65B 1KBL 70?8A0=0
2 2845
ł
F = "S"P = "S "r . (49)
r02
0?8H5< C@02=5=85 2B>@>3> 70:>=0 LNB>=0 2 ?@>5:F88 =0 @0480;L=>5 =0?@02;5=85
ł
"Sa = -"S "r , (50)
r02
345 a - CA:>@5=85 ?;>I04:8, :>B>@>5, >G5284=>, @02=> 2B>@>9 ?@>872>4=>9 >B C25;8G5=8O
@048CA0 a = ("r)3 . "5?5@L C@02=5=85 (50) ?@8>1@5B05B 7=0:><K9 284
ł
("r)3 = - "r , (51)
r02
C@02=5=8O 30@<>=8G5A:8E :>;510=89 A G0AB>B>9
0 1 ł
= = .
0
2Ą 2Ą r02
0:>=5F, 2A?><=8<, GB> 402;5=85, 459AB2CNI55 =0 ?>25@E=>ABL B@C1:8 ?@>?>@F8>=0;L=>
:204@0BC A8;K B>:0, A;54>20B5;L=>, G0AB>B0 87<5=5=8O 402;5=8O 2 420 @070 1>;LH5
G0AB>BK B>:0. B0:, @57>=0=A (8 53> A;54AB28O <0:A8<0;L=0O 0<?;8BC40 8 <0:A8<0;L=0O
3@><:>ABL) =0ABC?8B, :>340 G0AB>B0 B>:0 1C45B 2 420 @070 <5=LH5 G0AB>BK A>1AB25==KE
:>;510=89 B@C1:8, B> 5ABL ?@8
1 ł 1 1,56 "102
0
= = = H" 2,9 "102 F (52)
2
2 4Ą r02 4Ą
(1,0 "10-2) 0,119
38
'0ABL 6. !25@E?@>2>48<>ABL.
0G5AB25==> >1AC48<, :0:85 D878G5A:85 ?@>F5AAK 1C4CB ?@>8AE>48BL 2 40==><
M:A?5@8<5=B5. @8 ?>4:;NG5=88 :0BCH:8 : 8AB>G=8:C 2 >1>;>G:5 1C4CB 8=4CF8@>20BLAO
M;5:B@8G5A:85 B>:8, =>, B0: :0: ?;0AB8: >1;0405B M;5:B@8G5A:8< A>?@>B82;5=85<, MB8
B>:8 1KAB@> 70BCE=CB. @8 >E;0645=88 B@C1:8 8 ?5@5E>45 55 2 A25@E?@>2>4OI55
A>AB>O=85 2 ?>AB>O==>< <03=8B=>< ?>;5 B>:8 2 =59 =5 ?>O2OBAO. @8 872;5G5=88
A>;5=>840 A B>:>< 87 B@C1:8 <03=8B=K9 ?>B>: ?>;O A>;5=>840 G5@57 ?>?5@5G=>5 A5G5=85
B@C1:8 1C45B C<5=LH0BLAO, GB> ?@82545B : 2>7=8:=>25=8N =570BCE0NI53> B>:0 ?>
?>25@E=>AB8 B@C1:8. -B8 B>:8 A>7404CB <03=8B=>5 ?>;5, :>B>@>5, 2708<>459AB2CO A B>:><,
?@82545B : @0AH8@5=8N B@C1:8. "0:8< >1@07><, =0< =5>1E>48<> =09B8 ?>25@E=>AB=CN
?;>B=>ABL B>:>2, 8=4CF8@>20==KE 2 A25@E?@>2>4OI59 B@C1:5 ?>A;5 ?>;=>3> 872;5G5=8O
A>;5=>840.
0AA<>B@8< ?@>2>4OI89 :>=BC@, =0E>4OI89AO 2
<03=8B=>< ?>;5 ( 8A.15). @8 87<5=5=88 <03=8B=>3> ?>B>:0
2=5H=53> ?>;O 2 :>=BC@5 2>7=8: M;5:B@8G5A:89 B>:, <03=8B=>5
?>;5 :>B>@>3> 1C45B ?@5?OBAB2>20BL 87<5=5=8N ?>B>:0
2=5H=53> ?>;O. ;O B>:0 2 :>=BC@5 <>6=> 70?8A0BL C@02=5=85
70:>=0 <0 4;O ?>;=>9 F5?8
2 2
IR = -Ś0 - Ś1 , (53)
2
345 Ś0 - <03=8B=K9 ?>B>: 2=5H=53> ?>;O; (- Ś0 ) -
(?@>872>4=0O >B MB>3> ?>B>:0, 27OB0O A >1@0B=K< 7=0:><) -!
8=4C:F88, 2>7=8:0NI0O ?@8 87<5=5=88 ?>B>:0 2=5H=53> ?>;O,
2
0=0;>38G=>, (- Ś1) - -! A0<>8=4C:F88, 2>7=8:0NI0O ?@8
87<5=5=88 ?>;O, A>740==>3> B>:>< 2 :>=BC@5. A;8 :>=BC@
A25@E?@>2>4OI89, B> 87 C@02=5=8O (53) A;54C5B
2 2
0 = -Ś0 - Ś1 = -(Ś0 + Ś1)2 ,
B> 5ABL AC<<0@=K9 <03=8B=K9 ?>B>: G5@57 A25@E?@>2>4OI89 :>=BC@ >AB05BAO
=587<5==K<2.
>A?>;L7C5<AO MB8< CB25@645=85< 4;O @5H5=8O ?>AB02;5==>9 7040G8. >340 A>;5=>84 A
?@>B5:0NI8< B>:>< I0 =0E>48BAO 2=CB@8 B@C1:8, <03=8B=K9 ?>B>: G5@57 ;N1>5
?>?5@5G=>5 A5G5=85 @025=
Ś0 = 0nI0Ąa2 , (54)
?>A;5 872;5G5=8O A>;5=>840 MB>B 65 <03=8B=K9 ?>B>: A>7405BAO ?>25@E=>AB=K< B>:><
2
Ś0 = Ś1 = 0iĄr , (55)
745AL i - ?>25@E=>AB=0O ?;>B=>ABL 8=4CF8@>20==>3> B>:0 ?> ?>25@E=>AB8 B@C1:8, r -
@048CA 45D>@<8@>20==>9 B@C1:8. 59AB28B5;L=>, <03=8B=K9 ?>B>: 4>;65= A>E@0=OBAO 2
;N1>< A;CG05: 8 ?@8 87<5=5=88 2=5H=53> ?>;O 8 ?@8 87<5=5=88 ?;>I048 :>=BC@0, B> 5ABL
?@8 C25;8G5=88 @048CA0 :>=BC@0 A8;0 B>:0 2 =5< 1C45B C<5=LH0BLAO!
7 CA;>28O ?>AB>O=AB20 <03=8B=>3> ?>B>:0 ?>;CG05< C@02=5=85, A2O7K20NI55 A8;C B>:0 2
A>;5=>845 8 A8;C 8=4CF8@>20==>3> B>:0 ?> ?>25@E=>AB8 B@C1:8
2
nI0a2 = ir . (56)
MB>< C@02=5=88 425 =58725AB=KE 25;8G8=K ?;>B=>ABL B>:0 8 @048CA B@C1:8.
>?>;=8< 53> C@02=5=85< A2O7K20NI8< <03=8B=>5 402;5=85
0i2
P<03. = (57)
2
2
-B> CB25@645=85 =5 O2;O5BAO 01A>;NB=> AB@>38<, => 2K?>;=O5BAO A 2KA>:>9 B>G=>ABLN.
39
A @048CA>< 45D>@<8@>20==>9 B@C1:8 2 A>AB>O=88 @02=>25A8O (7) rP<03. = f . 7 C@02=5=8O
nI0a2
(56) 2K@078< i = , ?>;CG8< 2K@065=8O 4;O 402;5=8O
2
r
2
2
2
# ś#
0i2 0 nI0a2 0(na2) I0
ś# ź#
P<03. = = =
2 4
ś# ź#
2 2 r 2 r
# #
8 ?>4AB028< 53> 2 C@02=5=85 (7)
2
2
0(na2) I0
= f (58)
2 r3
6.1 'B>1K >?@545;8BL B@51C5<>5 7=0G5=85 B>:0 2 MB> 2K@065=85 A;54C5B ?>4AB028BL
?@545;L=K5 7=0G5=8O @048CA0 rmax 8 <5E0=8G5A:>3> =0?@O65=8O fmax 8 >?@545;8BL
?@545;L=>5 7=0G5=85 B>:0 2 A>;5=>845
3
1 2 fmaxrmax
I0 max = =
na2 0
(59)
3
1 2 " 93,7 "(1,4 "10-2)
= H" 0,25 A
2
4Ą "10-7
1,0 "103 "1,0 "103 "(0,90 "10-2)
6.2 0==>5 2 CA;>288 7=0G5=8O A8;K B>:0 <5=LH5 <0:A8<0;L=> 2>7<>6=>3>, ?>MB><C 2
MB>< A;CG05 B@C1:0 =5 @07>@25BAO, => MB> 7=0G5=85 A@02=8<> A ?@545;L=K<. 0945<
@5H5=85 C@02=5=8O (58), 8A?>;L7CO 4;O <5E0=8G5A:>3> =0?@O65=8O 70:>= C:0.
1>A=>20==>ABL 53> ?@8<5=5=8O ?@>25@8< ?>A;5 ?>;CG5=8O @5H5=8O. MB><
?@81;865=88 C@02=5=85 (58) 8<55B 284
2
2
0(na2) I0 r - r0
= ł ,
2 r3 r0
8 O2;O5BAO C@02=5=85< G5B25@B>9 AB5?5=8, ?>MB><C 53> A;54C5B @5H0BL G8A;5==>. K@078<
@048CA B@C1:8 G5@57 53> >B=>A8B5;L=>5 C4;8=5=85 r = r0(1+ ). >4AB0=>2:0 MB>3>
2K@065=8O 2 C@02=5=85 405B
2
2
0(na2) I0 1
= .
2ł r03 3
(1+ )
'8A;5==>5 7=0G5=85 ?>AB>O==>3> 157@07<5@=>3> <=>68B5;O MB>3> C@02=5=8O @02=>
2
2
0(na2) I0
H" 0,26 . "0:8< >1@07><, ?>;CG05< C@02=5=85 2 G8A;0E
2ł r03
0,26
= (60)
3
(1+ )
3> <>6=> @5H8BL ;N1K< G8A;5==K< <5B>4>< (4065 ?>41>@><). 48=AB25==K9
*
?>;>68B5;L=K9 :>@5=L MB>3> C@02=5=8O H" 0,16. ! C4>2>;LAB285< >B<5G05<, GB> 2 MB><
480?07>=5 ?@81;865=85 70:>=0 C:0 4>?CAB8<>. :>=G0B5;L=>, =0E>48< @048CA
45D>@<8@>20==>9 B@C1:8
*
r1 = r0(1+ )H" 1,2A< (61)
40
Wyszukiwarka
Podobne podstrony:
3 etap 05 solutions3 etap 00 solutions3 etap 02 solutions3 etap 07 solutions3 etap 04 solutions3 etap 01 solutions3 etap 08 experimental solutions3 etap 07 experimental solutions3 etap 11 experimental solutions3 etap 10 experimental solutionsTech tech chem11[31] Z5 06 usrodki ochrony 06[1]06 (184)06Test II III etap VIII OWoUEwięcej podobnych podstron