5A?C1;8:0=A:0O
D878G5A:0O
>;8<?8040
(III MB0?)
2007 3>4
"5>@5B8G5A:89 BC@
5H5=8O 7040G
1
9 :;0AA.
040=85 1. 01>G0O @07<8=:0
;O B>3>, GB>1K C?@>AB8BL @01>BC A> AE5<>9 >F5=820=8O, 2 @5H5=88 G8A;0<8 2 :@C3;KE
A:>1:0E ?@>=C<5@>20=K =5 B>;L:> D>@<C;K, => 8 G8A;5==K5 >B25BK 8 8458 (:>B>@K5,
:@><5 B>3>, ?>4G5@:=CBK).
5H5=85.
1.1.1 0 4;8=5 ?0@0;;5;5?8?540, 2 :>B>@K9 A:;04K20NB ?;8B:8, C;>68BAO
NA = A/ a =10 NB = B / b =10
?;8B>:, =0 H8@8=5 ?;8B>: 8 =0 2KA>B5
NC = C / c = 20
?;8B>:. ;8B:8 1C4CB ;560BL 2 NC A;>52 ?> NANB ?;8B>: 2 :064><.
1) 01>B0 ?@>B82 A8;K BO65AB8 ?@8 ?>4J5<5 3@C70 <0AA>9 m =0 2KA>BC h @02=0
A* = mgh
. (1)
0
;O B>3>, GB>1K ?>;>68BL ?;8B:C 2 ?5@2K9 A;>9, =5>1E>48<> 2K=CBL 55 87
<>AB>2>9 8 ?>4=OBL =0 2KA>BC c , A>25@H82 @01>BC mgc . 'B>1K ?>;>68BL ?;8B:C 2>
2B>@>9 A;>9, 55 =04> ?>4=OBL =0 2KA>BC 2c , B.5. A>25@H8BL @01>BC 2mgc . =0;>38G=>, 4;O
B>3>, GB>1K ?>;>68BL ?;8B:C 2 A;>9 =><5@ n , =04> A>25@H8BL @01>BC nmgc .
01>B0, A>25@H05<0O ?@8 C:;04:5:
?5@2>3> A;>O A* = mgcNANB ,
1
2B>@>3> A;>O A* = 2mgcNANB ,
2
& & & & & & & & & & & &
n ->3> A;>O A* = nmgcNANB ,
n
& & & & & & & & & & & &
NC
>;=0O @01>B0 ?> C:;04:5 :8@?8G59 A* = A* + A* + ...+ A* = A* .
1 2 NC " n
n=1
NC (NC +1) mgc A B C C
A* = mgcNANB (1+ 2 + ...+ NC ) = mgcNANB =# +1ś#
(2).
ś# ź#
22 a b c c
# #
>4AB0=>2:0 G8A;5==KE 7=0G5=89 ?@82>48B : @57C;LB0BC
ABC(C + c) 2,0 " 2,0 "1,0 "1,05
A = mg = 10 "10 = 105:6 . (3)
2abc 2 " 0,20 " 0,20 " 0,05
1.1.1 (0;LB5@=0B82=K9 20@80=B). 8=8<0;L=0O @01>B0 @02=0 87<5=5=8N ?>B5=F80;L=>9
M=5@388 A8AB5<K. 53:> 70<5B8BL, GB> ?0@0;;5;5?8?54 A>AB>8B 87 F5;KE ?;8B>:. 3>
?>;=0O <0AA0 @02=0
ABC
M = m .
abc
C c
E>45 AB@>8B5;LAB20 HB015;O F5=B@ <0AA A8AB5<K ?>4=8<05BAO =0 2KA>BC "h = + ,
2 2
?>MB><C 87<5=5=85 ?>B5=F80;L=>9 M=5@388 (B> 5ABL A>25@H5==0O @01>B0) @02=>
ABC(C + c)
A = Mg"h = mg .
2abc
2
1.1.2. ;O B>3> GB>1K ?>;>68BL ;N1CN ?;8B:C 2 OI8:,
=04> ?>4=OBL 55 =04 1>@B8:>< (4), ?>MB><C :064CN
?;8B:C =04> ?>4=OBL =0 2KA>BC ! + A (2=87 8E >?CAB8B
!
A8;0 BO65AB8) 8 A>25@H8BL @01>BC mg(C + c) . A53>
?;8B>: NANBNC , ?>MB><C ?>;=0O @01>B0 1C45B @02=0
A
ABC
A* = NANBNCmg(C + c) = mg(C + c)
(5)
a b c
A* = 210000 6 = 210:6
(6)
1.1.3. 064K9 A;>9 ?8@0<84K 1C45B :204@0B=K<, ?>A:>;L:C =0 2KA>B5 ?8@0<84K
H L L
C;>68BAO NH = =10 ?;8B>: 8 =0 >158E AB>@>=0E >A=>20=8O C;>68BAO = =10
c a b
?;8B>:.
0 4;8=5 >A=>20=8O 1>3> A;>O C<5AB8BAO 10 ?;8B>:.
0 4;8=5 >A=>20=8O 2>3> A;>O C<5AB8BAO 9 ?;8B>:.
0 4;8=5 >A=>20=8O n ->3> A;>O C<5AB8BAO 11- n ?;8B>:.
n ->< A;>5 1C45B (11- n)2 ?;8B>:, (7)
?@8G5< :064CN 87 =8E =5>1E>48<> ?>4=OBL =0 2KA>BC nc , B>340 @01>B0 =0 C:;04:C ?;8B>:
n ->3> A;>O 1C45B @02=0
A* = mgcn(11- n)2
(8)
n
>;=0O @01>B0, :>B>@CN =04> 70B@0B8BL =0 C:;04:C ?8@0<84K, @02=0
NH NH NH NHNH NH
3 3 2
A* = A* =
.
" "mgcn(11- n)2 = mgc"(n - 22n2 +121n) = mgc("n - 22"n +121"n)
n
n=1 n=1 n=1 n=1 n=1 n=1
5>1E>48<K5 AC<<K ?@82545=K 2 CA;>288 7040G8, ?>MB><C
NH 2(NH +1)2 NH (NH +1)(NH + 2) NH (NH +1)
A* = mgc( - 22 +121 ) =
462
NH (NH +1) NH (NH +1) 22
Ą#
= mgc - (NH + 2) +121ń# =
ó# Ą#
22 3
Ł# Ś#
H H
Ą# ń#
(H +1) (H c +1)
22
c c c
ó#
= mgc - (H c + 2) +121Ą# (9)
22 3
ó# Ą#
Ł# Ś#
A* = 24200 6 = 24, 2:6
(10)
3
2.1. 0A>A <>65B A>74020BL 402;5=85 P = 5"103 0 . -B> >7=0G05B, GB> >= A<>65B ?>4=OBL
2>4C =0 2KA>BC =5 1>;LHCN, G5<
P
hmax = . (11)
g
>4=OBL =0 1>;LHCN 2KA>BC =5 ?>72>;8B 384@>AB0B8G5A:>5 402;5=85.
A;CG05 2.1.1 =0A>A A<>65B =0?>;=8BL 10AA59= 4> C@>2=O hmax , ?@8 MB>< >1J5<
70:0G0==>9 2>4K 1C45B @025=
P
S
V = ABhmax = AB
. (12)
g
p
;O B>3> GB>1K =09B8 @01>BC, A>25@H5==CN =0A>A><,
"x
@0AA<>B@8< =51>;LH>9 ?@><56CB>: 2@5<5=8, 70 :>B>@K9 =0A>A ?>
H;0=3C 70:0G0; <0;K9 >1J5< 2>4K "V = S"x , 345 S - ?;>I04L ?>?5@5G=>3> A5G5=8O
F = pS
H;0=30. 0 2>4C ?@8 MB>< 459AB2C5B A> AB>@>=K =0A>A0 A8;0 ,
"A* = F"x = pS"x = p"V
0 55 @01>B0 ?@8 MB>< @02=0 .
>;=0O @01>B0, :>B>@CN A>25@H8B =0A>A,
P ABP2
A* = P = PV = PAB =
(13)
""V
g g
A* =100000 6 =100:6
(14)
!B>8B >B<5B8BL, GB> @01>B0, A>25@H5==0O =0A>A>< 2 2 @070 1>;LH5 87<5=5=8O
?>B5=F80;L=>9 M=5@388 2>4K
hmax hmax AB gP2 ABP2
"W = Mg = ABhmax g = = .
22 22g2 2 g
5;> 2 B><, GB> =0 2>4C 2 H;0=35 459AB2C5B =5 B>;L:> A8;0 A> AB>@>=K =0A>A0,
=> 8 A8;0 384@>AB0B8G5A:>3> 402;5=8O A> AB>@>=K 2>4K, C65 70:0G0==>9 2 10AA59=.
02=>459AB2CNI0O MB8E A8; =5 @02=0 =C;N (AB0=>28BAO @02=>9 =C;N B>;L:>, :>340
2KA>B0 2>4K 2 10AA59=5 4>AB83=5B hmax ), ?>MB><C 2>40 2 H;0=35 42865BAO A CA:>@5=85<
8 @01>B0 =0A>A0 845B =0 C25;8G5=85 ?>B5=F80;L=>9 8 :8=5B8G5A:>9 M=5@388 2>4K. (15)
>40 2 10AA59=5 1C45B 42830BLAO, =>, 2 :>=F5 :>=F>2, CA?>:>8BAO 1;03>40@O A8;0<
2O7:>3> B@5=8O.
2.2. A;CG05 2) 70:0G:0 2>4K ?@>AB> =5 =0G=5BAO. > H;0=3C, ?5@51@>H5==><C G5@57 1>@B
10AA59=0, =04> ?>4=OBL 2>4C :0: <8=8<C< =0 2KA>BC C = 1< , 0 =0A>A <>65B ?>4=OBL 2>4C
=5 2KH5, G5< =0 hmax = 0,5< . (12)
!>>B25BAB25==>, =8:0:>9 @01>BK =0A>A A>25@H8BL =5 A<>65B, ?>MB><C
A* = 0 6
(16)
4
1.3 'B>1K A<5AB8 ?5A>: 2 ?8@0<84C, =5>1E>48<> A>25@H8BL =5 B>;L:> @01>BC ?@>B82
A8;K BO65AB8 A* , => 5IQ 8 @01>BC A* ?@>B82 A8;K B@5=8O, 459AB2CNI59 =0 ?5AG8=:8
B B@
?@8 A<5B0=88.
01>B0 ?@>B82 A8;K BO65AB8, 84CI0O =0 C25;8G5=85 ?>B5=F80;L=>9 M=5@388
?5AG8=>:, @02=0
A* = Mghc
, (18)
B
345 M - <0AA0 ?8@0<84K, hc - 2KA>B0 F5=B@0 <0AA ?8@0<84K >B >A=>20=8O.
2
1 H gL2H
A* = Vghc = SHg =
. (19)
B
3 4 12
0<5B8<, GB> @01>B0 A8;K B@5=8O ?@8 42865=88 ?>
=0:;>==>9 ?;>A:>AB8 ?>;=>ABLN >?@545;O5BAO
3>@87>=B0;L=K< A<5I5=85<. 59AB28B5;L=>, ?CABL
B5;> =0E>48BAO =0 =0:;>==>9 ?;>A:>AB8. ">340 A8;0
B@5=8O, 459AB2CNI0O =0 =53> @02=0
F = mg cosą .
">340 ?@8 A<5I5=88 B5;0 =0 @0AAB>O=85 "L 24>;L =0:;>==>9 ?;>A:>AB8 1C45B A>25@H5=0
@01>B0
A = F"L = mg"L cosą = mg"x (20)
8 =5 7028A8B >B C3;0 =0:;>=0. 1>1I0O 40==K9 @57C;LB0B, A;54C5B 70:;NG8BL, GB> ?@8
42865=88 ?> ?>25@E=>AB8 ;N1>3> ?@>D8;O @01>B0 A8;K B@5=8O (8;8 @02=0O 59 @01>B0
2=5H=59 A8;K ?> ?@5>4>;5=8N B@5=8O) ?>;=>ABLN >?@545;O5BAO 3>@87>=B0;L=K<
A<5I5=85< ?> D>@<C;5 (20), 5A;8, :>=5G=>, =5 CG8BK20BL CA:>@5=8O, <>3CI85 2>7=8:0BL
?@8 42865=88 B5;0.
0945< @01>BC, A>25@H05<CN ?@>B82 A8;K B@5=8O ?@8
h
A<5B0=88 ?5A:0. G5284=>, GB> 4;O A<5B0=8O ?5AG8=>:,
=0E>4OI8EAO =0 @07=>< @0AAB>O=88 >B F5=B@0 :@C30,
=5>1E>48<> A>25@H8BL @07=CN @01>BC. 07>1L5< :@C3 =0
k
R
h =
1>;LH>5 :>;8G5AB2> NR :>;5F B>;I8=>9
NR . (21)
rk = kh
>;LF> =><5@ k , AG8B0O >B F5=B@0, 1C45B 8<5BL @048CA
Sk = 2Ą rkh = 2Ą h2k
8 ?;>I04L .
0AA0 2A53> ?5A:0 2=CB@8 :@C30 @02=0 <0AA5 ?5A:0 2
?8@0<845
SH L2H
M = V = =
,
3 3
?@8G5< MB>B ?5A>: =0E>48BAO 2 :@C35 ?;>I04LN S = Ą R2 .
:>;LF5 =><5@ k =0E>48BAO ?5A>: <0AA>9
Sk L2H 2Ą h2k 2L2Hh2
mk = M = = k
(22)
S 3 Ą R2 3R2
@8 A<5B0=88 ?5A:0 87 :>;LF0 k =0 =53> 459AB2C5B A8;0 B@5=8O Fk = gmk , 0
<8=8<0;L=0O @01>B0 ?@>B82 =55 (5A;8 A<5B0BL ?> @048CAC) @01>B0 @02=0
2L2Hh2 2gL2Hh3
A* = gmkrk = g khk = k2
(23)
k
3R2 3R2
5
B<5B8<, GB> @048CA ?;>I04:8 7=0G8B5;L=> ?@52KH05B 4;8=C >A=>20=8O :CG8, ?>MB><C
<>6=> AG8B0BL, GB> 2A5 ?5AG8=:8 A<5B0NBAO 2 F5=B@ :@C30.
>;=0O @01>B0 ?> ?@5>4>;5=8N A8; B@5=8O ?@8 A<5B0=8N ?5A:0 : F5=B@C @02=0
NR NR
2gL2Hh3 2gL2Hh3 NR 2 2gL2Hh3 NR (NR +1)(NR + 2)
A* = A* = k2 =
. (24)
B@ " " "k =
k
3R2 3R2 k =1 3R2 6
k =1 k =1
"0: :0: <K @0718;8 :@C3 =0 >G5=L 1>;LH>5 :>;8G5AB2> :>;5F NR >> 1, B> 548=8F59 8
42>9:>9 ?> A@02=5=8N A NR <>6=> ?@5=51@5GL.
R
@><5 B>3>, =04> 2A?><=8BL, GB> h = , B>340
NR
3
2gL2H (h3 N ) 2gL2HR3 gL2HR
R
A* == =
(25)
B@
3R2 6 18R2 9
>;=0O @01>B0 ?> A<5B0=8N ?5A:0 2 ?8@0<84C @02=0
2
gL2H gL2HR gL2H H R
#ś#
A* =+ = +
ś#ź#
12 9 3 4 3
# #
A* = 820000 6 = 820:6
(26)
040=85 2. >4=0O D55@8O
2.1 @8 >B:@K20=88 :@KH:8 402;5=85 C?045B 4> =>@<0;L=>3> 0B<>AD5@=>3>, ?@8 MB><
2>40 >:065BAO ?5@53@5B>9 =0G=5BAO 2A:8?0=85, :>B>@>5 1C45B ?@>4>;60BLAO 4> B5E ?>@,
55 B5<?5@0BC@0 =5 ?>=878BAO 4> B5<?5@0BC@K :8?5=8O ?@8 =>@<0;L=>< 0B<>AD5@=><
402;5=88, B> 5ABL 4> t1 = 100C . @8 MB> B5?;>B0, 2K45;82H0OAO ?@8 >ABK20=88 2>4K
?>945B =0 8A?0@5=85 55 G0AB8 "m . #@02=5=85 B5?;>2>3> 10;0=A0 2 MB>< A;CG05 ?@8<5B
284
L"m = c1m(t0 - t1), (1)
87 :>B>@>3> ;53:> >?@545;8BL 4>;N 2K:8?52H59 2>4K
"m c1(t0 - t1) 4,2 "103 "(120 -100)
= = H" 3,8 "10-2 . (2)
m L 2,2 "106
=K<8 A;>20<8, 2K:8?8B >:>;> 4% 2>4K.
2.2 @8 :@8AB0;;870F88 2K45;O5BAO B5?;>B0, :>B>@0O @0AE>4C5BAO =0 =03@520=85
>AB02H59AO 2>4K. @>F5AA :@8AB0;;870F88 1C45B ?@>4>;60BLAO 4> B5E ?>@, ?>:0
B5<?5@0BC@0 2>4K =5 AB0=5B @02=>9 t1 = 0C . #@02=5=85 B5?;>2>3> 10;0=A0 2 40==><
A;CG05 ?@8=8<05B 284
"m = c1m(t1 - t0 ). (1)
B:C40 =0E>48<
"m c1(t1 - t0 ) 4,2 "103 "(0 - (- 5))
= = H" 6,3"10-2 . (2)
m 330 "103
@8<5G0=85: 0;>ABL 4>;8 2>4K, ?@5B5@?520NI59 D07>2K9 ?5@5E>4, ?>72>;O5B
?@5=51@5GL 87<5=5=85< B5?;>5<:>AB8 A<5A8 ?@8 D07>2KE ?@52@0I5=8OE.
6
2.3 7028A8<>AB8 >B :>;8G5AB20 2?CI5==>3> ?0@0 <>3CB @50;87>2K20BLAO @07;8G=K5
:>=5G=K5 A>AB>O=8O 2>4K 2 A>AC45: B>;L:> ;54, ;54 8 684:>ABL, 684:>ABL, 684:>ABL 8 ?0@.
>A;54>20B5;L=> @0AA<>B@8< 2>7<>6=K5 ?@>F5AAK 8 :>=5G=K5 @02=>25A=K5 A>AB>O=8O
?@8 C25;8G5=88 :>;8G5AB20 2?CI5==>3> ?0@0.
1. 0@ A:>=45=A8@>20;AO, >1@07>202H0OAO 2>40 >ABK;0 4> B5<?5@0BC@K 70<5@70=8O 8
G0AB8G=> 70<5@7;0 ?@8 MB>< B5<?5@0BC@0 ;L40 =5 4>AB83;0 B5<?5@0BC@K ?;02;5=8O.
MB>< A;CG05 C@02=5=85 B5?;>2>3> 10;0=A0 8<55B 284
Lm + c1m(t1 - t?;.)+ m + c0m(t?;. - tx ) = c0m0(tx - t0 ), (1)
745AL 8 40;55: m - <0AA0 2?CI5==>3> ?0@0, tx - B5<?5@0BC@0, CAB0=>282H0OAO 2 A>AC45
?>A;5 CAB0=>2;5=8O B5?;>2>3> @02=>25A8O, t?;. = 0,0C - B5<?5@0BC@0 ?;02;5=8O ;L40.
7 C@02=5=8O (1) =0E>48< B@51C5<CN 7028A8<>ABL:
Lm + c1m(t1 - t?;.)+ m + c0mt?;. + c0m0t0
tx = . (2)
c0m + c0m0
>4AB02;OO G8A;5==K5 7=0G5=8O E0@0:B5@8AB8: 2>4K (C4>1=> B5?;>BC 87<5@OBL 2 :6, 0
<0AAK 2 3@0<<0E), ?>;CG8< DC=:F8N
Lm + c1m(t1 - t?;.)+ m + c0mt?;. + c0m0t0 (2200 + 4,2 "100 + 330)m + 2,1" 300 "(-10)
tx = = =
c0m + c0m0 2,1"(300 + m)
(3)
2950m - 6300
=
6300 + 2,1m
6300
"5<?5@0BC@0 ;L40 4>AB83=5B =C;O, ?@8 <0AA5 2?CI5==>3> ?0@0 @02=>9 m1 = H" 2,13 .
2950
"0: :0: MB0 <0AA0 <0;0, B> 7028A8<>ABL (3) O2;O5BAO ?@8<5@=> ;8=59=>9. -B>B CG0AB>: =0
3@0D8:5 >1>7=0G5= 0-1.
2. >;8G5AB2> ?0@0 ?@52KA8;> =0945==>9 7=0G5=85 m1 H" 2,13 . @8 MB>< ;54 =0G0;
?;028BLAO, => ?0@0 =5 E20B05B, GB>1K @0A?;028BL 25AL ;54. MB>< A;CG05 2 A>AC45 2
A>AB>O=88 @02=>25A8O >:065BAO A<5AL ;L40 8 2>4K, =0E>4OI0OAO ?@8 B5<?5@0BC@5
t?;. = 0,0C . 0945< <0AAC ?0@0 m2 , ?@8 :>B>@>9 :>;8G5AB2> B5?;>BK, 2K45;82H59AO ?@8
:>=45=A0F88 ?0@0 8 >ABK20=88 >1@07>202H59AO 2>4K, 1C45B 4>AB0B>G=>, GB>1K =03@5BL
;54 4> B5<?5@0BC@K ?;02;5=8O 8 ?>;=>ABLN 53> @0A?;028BL. 7 C@02=5=8O B5?;>2>3>
10;0=A0
Lm2 + c1m2(t1 - t?;.)= c0m0(t?;. - t0)+ m0 (4)
=0E>48<
c0m0(t?;. - t0)+ m0 2,1"300"10 + 330"300
m2 = = H" 40 3 . (5)
L + c1(t1 - t?;.) 2200 + 4,2"100
"0:8< >1@07><, ?@8 <0AA5 ?0@0 >B m1 H" 2,13 4> m2 H" 40 3 B5<?5@0BC@0 CAB0=>282H0OAO 2
A>AC45 1C45B @02=0 t?;. = 0,0C (CG0AB>: 1-2 =0 3@0D8:5).
3. 5AL ;54 @0A?;028;AO, >1@07>202H0OAO ?@8 MB>< 2>40 AB0;0 =03@520BLAO. MB>< A;CG05
B5?;>2>9 10;0=A 8<55B 284: B5?;>B0, 2K45;82H0OAO ?@8 :>=45=A0F88 ?0@0 8 >ABK20=88
>1@07>202H59AO 2>4K, @0AE>4C5BAO =0 =03@520=85 ;L40, 53> ?;02;5=85 8 =03@520=85 B0;>9
2>4K 4> @02=>25A=>9 B5<?5@0BC@K tx , 8;8 =0 O7K:5 C@02=5=8O:
Lm + c1m(t1 - tx ) = c0m0(t?;. - t0 )+ m0 + c1m0(tx - t?;.). (6)
7 MB>3> C@02=5=8O >?@545;O5<
7
Lm + c1mt1 - c0m0(t?;. - t0)- m0
tx = =
c1m + c1m0
(7)
2200m + 4,2"100m - 2,1"300"10 - 330"300 2620m -105300
= =
4,2m + 4,2"300 4,2m +1260
-B> CG0AB>: =0 3@0D8:5 >1>7=0G5= 2-3. >=5G=0O B5<?5@0BC@0 4>AB83=5B B5<?5@0BC@K
:>=45=A0F88 t1 = 100C , 5A;8 <0AA0 2?CA:05<>3> ?0@0 ?@52KA8B 7=0G5=85 m3 , :>B>@>5
B0:65 <>6=> >?@545;8BL 87 C@02=5=8O (6), 2 :>B>@>< A;54C5B ?>;>68BL tx = t1 :
Lm3 = c0m0(t?;. - t0)+ m0 + c1m0(t1 - t?;.). (8)
;8
Lm3 = c0m0(t?;. - t0)+ m0 + c1m0(t1 - t?;.)
c0m0(t?;. - t0)+ m0 + c1m0(t1 - t?;.)
m3 = = (9)
L
2,1"300"10 + 330"300 + 4,2"300"100
= H"105 3
2200
@8 40;L=59H5< C25;8G5=88 <0AAK ?0@0 :>=5G=0O B5<?5@0BC@0 =5 ?@52KA8B t1 = 100C .
"@51C5<K9 3@0D8: ?>:070= =0 @8AC=:5.
B<5B8<, GB> =0:;>==K5 CG0AB:8, AB@>3> 3>2>@O, =5 ?@O<>;8=59=K. 4=0:> MB8
>B:;>=5=8O =57=0G8B5;L=K.
040=85 3. ?KB @03>
1. 5>1E>48<>, GB>1K ;CG8 >B@06Q==K5 >B 75@:0;LF0 1, ?>?0;8 =0 75@:0;> 2 . -B>
2>7<>6=> ?@8
90o -
< < 45o (1),
2
40 < < 45o (2),
8, B.:. 75@:0;LF5 42CEAB>@>==55,
90o -
+180o < < 45o +180o (3),
2
220 < < 225o (4).
8
2. %>4 ;CG59 ?>:070= =0 @8AC=:5.
>A;5 >B@065=8O >B AD5@8G5A:>3> 75@:0;0 ;CG8 84CB ?>4 C3;><
d
= (5)
R
?> >B=>H5=8N : ?5@2>=0G0;L=><C =0?@02;5=8N. >A;5 2B>@8G=>3> >B@065=8O >B 75@:0;LF0,
?CG>: @0AE>48BAO ?>4 C3;>< @02=K< 2 . ">340 480<5B@ ?OB=0 =0 M:@0=5
d
D = d + 2 L = 25<< (6)
R
3. @8 2@0I5=88 75@:0;LF0, 70 2@5<O , =5>1E>48<>5
A25BC 4;O B>3>, GB>1K A;5B0BL >B 75@:0;LF0 1 4> 75@:0;0 2 8
>1@0B=>, 75@:0;LF5 ?>25@=QBAO =0 C3>; . 075@=K9 ;CG
?>25@=QBAO =0 C3>; 2 8 ?OB=> A<5AB8BLAO =0 =5:>B>@>5
@0AAB>O=85. 0?@02;5=85 A<5I5=8O 7028A8B >B =0?@02;5=8O
2@0I5=8O 75@:0;LF0. 5;8G8=0
R
= 2 (7),
c
B>340
= 2Ą " " (8).
5;8G8=0 A<5I5=8O:
R
x = 2L = 8Ą " " L = 8,4<< (9).
c
4. @8 2@0I5=88 75@:0;LF0 ?OB=> 1C45B A>AB>OBL 87 42CE >48=0:>2KE ?OB5=, F5=B@K
:>B>@KE =5 A>2?040NB. @8 C25;8G5=88 A:>@>AB8 2@0I5=8O @0AAB>O=85 <564C F5=B@0<8
?OB5= 1C45B C25;8G820BLAO. OB=0 1C4CB A42830BLAO 2 >4=C 8 BC 65 AB>@>=C.
5. OB=0 1C4CB @0745;5=K, 5A;8 @0AAB>O=85 <564C 8E F5=B@0<8 AB0=5B @02=> 480<5B@C
D .
R
2
"x = 8Ą " " L "(n -1) = D (10).
c
B:C40 8 ?>;CG05< 7=0G5=85 G0AB>BK 2@0I5=8O:
D " c
2
= = 5,0 "103 >1 (11).
A
8Ą(n -1)" R " L
9
10 :;0AA.
040=85 1. @>AB> @07<8=:0
V0 <
1.1 G5284=>, GB> A:>@>ABL B@0:B>@0 @02=0 V = = 10 .
,
2 A
1.2 0;L=>ABL ?>;5B0 A=0@O40, 2K?CI5==>3> 87 =5?>4286=>3> >@C48O, >?@545;O5BAO
D>@<C;>9
2V02
S0 = siną cosą . (1)
g
345 V0 - A:>@>ABL A=0@O40 >B=>A8B5;L=> AB2>;0.
'B>1K 4>AB8GL <0:A8<0;L=>9 40;L=>AB8, 2KAB@5;, >G5284=>, =5>1E>48<> ?@>872>48BL 2
=0?@02;5=88 42865=8O ?;0BD>@<K. MB>< A;CG05 : 3>@87>=B0;L=>9 A>AB02;ONI59
A:>@>AB8 A=0@O40 >B=>A8B5;L=> ?CH:8 4>1028BAO A:>@>ABL 42865=8O ?;0BD>@<K, ?>MB><C
70:>= 42865=8O A=0@O40 1C45B 8<5BL 284
x = (V0 cosą + v1)t
(2)
gt2 .
y = V0t siną -
2
2V0 siną
?@545;8< 2@5<O ?>;5B0 A=0@O40 (87 CA;>28O y = 0 ): t1 = 8 ?>4AB028< 53> 2
g
2K@065=85 4;O 3>@87>=B0;L=>9 :>>@48=0BK 4;O >?@545;5=8O 40;L=>AB8 ?>;5B0
2V02 siną cosą # ś#
v1
0
ś#
S1 = (V0 cosą + v1)2V siną = .
ś#1+ V0 cosą ź#
ź#
g g
# #
"0:8< >1@07><, >B=>A8B5;L=>5 C25;8G5=85 40;L=>AB8 ?>;5B0 A>AB028B
S1 - S0 v1
= = 10%
S0 V0 cosą
@8 =0;8G88 ?>=8<0=8O, @5H5=85 <>65B 1KBL ?>;CG5=> ?>GB8 <3=>25==>:
- B0: :0: 2@5<O ?>;5B0 A=0@O40 >?@545;O5BAO B>;L:> 25@B8:0;L=>9 A>AB02;ONI59
A:>@>AB8, B> >=> =5 7028A8B >B A:>@>AB8 ?;0BD>@<K;
- 40;L=>ABL ?>;5B0 (?@8 ?>AB>O==>< 2@5<5=8 42865=8O) ?@>?>@F8>=0;L=0
3>@87>=B0;L=>9 A>AB02;ONI59 A:>@>AB8 A=0@O40;
S1 - S0 (V0 cosą + v1)-V0 cosą v1
- A;54>20B5;L=>, = = = 10% .
S0 V0 cosą V0 cosą
1.3 'B>1K A428=CBL F8;8=4@, : =5<C =5>1E>48<> ?@8;>68BL A8;C, ?@52KH0NICN A8;C
B@5=8O.
7>1@078< A8;K, 459AB2CNI85 =0 F8;8=4@ 2 ?@>5:F88 =0
25@B8:0;L=CN ?;>A:>ABL. "0: :0: A8;K @50:F88 =0?@02;5=K
?5@?5=48:C;O@=> ?>25@E=>AB8, B> B@>9:0 A8;: 425 A8;K @50:F88 8
A8;0 BO65AB8 =0?@02;5=K ?>4 @02=K<8 C3;0<8 120 4@C3 : 4@C3C.
"0: ?@>5:F8O CA:>@5=8O =0 25@B8:0;L=CN ?;>A:>ABL @02=0 =C;N,
B> AC<<0 87>1@065==KE A8; B0:65 @02=0 =C;N. !;54>20B5;L=>,
<>4C;8 MB8E A8; @02=K, B> 5ABL N = mg . "0:8< >1@07><,
AC<<0@=0O A8;0 B@5=8O (8 8A:><0O A8;0) @02=0
F @ = 2N = 2mg =12H .
10
1.4 !C<<0@=0O A8;0 402;5=8O 2>4K =0 4=> 8 AB5=:8 A>AC40 ?> <>4C;N @02=0 A8;5 BO65AB8
8 =0?@02;5=0 2 ?@>B82>?>;>6=CN AB>@>=C: ghS + FAB. = mg . 7 MB>3> C@02=5=8O =0E>48<
FAB. = mg - ghS H" -10H , B.5. MB0 A8;0 =0?@02;5=0 225@E.
1.5 1>7=0G8< =0G0;L=CN A:>@>ABL 2B>@>3> H0@8:0 u0 8 =0945< A:>@>AB8 H0@8:>2 v1 8
v2 ?>A;5 ?5@2>3> AB>;:=>25=8O ( 8A.1), 8A?>;L7CO C@02=5=8O 70:>=>2 A>E@0=5=8O
8<?C;LA0 8 <5E0=8G5A:>9 M=5@388
m2u0 = m1v1 + m2v2 , (1)
2 2 2
m2u0 m1v1 m2v2
= + . (2)
2 2 2
;O 1KAB@>3> @5H5=8O MB>9 A8AB5<K C@02=5=89 ?5@5?8H5< 55
2 2845
m2(u0 - v2)= m1v1 , (3)
2 2 2
m2(u0 - v2 )= m1v1 . (4)
8 @0745;8< 2B>@>5 C@02=5=85 =0 ?5@2>5. @8 MB>< <K, :>=5G=>,
B5@O5< >4=> @5H5=85 A8AB5<K
v2 = u0, v1 = 0 , (5)
=> MB> @5H5=85 A>>B25BAB2C5B =0G0;L=K< CA;>28O< A:>@>ABO< H0@8:0 4> C40@0, 0 =0<
=5>1E>48<> 2B>@>5 @5H5=85:
u0 + v2 = v1 .
>4AB0=>2:0 MB>3> A>>B=>H5=85 2 C@02=5=85 (1) ?>72>;O5B ;53:> =09B8 8=B5@5ACNI55 =0A
@5H5=85
m2 - m1 2m2
v2 = u0, v1 = u0 . (6)
m2 + m1 m2 + m1
>=OB=>, GB> A:>@>ABL ?5@2>3> H0@8:0 1>;LH5 A:>@>AB8 2B>@>3>,
?>MB><C, A>25@H82 ?>;=K9 >1>@>B, ?5@2K9 H0@8: 4>3>=8B
2B>@>9 2 =5:>B>@>9 B>G:5 A1 , ?>25@=CB>9 >B=>A8B5;L=> 8AE>4=>9
=0 =5:>B>@K9 C3>; ( 8A. 1).
;O >?@545;5=8O MB>3> C3;0 70<5B8<, GB> 4> 2B>@>3> C40@0
?5@2K9 H0@8: ?@>945B ?CBL
S1 = (2Ą +)R (7)
70 2@5<O
(2Ą +)R
t1 = . (8)
v1
=0;>38G=>, 4;O 2B>@>3> H0@8:0 <>6=> 70?8A0BL
R
S2 = R, t2 = . (9)
v1
@8@02=820O 2@5<5=0 42865=8O H0@8:>2, ?>;CG8< C@02=5=85
(2Ą +)R R
= . (10)
v1 v2
87 :>B>@>3> >?@545;8< 8A:><K9 C3>; , 7040NI89 ?>;>65=85 B>G:8 AB>;:=>25=8O
2Ą
= . (11)
v1
-1
v2
KG8A;8< 7=0G5=85 MB>3> C3;0, 8A?>;L7CO 2K@065=8O 4;O A:>@>AB59 (6)
11
2Ą 2Ą 2Ą 2
= = = H" Ą . (12)
v1 2m2 2"30
5
-1 -1 -1
v2 m2 - m1 30 - 20
"0:8< >1@07><, <K >?@545;8;8 B>G:C 2B>@>3> AB>;:=>25=8O. ;O >?@545;5=8O A:>@>AB59
H0@8:>2 ?>A;5 MB>3> C40@0, A;54C5B A>AB028BL A8AB5<C C@02=5=89, 0=0;>38G=CN (1)-(2).
4=0:> ?>4@>1=> 55 @5H0BL =5B =5>1E>48<>AB8, B0: :0: >B25B ?>GB8 >G52845=: 5A;8 ?@8
=0G0;L=KE CA;>28OE (5) @5H5=85< 2K@060NBAO D>@<C;0<8 (6), B> B5?5@L (6) O2;ONBAO
=0G0;L=K<8 CA;>28O<8, A;54>20B5;L=>, @5H5=85 2K@0605BAO D>@<C;0<8 (5). =K<8
A;>20<8, ?>A;5 2B>@>3> C40@0 H0@8:8 2>AAB0=>2OB A2>8 =0G0;L=K5 A:>@>AB8 2B>@>9
1C45B 42830BLAO A> A:>@>ABLN u0 , 0 ?5@2K9
>AB0=>28BAO! >A;5 2B>@>3> C40@0 2B>@>9 H0@8:
A45;05B ?>;=K9 >1>@>B 8 AB>;:=5BAO A ?5@2K< 2 B>9
65 B>G:5 A1 . >A;5 MB>3> ?>2B>@8BAO @0AA<>B@5==0O
A8BC0F8O (B>;L:> ?>25@=CB0O =0 C3>; ): ?5@2K9
4>3>=O5B 2B>@>9 8 B.4. -B8 @0AAC645=8O ?>72>;ONB
C:070BL B>G:8 2A5E AB>;:=>25=89 H0@8:>2 ( 8A.3) .
@>ABK< ?5@5G8A;5=85< ;53:> >?@545;8BL, GB>
B@8=04F0B>5 AB>;:=>25=85 ?@>87>945B 2 B>G:5,
?>25@=CB>9 >B=>A8B5;L=> 8AE>4=>3> ?>;>65=8O =0
C3>;, @02=K9
1
13 = Ą H" 72 .
5
12
040=85 2. 5A 8 A60B85
1. !0<>A60B85
1.1 0AA<>B@8< A;>9 2>4K <0;>9 B>;I8=K " hi , =0E>4OI89AO =0 3;C18=5 hi .
#<5=LH5=85 >1J5<0 "Vi @0AA<0B@8205<>3> A;>O 2A;54AB285 A68<05<>AB8 684:>AB8
70?8H5< :0:
"Vi = V0(1- " p )-V0 = -V0 p = -S"hi p ,
345 S ?;>I04L @0AA<0B@8205<>3> A;>O.
;O >F5=:8 1C45< AG8B0BL, GB> ?;>B=>ABL 684:>AB8 >AB05BAO ?@81;878B5;L=>
?>AB>O==>9, B>340 B>;I8=0 A60B>3> A;>O " h*
i
"h* = "h(1- " p) = "h(1- " 0ghi ),
i i i
345 0 ?;>B=>ABL =5A60B>9 684:>AB8, g CA:>@5=85 A2>1>4=>3> ?045=8O.
!>>B25BAB25==>, C<5=LH5=85 hi 53> 2KA>BK 2A;54AB285 MDD5:B0 A0<>A60B8O
=0945< :0:
hi = 0g hi"hi (1)
;O =0E>645=8O ?>;=>3> A60B8O "H 2A59 684:>AB8 ?@>AC<<8@C5< (1) ?> 2A5<
3;C18=0< hi
2
"H = hi = g hi"hi = 0g "hi =0g . (2)
" " "h H
0 i
2
ii i
0AG5B 4;O <8@>2>3> >:50=0 405B
"H = 238< .
1.2 ;>B=>ABL ! CG5B>< MDD5:B0 A0<>A60B8O =0945< ?;>B=>ABL @50;L=>9
684:>AB8 =0 3;C18=5 h
mi mi mi
(h) = = = H" 0(1+ 0gh) . (3)
Vi V0(1- pi ) V0(1- 0gh )
(2) CGB5=>, GB> ?0@0<5B@ (A68<05<>ABL 2>4K) >G5=L <0;, ?>MB><C <>6=> A
4>AB0B>G=>9 B>G=>ABLN 8A?>;L7>20BL ?@81;865=85 4;O <0;KE x
1
H"1+ x .
1- x
!>>B25BAB25==>, >B=>A8B5;L=>5 C25;8G5=85 ?;>B=>AB8 2>4K 2 ?@>F5=B0E =0
3;C18=5 H =0945< :0:
"
= "100% = 0gH "100% = 4,76% . (4)
0
>A:>;L:C 40==K5 2 CA;>288 7040G8 ?@82545=K A B>G=>ABLN 4> B@5E 7=0G0I8E
F8D@, B> 2 >B25B5 (4) (8 40;55!) <K B0:65 1C45< >AB02;OBL B@8 7=0G0I85 F8D@K.
0: 2848< 87 (4) 4065 ?@8 1>;LH8E 3;C18=0E <8@>2>3> >:50=0 >B=>A8B5;L=>5
87<5=5=85 ?;>B=>AB8 <>@A:>9 2>4K =57=0G8B5;L=> 8 A>AB02;O5B =5A:>;L:> ?@>F5=B>2. -B>
?>72>;O5B AG8B0BL 684:>AB8 ?@0:B8G5A:8 =5A68<05<K<8 ?@8 @5H5=88 @O40 ?@8:;04=KE
7040G.
1.3 02;5=85 ;O 2KG8A;5=8O 402;5=8O p(h) @50;L=>9 684:>AB8 =0 3;C18=5 h
=5>1E>48<> ?@>AC<<8@>20BL 402;5=8O, A>740205<K5 1>;55 2KA>:8<8 B>=:8<8 A;>O<8
"hi , :>B>@K5, A>3;0A=> 70:>=C 0A:0;O, ?5@540NBAO =86=8< A;>O<
13
p(h) = pi = g"hi = g "hi , (5)
" " "
i i
i i i
345 i ?;>B=>ABL 684:>AB8 =0 3;C18=5 i - 3>
A;>O.
!C<<0 2 2K@065=88 (4), 5ABL ?;>I04L S ?>4
3@0D8:>< (h) 7028A8<>AB8 (2) ?;>B=>AB8 684:>AB8 >B
3;C18=K, B.5. ?;>I04L B@0?5F88 (@8A. 02).
!;54>20B5;L=>
0 + 0(1+ 0gh ) 0g
p(h) = h = 0gh(1+ h). (6)
8A. 02
22
7 (6) A;54C5B, GB> 8A:><0O 7028A8<>ABL
?@54AB02;O5B A>1>9 ?0@01>;C A <0;K< :>MDD8F85=B>< =5;8=59=>AB8.
! CG5B>< B>3>, GB> 0B<>AD5@=>5 402;5=85 p0 B0:65 ?5@5405BAO ?> 70:>=C 0A:0;O,
>:>=G0B5;L=> ?>;CG8<
0g
p(h) = p0 + 0gh(1+ h) . (7)
2
1.4 #B>=C2H89 ;5BCG89 3>;;0=45F @54<5B ?5@5AB0=5B B>=CBL 8 1C45B
=0E>48BAO 2 A>AB>O=88 CAB>9G82>3> @02=>25A8O =0 =5:>B>@>9 3;C18=5 ?@8 CA;>288, GB>
53> ?;>B=>ABL 1C45B @02=0 ?;>B=>AB8 <>@A:>9 2>4K =0 MB>9 3;C18=5.
!;54>20B5;L=>, ?;>B=>ABL CB>=C2H53> ;5BCG53> 3>;;0=4F0 <>6=> =09B8, ?>4AB0282
2 2K@065=85 (3) 8A:><CN 3;C18=C h = 5,00:<
(h) = 0(1+ 0gh).
0AG5B A B>G=>ABLN 4> B@5E 7=0G0I8E F8D@ 405B
:3
1 = 105"103 .
,
<3
2. 0@O65==0O 684:>ABL 40==>< ?C=:B5 @0AA<0B@8205BAO =5A68<05<0O
684:>ABL.
2.1 0AA<>B@8< 25@E=89 A;>9 684:>AB8 3;C18=>9 h (@8A. 03). 59AB2CNI0O =0
=53> A25@EC A8;0 0B<>AD5@=>3> 402;5=8O p0S 8 A8;0
53> BO65AB8 F1 = ghS C@02=>25H820NBAO A8;>9
402;5=8O A=87C p(h)S 8 A8;>9 M;5:B@>AB0B8G5A:>3>
r
>BB0;:820=8O F2 A;>O B>;I8=>9 h A;>5< B>;I8=>9
( H - h ).
0?@O65==>ABL M;5:B@>AB0B8G5A:>3> ?>;O,
A>740205<>3> A;>5< B>;I8=>9 ( H - h ) 2=5 53> <>65B
1KBL =0945=0, :0:
ł ( H - h)
E( H - h ) = = .
20 20
8A. 03
0@O4 A;>O B>;I8=>9 h =0945< ?> >?@545;5=8N >1J5<=>9 ?;>B=>AB8 70@O40
q( h ) = ł "V = ł hS .
!;54>20B5;L=>, A8;0 M;5:B@>AB0B8G5A:>3> >BB0;:820=8O 25@E=53> A;>O =86=8<
@02=0
14
( H - h)
2
F2 = E( H - h )" q( h ) = ł hS . (8)
20
@57C;LB0B5 CA;>285 @02=>25A8O A;>O 3;C18=>9 h ?@8<5B 284
( H - h)
2
p0S + ghS = ł hS + p( h )S .
20
!>:@0I0O 2 ?>A;54=5< 2K@065=88 =0 ?;>I04L S A;>O, ?>;CG8< 8A:><CN
7028A8<>ABL 402;5=8O 2 70@O65==>9 684:>AB8 >B 3;C18=K
22
ł H ł
p(h) = p0 + ( g - )h + h2 . (9)
20 20
2.2 K@065=85 (8) ?@54AB02;O5B A>1>9 :204@0B8G=CN 7028A8<>ABL (?0@01>;C),
25B28 :>B>@>9 =0?@02;5=K 225@E.
@8 <0;KE h ( h 0 ) 55 ?>2545=85 >?@545;O5BAO ;8=59=K< G;5=><, :>B>@K9, 2
7028A8<>AB8 >B B>;I8=K ?;0AB8=K H <>65B 1KBL :0: ?>;>68B5;L=K<, B0: 8
>B@8F0B5;L=K<. @8 <0:A8<0;L=> 2>7<>6=>9 B>;I8=5 ?;0AB8=K Hmax ;8=59=K9 G;5=
4>;65= 1KBL @025= =C;N.
@8 MB>< 25@E=85 A;>8 684:>AB8 AB0=CB =525A><K, B.5. p(h 0 ) H" p0 . BAN40
=0945<
2
ł H 2g0
g =! Hmax = . (10)
2
20 ł
@8 40;L=59H5< C25;8G5=88 B>;I8=K 70@O65==>3> A;>O 25@E=85 G0AB8FK
684:>AB8 1C4CB >B@K20BLAO 8 C;5B0BL 225@E, >1@07CO M;5:B@8G5A:>5 8A?0@5=85.
2.3 C18: =5:>B>@>9 <0AAK, ?>3@C65==K9 2 70@O65==CN 684:>ABL, 1C45B
=0E>48BLAO 2 @02=>25A88, 5A;8 A8;0 53> BO65AB8 1C45B C@02=>25H5=0 A8;>9 @E8<540.
>3@C65=85 :C18:0 2 70@O65==CN 684:>ABL =5 87<5=8B @0A?@545;5=8O 55 402;5=8O
2 =865;560I8E A;>OE.
-B> ?>72>;O5B 70?8A0BL A;54CNI55 C@02=5=85
22
ł H ł
mg = (( g - )a + a2 )" a2 .
20 20
BAN40 =0945< 8A:><>5 7=0G5=85 <0AAK :C18:0
22
ł H ł
(( g - )a + a2 )" a2
20 20
m = . (11)
g
15
040=85 3. AB>@>6=59 =0 ?>2>@>B0E
1. 53:> 70<5B8BL 87 @8AC=:0 1, GB> =0;8G85 C3;0 C2>40 C
?5@54=8E :>;QA C<5=LH05B C3>; ?>2>@>B0, 0 C 704=8E
C25;8G8205B. ".5.
= - + (1)
2. 03@C7:8 =0 :>;QA0 @02=K A8;0< @50:F88 4>@>38 =0 MB8
:>;QA0. ;O =0E>645=8O MB8E 25;8G8= =5>1E>48<> 70?8A0BL
2B>@>9 70:>= LNB>=0 2 ?@>5:F88 =0 25@B8:0;L=CN >AL, 0
B0:65 C@02=5=8O 4;O @025=AB20 <><5=B>2 >B=>A8B5;L=>
=5A:>;L:8E >A59, GB>1K ?>;CG8BL G5BK@5 =57028A8<KE
C@02=5=8O. 4=0:> <>6=> ?>ABC?8BL ?@>I5. A;8
02B><>18;L ?>:>8BAO, B> A8;K @50:F88 >?@545;8BL =5
8A. 1
B@C4=>:
1 Mgb
0 0
N = N =
2 L
(2).
1 Mg0
0 0
N = N =
2 L
@8 ;52>< ?>2>@>B5, : ?@02K< :>;QA0< ?@8;>68BAO 4>?>;=8B5;L=0O =03@C7:0
1 Mv2 h
"N = , 0 A8;0, 459AB2CNI0O =0 ;52K5 :>;QA0, =0>1>@>B C<5=LH8BAO =0 MBC
2 R d
25;8G8=C. >;CG8<:
1 Mgb 1 Mv2 h
N = -
2 L 2 R d
1 Mgb 1 Mv2 h
N = +
2 L 2 R d
(3).
1 Mg0 1 Mv2 h
N = -
2 L 2 R d
1 Mg0 1 Mv2 h
N = +
2 L 2 R d
3. 0?8H5< CA;>285 @025=AB20 <><5=B>2 A8; >B=>A8B5;L=> 25@B8:0;L=>9 >A8, ?@>E>4OI59
G5@57 F5=B@ BO65AB8 02B><>18;O:
(F + F )" a = (F + F )" b (4),
B.5. 02B><>18;L =5 2@0I05BAO. >4AB02;OO 7=0G5=8O A8;, ?>;CG8<:
(k N + k N )" a = (k N + k N )" b (5)
8;8
" a "(N + N ) = " b "(N + N ) (6).
>4AB0282 7=0G5=8O =03@C7>:, ?>;CG8<:
= (7),
0 7=0G8B
= (8)
=59B@0;L=0O ?>2>@0G8205<>ABL.
4. 0?8H5< 2B>@>9 70:>= LNB>=0 2 ?@>5:F88 =0 >AL F5=B@ BO65AB8 02B><>18;O F5=B@
:@8287=K ?>2>@>B0:
Mv2
F + F + F + F = (9)
R
8;8, ?>4AB02;OO 7=0G5=8O A8;, 8 CG8BK20O, GB> = = :@ ,
16
Mv2
kMg:@ = (10).
R
0:A8<0;L=0O A:>@>ABL @02=0
vmax = k:@ Rg (11).
5. 0AA<>B@8< 42865=85 2 ;52>< ?>2>@>B5. @8 =0;8G88 AE>645=8O ?5@54=8E :>;QA, C3;K
C2>40 ;52>3> 8 ?@02>3> :>;QA >B;8G0NBAO >B =5:>B>@>3> A@54=53> 7=0G5=8O =0
25;8G8=C , B.5.
= +
(12).
= -
0?8H5< 2B>@>9 70:>= LNB>=0 2 ?@>5:F88 =0 >AL F5=B@ BO65AB8 02B><>18;O F5=B@
:@8287=K ?>2>@>B0 8 CA;>285 @025=AB20 <><5=B>2 A8; >B=>A8B5;L=> 25@B8:0;L=>9 >A8,
?@>E>4OI59 G5@57 F5=B@ BO65AB8 02B><>18;O:
ż#
#kN - kN ( + k(N + N Mv2
( )+ )+ )=
(13).
#
R
#
(kN ( - )+ kN ( + ))" a = k(N + N )" b
#
5H5=85 A8AB5<K:
v2 v2 v2hL
= 8 = - (14).
gkR gkR Rdgb
#3>; ?>2>@>B0:
v2hL
= - + = + (15).
Rdgb
L
5>1E>48<> CG5ABL, GB> R = , B>340
v2h
= + (16).
dgb
K@060O , ?>;CG8<:
= (17).
v2h
1-
dgb
5;05< 2K2>4: ?@8 ?>;>68B5;L=>< > 02B><>18;L >1;0405B 871KB>G=>9
?>2>@0G8205<>ABLN, ?@8 < 0 =54>AB0B>G=>9.
6. @8 >?@545;Q==>9 A:>@>AB8 42865=8O 7=0<5=0B5;L 2 D>@<C;5 (17) >1@0I05BAO 2 =C;L.
-B> 7=0G8B, GB> ?@8 <0;59H5< 42865=88 @C;O, C3>; ?>2>@>B0 AB0=>28BAO >G5=L 1>;LH8< 8
02B><>18;L @072>@0G8205BAO. =0G5=85 :@8B8G5A:>9 A:>@>AB8:
dgb
vcrit = (18).
" h
7. >-?@56=5<C @0AA<0B@8205< ;52K9 ?>2>@>B. 5>1E>48<> >?@545;8BL, :0:>5 :>;5A>
?5@2K< ?>B5@O5B AF5?;5=8O A 4>@>3>9. @54?>;>68<, GB> MB> 1C4CB 704=85 :>;QA0. ;O
B>3> GB>1K MB> ?@>87>H;>, =5>1E>48<>, GB>1K
v2 = :@ gkR (19).
MB>< A;CG05 7=0G5=85 <0:A8<0;L=>9 A:>@>AB8 A>2?0405B A> 7=0G5=85<, ?>;CG5==K< 2
?C=:B5 4:
vmax = k:@ Rg (20)
4=0:> =5>1E>48<> ?@>0=0;878@>20BL, 459AB28B5;L=> ;8 704=85 :>;QA0 A>@2CBAO
?5@2K<8. 48=AB25==K< :>=:C@5=B>< O2;O5BAO ?5@54=55 ?@02>5 :>;5A>.
17
>4AB028< 7=0G5=85 (19) 2 D>@<C;C (14) 4;O .
:@khL
= :@ - (21).
db
;O ?5@54=53> ?@02>3> :>;5A0:
:@khL :@khL
# ś#
ś# ź#
= + = :@ - + = :@ + (22).
ś#1- ź#
db db
# #
7 D>@<C;K (21) 45;05< 2K2>4, GB> ?@8
:@khL < db (23)
7=0G5=85 1C45B 1>;LH5 :@ , B.5. ?5@2K< A>@25BAO ?5@54=55 ?@02>5 :>;5A>, ?>A;5 G53>
?@>F5AA AB0=5B =5>1@0B8<K<. ;O =0E>645=8O :@8B8G5A:>9 A:>@>AB8, 70?8H5< CA;>285
4>AB865=8O :@8B8G5A:>3> C3;0 C2>40 ?5@54=8< ?@02K< :>;5A><.
# ś#
v2 v2hL
ś#
:@ = + (24).
ś#1- ź#
ź#
gkR Rdgb
# #
K@060O v , ?>;CG8<:
kRg(:@ - )
2
vmax = (25).
hLk
1-
db
0<5B8<, GB> 5A;8 :@khL < db , B> 8 " khL < db .
"0:8< >1@07><, <0:A8<0;L=0O A:>@>ABL >?@545;O5BAO ;81> A@K2>< 704=8E :>;QA, ;81>
A@K2>< ?5@54=53> ?@02>3> :>;5A0 2 7028A8<>AB8 >B A>>B=>H5=8O <564C 25;8G8=0<8 (23).
18
11 :;0AA.
040G0 1. 7@K2=0O M<8AA8O
1. >=F5=B@0F8O M;5:B@>=>2.
48= <>;L ?;0B8=K 8<55B <0AAC 8 70=8<05B >1J5< V = , ?@8 MB>< 2 =5<
=0E>48BAO NA 0B><>2. A;8 >B :064>3> 0B><0 2 7>=C ?@>2>48<>AB8 ?5@5H5; >48=
M;5:B@>=, B> 2 >4=>< <>;5 ?;0B8=K 1C45B NA M;5:B@>=>2, 0 8E :>=F5=B@0F8O 1C45B @02=0
NA NA
n = = (1)
Vm
n = 6,62"1028 <-3 (2)
@8 B0:>9 1>;LH>9 :>=F5=B@0F88 M;5:B@>=K =5 ?>:840NB <5B0;;, ?>B><C GB> =0
=8E 459AB2CNB A8;K ?@8BO65=8O A> AB>@>=K ?>;>68B5;L=> 70@O65==KE 8>=>2 <5B0;;0.
@C38<8 A;>20<8, M=5@38O M;5:B@>=>2 2 <5B0;;5 <5=LH5, G5< 8E M=5@38O 2 20:CC<5.
;O B>3> GB>1K M;5:B@>= ?>:8=C; <5B0;;, =04> A>>1I8BL 5<C 4>?>;=8B5;L=CN
M=5@38N, =0?@8<5@, ?>2KA82 B5<?5@0BC@C <5B0;;0. -<8AA8N M;5:B@>=>2 ?>4 459AB285<
M;5:B@8G5A:>3> ?>;O ?>72>;O5B >1JOA=8BL B>;L:> :20=B>20O <5E0=8:0.
2. -;5:B@8G5A:>5 ?>;5.
">G=>5 =0E>645=85 M;5:B@8G5A:>3> ?>;O 21;878 ?>25@E=>AB8 <5B0;;0 ?@8
8725AB=>< ?>B5=F80;5 4>AB0B>G=> A;>6=0O 7040G0, => ?@81;865==> <>6=> ?>:070BL
A2O7L =0?@O65==>AB8 8 ?>B5=F80;0 =0 ?@8<5@5 <5B0;;8G5A:>9 AD5@K. A;8 =0 AD5@5
@048CA>< r =0E>48BAO 70@O4 Q , B> =0?@O65==>ABL M;5:B@8G5A:>3> ?>;O C ?>25@E=>AB8
Q
E = ,
4Ą0r2
0 ?>B5=F80;
Q
=
4Ą0r
0: 284=>,
E = . (3)
r
1;878 >AB@8O <5B0;;8G5A:>9 83>;>G:8 40==>5 @025=AB2> 2K?>;=O5BAO B>;L:>
?@81;865==>, => <>6=> 8A?>;L7>20BL 53> 4;O >F5=:8 =0?@O65==>AB8 M;5:B@8G5A:>3>
?>;O.
3. "5?;>?@>2>4=>ABL.
T
3.1. "0: :0: AB5@65=L >4=>@>4=K9 8 =5B 8AB>G=8:>2 B5?;>BK
2=CB@8 AB5@6=O, B> ?>B>: B5?;>BK 2 ;N1>< A5G5=88 53> >48=0:>2,
T1
B.5.
dT
T0
q =- = const
dx
x
dT
0 l
= C (4)
dx
Tx
dT = C
+"+"dx
T0 0
19
T (x) = T0 + Cx ,
T1 -T2
?@8G5< =0 B>@F5 x = l T(l) = T1 , ?>MB><C C =
l
x
T (x) = T0 + (T1 -T0) (5)
l
@0D8: MB>9 7028A8<>AB8 ?@O<0O ;8=8O.
T
3.2. A;8 AB5@65=L B5?;>87>;8@>20= 8 =0 >4=>< 87 :>=F>2
?>445@68205BAO ?>AB>O==0O B5<?5@0BC@0 T0 , B> =8:0:>3> ?>B>:0 B5?;0
T0
2 MB>< AB@56=5 =5B 8 B5<?5@0BC@0 ?> 2A59 4;8=5 AB5@6=O ?>AB>O==0 8
x
@02=0
T (x) = T0 . (6) 0 l
3.3. >A:>;L:C AB5@65=L B5?;>87>;8@>20=, B> 2 CAB0=>282H5<AO @568<5 (B> 5ABL, :>340
B5<?5@0BC@0 ?5@5AB0=5B 87<5=OBLAO) 4;O B5?;>2>3> 10;0=A0 =5>1E>48<>, GB>1K 2AO
B5?;>B0, 2K45;ONI0OAO ?@0255 =5:>B>@>9 B>G:8 x AB5@6=O (B> 5ABL, A G0AB8 AB5@6=O
4;8=>9 l - x ), >B2>48;0AL ?CB5< B5?;>?5@540G8.
#@02=5=85 B5?;>2>3> 10;0=A0
PB5?; + qS = 0 (7)
0 x l x
dT
(l - x)Sw - S = 0
dx
dT w
= (l - x)
dx
T x
w
dT = (l - x)dx
+" +"
T0 0
w w x2 w x
T (x) = T0 + lx - = T0 + x(l - ) (8)
2 2
"5<?5@0BC@0 ?@02>3> B>@F0 AB5@6=O @02=0
wl2
Tl = T0 + (9)
2
3.4. A;8 ?> AB5@6=N B5G5B B>: ?;>B=>ABLN j , B> ?> 70:>=C 6>C;O-5=F0 2
48DD5@5=F80;L=>9 D>@<5 2 548=8F5 >1J5<0 2 548=8FC 2@5<5=8 2K45;O5BAO B5?;>B0
w = ł j2 . (10)
!>>B25BAB25==>, B5<?5@0BC@0 ?@02>3> B>@F0 AB@56=O 1C45B @02=0
ł j2l2
Tl = T0 + (11)
2
20
4. -<8AA8O M;5:B@>=>2.
a, T < b
ż#
4.1. 028A8<>ABL j(T ) =
#a + k(T - b),T e" b -
#
j
:CA>G=>-;8=59=0O, ?@8G5< :>MDD8F85=B a
?>:07K205B, =0A:>;L:> 3@0D8: ?@8?>4=OB =04
>ALN 01AF8AA, :>MDD8F85=B b ?>:07K205B
B5<?5@0BC@C 87;><0, 0 :>MDD8F85=B k - C3;>2>9
:>MDD8F85=B 4;O =0:;>==>3> CG0AB:0 3@0D8:0. !
@>AB>< =0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O
21;878 ?>25@E=>AB8 <5B0;;0 :>MDD8F85=B a
C25;8G8205BAO, :>MDD8F85=B b C<5=LH05BAO, 0
a
:>MDD8F85=B k C25;8G8205BAO.
@0D8: 7028A8<>AB8 j(T ) ?@8 42CE
b
T
7=0G5=8OE =0?@O65==>AB8 ?>:070= =0 @8AC=:5.
4.2. A;8 ?>B5=F80; :0B>40 @025= =-50"103 , B> =0?@O65==>ABL ?>;O 21;878 >AB@8O
83>;>G:8 @02=0 E = 5"109 . =0G5=8O :>MDD8F85=B>2 @02=K a = 4,06"107 ,
<
<2
b = 1900 , k = 3, 49"104 .
<2
"5?;>, 2K45;ONI55AO 2 83>;>G:5 ?@8 ?@>B5:0=88 ?> =59 B>:0, 2K7K205B =03@52.
CAB0=>282H5<AO @568<5 2AO 2K45;ONI0OAO B5?;>B0 >B2>48BAO 70 AG5B B5?;>?5@540G8. @8
ł j2l2
?@>B5:0=88 B>:0 ?;>B=>ABLN j B5<?5@0BC@0 >AB@8O 1C45B @02=0 T = T0 + (A<. ?. 3.4).
2
>65< 2K@078BL ?;>B=>ABL B>:0 2 7028A8<>AB8 >B B5<?5@0BC@K >AB@8O
2
j = (T -T0)
ł l2
!<KA; MB>3> A>>B=>H5=8O: GB>1K B5<?5@0BC@0 >AB@8O 1K;0 @02=0 T , ?> 83>;>G:5
4>;65= B5GL B>: ?;>B=>ABLN j .
>A:>;L:C M<8AA8O ?@>8AE>48B A ?>25@E=>AB8 ?>;CAD5@K ?;>I04LN 2Ą r2 , 0
?>?5@5G=>5 A5G5=85 83>;>G:8 @02=> Ą r2 , ?@8 MB>< A8;0 B>:0 >48=0:>20 GB> 2=CB@8
83>;:8, GB> A ?>25@E=>AB8 >AB@8O, B> ?;>B=>ABL B>:0 2=CB@8 83>;:8 8 =0 ?>25@E=>AB8
>AB@8O >B;8G0NBAO 2 420 @070:
Ą r2 j2=CB@8 = 2Ą r2 j?>25@E=
j2=CB@8 = 2 j?>25@E= . (12)
!>>B25BAB25==>, 4;O B>3>, GB>1K B5<?5@0BC@0 >AB@8O 1K;0 @02=0 T , A ?>25@E=>AB8
4>;65= AB5:0BL B0:>9 B>:, GB>
2
2 j?>25@E= = (T -T0)
ł l2
j?>25@E= = (T -T0)
2ł l2
;O C4>1AB20 40;L=59H59 @01>BK >1>7=0G8< :>MDD8F85=B
== 3,61"107 1
2ł l2
<22
;>B=>ABL B>:0 M<8AA88 j1 >?@545;O5BAO B5<?5@0BC@>9 >AB@8O T
21
a, T < b
ż#
j1(T ) =
#a + k(T - b),T e" b
#
?;>B=>ABL B>:0 j2 , :>B>@K9 4>;65= AB5:0BL A >AB@8O, GB>1K =0 =5< 1K;0
B5<?5@0BC@0 T , >?@545;O5<0O B5?;>?5@540G59
j2(T ) = T -T0
G5284=>, GB> MB> >4=0 8 B0 65 ?;>B=>ABL B>:0, ?>MB><C B5<?5@0BC@0 AB0F8>=0@=>3>
A>AB>O=8O >?@545;O5BAO C@02=5=85<
j1(T ) = j2(T ) (13)
>AB@>8< >15 DC=:F88 =0 3@0D8:5, B>G:8 ?5@5A5G5=8O 1C4CB A>>B25BAB2>20BL @5H5=8O<.
5H5=8O 420. 0AA<>B@8<
?5@2>5, :>B>@>5 A>>B25BAB2C5B
j1
1,60E+009
<5=LH59 B5<?5@0BC@5. A;8
j2
24@C3 A;CG09=> B5<?5@0BC@0
1,40E+009
C25;8G8BAO, B> ?;>B=>ABL B>:0
1,20E+009
M<8AA88 =5 87<5=8BAO, 0 2>B
1,00E+009
?>B5@8 B5?;>BK 2>7@0ABCB (4;O
8,00E+008
?>445@60=8O B5<?5@0BC@K
6,00E+008
B@51C5BAO 1>;LH0O ?;>B=>ABL
B>:0), 8 >AB@85 83>;:8 =0G=5B
4,00E+008
>E;0640BLAO. A;8 24@C3
2,00E+008
B5<?5@0BC@0 >AB@8O C<5=LH8BAO,
0,00E+000
B> ?>B5@8 B5?;>BK B>65 400 600 800 1000 1200 1400 1600 1800 2000 2200
T
C<5=LH0BAO, => B>: M<8AA88
1C45B B5< 65, ?>MB><C 83>;:0
=0G=5B =03@520BLAO. -B> @5H5=85 CAB>9G82>, ?>B><C GB> >B:;>=5=8O B5<?5@0BC@K >B
AB0F8>=0@=>9 ?@82>4OB : ?@>F5AA0<, 2>72@0I0NI8< 83>;:C : @02=>25A=>9 B5<?5@0BC@5.
5H5=85 65, A>>B25BAB2CNI55 1>;LH59 B5<?5@0BC@5, =5CAB>9G82>. !B>8B B>;L:>
B5<?5@0BC@5 A;CG09=> C25;8G8BLAO, :0: B>: M<8AA88 AB0=5B 1>;LH5, 0 C25;8G5=85 ?>B5@L
B5?;>BK =5 A<>65B :><?5=A8@>20BL C25;8G5=85 2K45;ONI53>AO 46>C;520 B5?;0. -B>
?@82545B 5IQ 1>;LH5<C @07>3@52C 83>;>G:8 B5<?5@0BC@0 CE>48B >B @02=>25A=>3>
7=0G5=8O.
0 ?@0:B8:5, 5A;8 =5 A>740BL A?5F80;L=KE CA;>289, ?@8 =51>;LH8E ?;>B=>ABOE
B>:0 @50;87C5BAO B>;L:> CAB>9G82>5 @5H5=85, 254L =03@52 83>;:8 >B :><=0B=>9
B5<?5@0BC@K ?@>872>48BAO ?@8 ?><>I8 ?@>B5:0NI53> B>:0.
0945< MB> 7=0G5=85 B5<?5@0BC@K Tl . #A;>285 (13) ?@52@0I05BAO 2
a = Tl -T0
a2
Tl = T0 +2 (14)
Tl = 301,3 (15)
@8<5G0=85.
5 >1O70B5;L=> AB@>8BL 8<5==> 3@0D8:8 j1(T ) 8 j2(T ) , <>6=> ?>AB@>8BL 8E
:204@0BK, => @0AAC645=8O ?@8 MB>< 1C4CB 0=0;>38G=K<8, 8 G8A;5==K5 >B25BK
?>;CG0BAO B0:8<8 65.
4.3. =0G5=8O :>MDD8F85=B>2 a,b,k 7028AOB >B =0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O,
A>>B25BAB25==>, >B ?>B5=F80;0 :0B>40. @8 =5:>B>@>< 7=0G5=88 ?>B5=F80;0 2>7=8:=5B
:@8B8G5A:0O A8BC0F8O, :>340 ?>B5@8 B5?;0 =0 B5?;>?@>2>4=>ABL =5 A<>3CB >AB0=>28BL
@07>3@52, B5<?5@0BC@0 4>AB83=5B B5<?5@0BC@K ?;02;5=8O 8 =0ABC?8B @07@CH5=85 >AB@8O
22
1
2
j , j
83>;:8. -B> <>65B =0ABC?8BL, 5A;8 CAB>9G82>5 @5H5=85 C@02=5=8O (13) AB0=5B @02=K<
B5<?5@0BC@5 ?;02;5=8O (A;CG09 ) 8;8 5A;8 j1(T ) e" j2(T ) 4;O ;N1>9 B5<?5@0BC@K T , GB>
A>>B25BAB2C5B :0A0=8N 3@0D8:>2 2 >4=>9 B>G:5, GB> <>65B ?@>87>9B8 42C<O A?>A>10<8
(A;CG09 8 ).
1,80E+009
2,00E+009
1,80E+009
A
1,90E+009
j1 1,60E+009
1,80E+009
1,60E+009
1,70E+009
j1
1,40E+009
1,60E+009
1,40E+009
1,50E+009
1,20E+009
1,40E+009
1,20E+009
j2
1,30E+009
1,00E+009
j2 1,20E+009
1,00E+009
1,10E+009
j2
8,00E+008 1,00E+009 j1
8,00E+008
9,00E+008
6,00E+008 8,00E+008
6,00E+008
7,00E+008
6,00E+008
4,00E+008
4,00E+008
5,00E+008
4,00E+008
2,00E+008
2,00E+008
T?;
3,00E+008
2,00E+008
0,00E+000
0,00E+000
1,00E+008
400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
0,00E+000
T T
500 1000 1500 2000 2500
T
KOA=8<, :0:>9 A;CG09 @50;87C5BAO 2 =0H59 7040G5. 50;87>20BLAO <>65B B>;L:>
>48= 87 B@5E 2>7<>6=KE A;CG052, ?>A:>;L:C DC=:F88 a(E),b(E),k(E) <>=>B>==K5,
?@8G5< a(E),k(E) M:A?>=5=F80;L=> 1KAB@> 2>7@0AB0NB.
@0D8: j1(T ) - MB> C3>;>:. KOA=8<, :0:85 ?>;>65=8O =0 3@0D8:5 <>65B
70=8<0BL 25@H8=0 C3;0. @8 =0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O E ?>;>65=85 25@H8=K
C3;0 =0 3@0D8:5 MB> B>G:0 (b(E),a(E)) .
!8AB5<0
a = a1 exp(a2E)
b = b1 - b2E
70405B ?>;>65=85 25@H8=K C3>;:0 2 ?0@0<5B@8G5A:>< 2845. >6=> ?>AB@>8BL ?>
B>G:0< 3@0D8: 2>7<>6=KE ?>;>65=89 25@H8=K C3;0, 704020O @07=K5 7=0G5=8O
=0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O. >6=> 70<5B8BL, GB> :>MDD8F85=B b , C :>B>@>3>
@07<5@=>ABL B5<?5@0BC@K, ;8=59=> A2O70=0 A E , ?>MB><C
b1 - b
E =
b2
#ś#
a2
a = a1 expś# (b1 - b)ź#
b2
# #
>AB@>8< =0 >4=>< 3@0D8:5 j2(b) 8 a(b) .
23
1
2
j , j
1
2
j , j
1
2
j , j
5,00E+009
4,00E+009
a(b)
3,00E+009
2,00E+009
j2(b)
1,00E+009
0,00E+000
400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
T
> 3@0D8:C A@07C 284=>, GB> A;CG09 2 MB>9 7040G5 =5 @50;87C5BAO, ?>A:>;L:C
25@H8=0 C3>;:0 ?5@5A5:05B 3@0D8: j2(b) ?@8 B5<?5@0BC@5, <5=LH59 B5<?5@0BC@K
?;02;5=8O ?;0B8=K T?; = 2045 .
!:>@55 2A53>, <K 8<55< 45;> A> A;CG05< , => MB> =04> 4>:070BL. ;O B>3> GB>1K
459AB28B5;L=> @50;87>20;AO A;CG09 , =5>1E>48<>, GB>1K 25@H8=0 C3>;:0 A>2?040;0 A
>4=>9 87 B>G5: j2(T ) , B> 5ABL
b = T*
*
a = T -T0
8 3@0D8: j1(T ) H5; :@CG5, G5< j2(T ) , B> 5ABL
dj2
k >= (16)
*
dT
T =T* 2 T -T0
"5<?5@0BC@C, A>>B25BAB2CNICN @5H5=8N T*, <>6=> =09B8 3@0D8G5A:8, 5A;8
0::C@0B=> ?>AB@>8BL 3@0D8:, 0 <>6=> =09B8 0=0;8B8G5A:8, 254L
#ś#
a2
a = a1 expś# (b1 - b)ź# .
b2
# #
">340
#ś#
a2 *
*
T -T0 = a1 expś# (b1 -T )ź#
b2
# #
#ś#
a12 2a2
*
T = T0 + expś# (b1 -T*)ź# .
2
b2
# #
-B> C@02=5=85 0=0;8B8G5A:8 =5 @5H05BAO, => <>6=> 2>A?>;L7>20BLAO <5B>4><
?>A;54>20B5;L=KE ?@81;865=89. @8 MB>< =5>1E>48<> A>1;N40BL >AB>@>6=>ABL,
#ś#
a12 2a2
?>A:>;L:C 8A?>;L7>20=85 <5B>40 ?> D>@<C;5 T*(k +1) = T0 + expś# (b1 -T*(k ) )ź# =5 405B
2
b2
# #
@5H5=85, ?>B><C GB> <5B>4 @0AE>48BAO. 5>1E>48<> A45;0BL =51>;LH85 <0B5<0B8G5A:85
2
#ś#
#ś#
b2
?@5>1@07>20=8O 8 ?@8<5=OBL <5B>4 ?> D>@<C;5 T*(k +1) = - ln (T*(k ) -T0)ź#ź# .
ś#b1
ś#
2a2 ś# a12
# #ź#
# #
24
2
j (b), a(b)
5B>4 >G5=L 1KAB@> AE>48BAO 8 4065 5A;8 27OBL =0G0;L=>5 ?@81;865=85 T*(0) =1800 ,
C65 ?>A;5 ?5@2>9 8B5@0F88 ?>;CG05BAO ?@028;L=K9 >B25B
T* = 1840,8 (17)
-B0 B5<?5@0BC@0 A>>B25BAB2C5B =0?@O65==>AB8 M;5:B@8G5A:>3> ?>;O
b1 -T*
E* == 8,5"109
<
b2
?@8 MB>< 25;8G8=0 :@8B8G5A:>3> ?>B5=F80;0 @02=0 (A B>G=>ABLN 4> 7=0:0)
:@ = 85: (18)
> 5IQ =5>1E>48<> ?@>25@8BL CA;>285 (). @8 =0?@O65==>AB8 ?>;O
E* = 8,5"109 , :>MDD8F85=B k = 9,33"105 , 0 = 4,60"105 .
<
<2 <2
2 T* -T0
>A:>;L:C
dj2
k >= , (19)
*
dT
T =T* 2 T -T0
B> 459AB28B5;L=> @50;87C5BAO A;CG09 .
4.4. @8 ?>40G5 =0 :0B>4 ?>B5=F80;0 =130: =0?@O65==>ABL M;5:B@8G5A:>3> ?>;O
21;878 >AB@8O ?@8=8<05B 7=0G5=85 E =1,3"1010 . =0G5=8O :>MDD8F85=B>2 ?@8 40==>9
<
=0?@O65==>AB8 @02=K a = 1,31"1011 , b = 1721 , k = 6,39"107 .
<2 <2
;>B=>ABL B>:0 A@07C ?>A;5 2:;NG5=8O, ?>:0 83>;>G:0 =5 CA?5;0 @07>3@5BLAO 4>
B5<?5@0BC@K b , @02=O5BAO
j1 = a = 1,31"1011 . (20)
<2
> 4065 ?@8 B5<?5@0BC@5 ?;02;5=8O ?;0B8=K ?;>B=>ABL B>:0
j2 = 1,51"109 , (21)
<2
GB> =0 420 ?>@O4:0 <5=LH5 =0G0;L=>9 ?;>B=>AB8 B>:0 j1 . -B> >7=0G05B, GB> ?>B5@O<8
B5?;>BK =0 B5?;>?@>2>4=>ABL <>6=> ?@5=51@5GL 8 AG8B0BL 83>;>G:C ?>;=>ABLN
B5?;>87>;8@>20==>9.
K45;ONI0OAO 46>C;520 B5?;>B0 845B B>;L:> =0 =03@520=85 83>;>G:8. B0:><
A;CG05 C@02=5=85 B5?;>2>3> 10;0=A0
cm"T = P6"t
cV "T = ł j2V "t
48DD5@5=F80;L=>< 2845
c
dt = dT (22)
ł j2(T )
T?;
c dT
t =
+"
ł j2(T )
T0
>A:>;L:C j(T) - :CA>G=>-;8=59=0O DC=:F8O, B> 8=B53@0; =04> AG8B0BL >B45;L=> 4>
B>G:8 b 8 ?>A;5 =5Q.
b
c dT c
t1 == b -T0
( )
+"
ł a2 ł a2
T0
25
"?;
T?; T?;
#ś#
c dT c d[a + k(T - b)] c 1 c 1 1
t2 == = - = -
ś#ź#
+"+"
ł [a + k(T - b)]2 ł k [a + k(T - b)]2 ł k [a + k(T - b)] ł k a [a + k(T?; - b)]
# #
bb b
@5<O 4> 27@K20 @02=>
c c 11
t = b -T0 + - (23)
( )#ś#
ś#ź#
ł a2 ł k a [a + k(T?; - b)]
# #
t = 2,6"10-6A = 2,6<:A (24)
B5G5=85 2@5<5=8 4> 27@K20 A >AB@8O 83>;>G:8 845B 4>AB0B>G=> 1>;LH>9 B>:.
/2;5=85 27@K2=>9 M<8AA88 8A?>;L7CNB 2 35=5@0B>@0E >G5=L <>I=KE 8 >G5=L :>@>B:8E
8<?C;LA>2 B>:0.
26
040G0 2 8=0<8:
1. !8;0 <?5@0.
G5284=>, GB> 5A;8 :0BCH:0 A B>:>< =0E>48BAO 2 >4=>@>4=>< 25@B8:0;L=><
<03=8B=>< ?>;5, B> @02=>459AB2CNI0O A8; <?5@0,
r
459AB2CNI8E =0 @07=K5 G0AB8 :0BCH:8, @02=0 =C;N.
B
;O 2>7=8:=>25=8O =5=C;52>9 25@B8:0;L=>9
A>AB02;ONI59 A8;K <?5@0 =5>1E>48<>, GB>1K
I
r
ACI5AB2>20;0 @0480;L=0O A>AB02;ONI0O <03=8B=>3> ?>;O
FA
<03=8B=>5 ?>;5 4>;6=> 1KBL =5>4=>@>4=K<. (1) r r
FA FA
?@545;8< @0480;L=CN A>AB02;ONICN <03=8B=>3>
?>;O. K45;8< 2 ?@>AB@0=AB25 2>>1@0605<K9 F8;8=4@, A>2?040NI89 A :0BCH:>9,
2KA>B>9 "z 8 @048CA>< r . >;=K9 <03=8B=K9 ?>B>: G5@57 70<:=CBCN ?>25@E=>ABL
F8;8=4@0 @025= =C;N
Bz (z + "z)
ŚB = 0 .
Bz (z +"z)Ą r2 + Br 2Ą r"z - Bz (z)Ą r2 = 0
Bz (z) - Bz (z + "z) r dBz
BAN40 Br = r = - .
Br
2"z 2 dz
ą B0r
Br = (2)
Bz (z)
2
">340 25@B8:0;L=0O A>AB02;ONI0O A8;K
<?5@0, 459AB2CNI0O =0 :0BCH:C, @02=0
ą Br
0
FA = 2Ą rNIBr = 2Ą rNI = ą B0Ą r2NI (3)
2
>A:>;L:C <03=8B=>5 ?>;5 A8<<5B@8G=> >B=>A8B5;L=> >A8 Oz , @02=>459AB2CNI0O
3>@87>=B0;L=KE A>AB02;ONI8E A8;K @02=0 =C;N.
2. <?;8BC40 :>;510=89.
5@5<5==>5 =0?@O65=85, ?>40==>5 =0 :;5<<K :0BCH:8, 2K7>25B ?5@5<5==K9 B>: 2
:0BCH:5, =0 =55 =0G=5B 459AB2>20BL A8;0 <?5@0, 8 :0BCH:0 ?@845B 2 42865=85.
0AA<>B@8< A8;K, 459AB2CNI85 =0 :0BCH:C.
1) A;8 A>1AB25==0O G0AB>B0 :>;510=89 A8AB5<K @02=0 0 , B> ?@8 >B:;>=5=88
:0BCH:8 87 ?>;>65=8O @02=>25A8O =0 =51>;LH>5 @0AAB>O=85 z , =0 =55 1C45B 459AB2>20BL
C?@C30O 2>72@0I0NI0O A8;0
FC?@ =-kz =-m02z .
!>1AB25==0O G0AB>B0 :>;510=89 4;O :0BCH:8 @02=0 0 = 2Ą f0 =188c-1 .
2) 0: 1K;> A:070=> 2 CA;>288, ?@8 42865=88 <5<1@0=K 2<5AB5 A :0BCH:>9,
A>740NBAO 72C:>2K5 2>;=K, ?@8 MB>< =0 <5<1@0=C 459AB2C5B A8;0 A>?@>B82;5=8O
FA>?@ = -v .
2ł PS
0
:3
40==>< A;CG05 :>MDD8F85=B == 26, 4 .
A
c
3) 5IQ =0 :0BCH:C 1C45B 459AB2>20BL A8;0 <?5@0, =0?@02;5=85 :>B>@>9 7028A8B
>B A?>A>10 =0<>B:8 :0BCH:8 8 =0?@02;5=8O B>:0 2 =59. ;O >?@545;5==>AB8 1C45<
AG8B0BL, GB> ?@8 ?@8;>65=88 ?>;>68B5;L=>3> ?>AB>O==>3> =0?@O65=8O B>: B5G5B B0:, GB>
A8;0 <?5@0 =0?@02;5=0 225@E.
FA = ą B0Ą r2NI .
27
;O C4>1AB20 >1>7=0G8< ?>AB>O==K9 <=>68B5;L ?5@54 A8;>9 B>:0 g = ą B0Ą r2N .
40==>< A;CG05 g = 3,14
.
> 2B>@><C 70:>=C LNB>=0
ma = FC?@ + FA>?@ + FA (4)
ma = -m02z - v + gI
&& &
mz + z + m02z = gI (5)
(B>G:0<8 A25@EC >1>7=0G5=K A>>B25BAB2CNI85 ?@>872>4=K5 ?> 2@5<5=8).
0AA<>B@8<, :0: A2O70=K <564C A>1>9 A8;0 B>:0 8 ?@8:;04K205<>5 : :0BCH:5
=0?@O65=85.
1) =5H=OO -!, ?@8:;04K205<0O : :0BCH:5 @02=0
= 0 cost .
2) 5@5<5==>5 =0?@O65=85 2K7K205B ?5@5<5==K9 B>:, 8 2 :0BCH:5, >1;040NI59
8=4C:B82=>ABLN L , 2>7=8:05B -! A0<>8=4C:F88
dI
A8 =-L .
dt
3) @8 42865=88 :0BCH:8 2 <03=8B=>< ?>;5, <03=8B=K9 ?>B>: G5@57 =55
87<5=O5BAO, GB> ?@82>48B : 2>7=8:=>25=8N -! 8=4C:F88
dŚB ddz
&
8=4 =- =- NĄ r2B0(1-ą z) = ą B0Ą r2N = gz
()
dt dt dt
> 70:>=C <0 4;O ?>;=>9 F5?8
RI = + A8 + 8=4 (6)
dI
&
RI = -L + gz +
dt
dI
&
L + RI = (t) + gz (7)
dt
B>3>, ?>;CG8;0AL A8AB5<0 87 42CE A2O70==KE 48DD5@5=F80;L=KE C@02=5=89
&& &
mz + z + m02z = gI
dI
&
L + RI = (t) + gz
dt
0A 8=B5@5AC5B @5H5=85, :>B>@>5 CAB0=>28BAO, 2 :>=F5 :>=F>2, ?@8 ?5@8>48G5A:>9
2=5H=59 -! (t) = 0 cost . /A=>, GB> :>340 A>1AB25==K5 :>;510=8O 70BCE=CB, A8;0 B>:0
8 :>>@48=0B0 1C4CB 87<5=OBLAO ?> 30@<>=8G5A:><C 70:>=C A B0:>9 65 G0AB>B>9, => 4@C3>9
=0G0;L=>9 D07>9.
?@8=F8?5, <>6=> B>G=> @5H8BL MB> C@02=5=85, ?>4AB0282 2 A8AB5<C
48DD5@5=F80;L=KE C@02=5=89
I = I0 cos(t +Ć) ,
z = Acos(t + ) ,
?>B><, ?@>48DD5@5=F8@>202, @07;>682 B@83>=><5B@8G5A:85 DC=:F88 8 ?@8@02=O2
:>MDD8F85=BK ?@8 A>>B25BAB2CNI8E A8=CA0E 8 :>A8=CA0E, => MB> 40AB A8AB5<C 87
G5BK@5E ;8=59=KE C@02=5=89 A G5BK@L<O =58725AB=K<8, @5H5=85 :>B>@>9 25AL<0
?>;CG8BAO 4>AB0B>G=> 3@><>74:8< 8 >B=8<5B <=>3> 2@5<5=8 8 A8;. > =5B@C4=> 70<5B8BL,
GB> 2 5A;8 B>: 87<5=O5BAO ?> 70:>=C I = I0 cos(t +Ć) , B> 0<?;8BC40 A;0305<>3> RI 1C45B
dI
@02=0 RI0 , 0 0<?;8BC40 A;0305<>3> L 1C45B @02=0 LI0 . 065 =0 ?@545;L=>9 4;O
dt
G5;>25G5A:>3> CE0 G0AB>B5 f = 20:F 25;8G8=0 L = 0.12 < , B>340 :0:
28
dI
R = 4,0 < ? 0,12 < . >MB><C A;0305<K< L <>6=> ?@5=51@5GL, 8 A8AB5<0 C@02=5=89
dt
?@52@0I05BAO 2
&& &
mz + z + m02z = gI
&
(t) + gz
I =
R
>4AB0282 B>: 2 ?5@2>5 C@02=5=85, ?>;CG8< >1K:=>25==>5 =5>4=>@>4=>5
48DD5@5=F80;L=>5 C@02=5=85 2B>@>3> ?>@O4:0 A ?>AB>O==K<8 :>MDD8F85=B0<8
C@02=5=85 2K=C645==KE :>;510=89 A 70BCE0=85<:
&
(t) + gz
&& &
mz + z + m02z = g
R
#ś#
g2 g
&& &
mz + - z + m02z = 0 cost
ś#ź#
RR
# #
#ś#
1 g2 g0
&& &
z + - z + 02z = cost
ś#ź#
m R mR
# #
#ś#
1 g2 g0
;O C4>1AB20 >1>7=0G8< = - , D = .
ś#ź#
m R mR
# #
<
=0G5=85 :>MDD8F85=B0 = 479c-1 . =0G5=85 :>MDD8F85=B0 D = 15,7 ?@8
A2
0<?;8BC45 2=5H=53> =0?@O65=8O 0 =1 .
8DD5@5=F80;L=>5 C@02=5=85 ?@8=8<05B 284
&& & (8)
z + z + 02z = D cost
0A 8=B5@5AC5B 0<?;8BC40 =0?@O65=85. 5H8BL MB> 48DD5@5=F80;L=>5 C@02=5=85
<>6=> @07=K<8 A?>A>10<8: ?>4AB0=>2:>9 z = Acos(t + ) , <5B>4>< 25:B>@=KE 4803@0<<
8;8 <5B>4>< :><?;5:A=KE 0<?;8BC4 MB> 45;> 2:CA0. @57C;LB0B5 ?>;CG05BAO
D
A() = . (9)
(2 -02)2 + ()2
A;8 ?>4AB028BL 2K@065=8O 4;O 2A5E :>MDD8F85=B>2, B> ?>;CG8BAO 25AL<0
3@><>74:>5 2K@065=85
ą B0Ą r2N0
mR
A() = , (10)
1
Ą#
#ś#
1 2ł PS (ą B0Ą r2N)2 2 ń# 2
2
ó#( - (2Ą f0)2)2 + (m c0 - ź#) Ą#
ś#
R
# #
Ł#Ś#
?>MB><C 4>?CAB8<> >AB028BL 2K@065=85 4;O 0<?;8BC4K 2 1>;55 ?@>AB>< 2845 (9),
?@5420@8B5;L=> >F5=82 2A5 :>MDD8F85=BK.
@8 2K=C645==KE :>;510=8OE A 70BCE0=85< =0 =5:>B>@>9 G0AB>B5 21;878
A>1AB25==>9 =01;N405BAO @57>=0=A (<0:A8<C< 0<?;8BC4K), :>B>@K9 =0ABC?05B ?@8
<8=8<C<5 7=0<5=0B5;O.
(2 -02)2 + ()2 = min
AA;54C5< 2K@065=85 =0 M:AB@5<C< AB0=40@B=K< A?>A>1><.
d
[(2 -02)2 + ()2] = 0
d
2
2(2 -02)2 + 2 = 0
2
(2 - (02 - )) = 0
2
29
B>@0O ?@>872>4=0O 8AA;54C5<>9 DC=:F88 @02=0
22
d
[(2 - (02 - ))] = 32 - (02 - )
22
d
2
# MB>3> C@02=5=8O 3 :>@=O 1 = 0 8 2,3 =ą 02 - , ?@0240, >B@8F0B5;L=K9
2
:>@5=L D878G5A:>3> A<KA;0 2 A515 =5 =5A5B.
2
@8 =51>;LH>< 70BCE0=88 (02 > ) 2 B>G:5 1 = 0 ;>:0;L=K9 <0:A8<C<
2
(A>>B25BAB25==>, <8=8<C< 0<?;8BC4K), 254L 2B>@0O ?@>872>4=0O <5=LH5 =C;O, 0 2 B>G:5
2
2 = 02 - - ;>:0;L=K9 <8=8<C< (A>>B25BAB25==> <0:A8<C< 0<?;8BC4K).
2
2
A;CG05 65 A8;L=>3> 70BCE0=8O (02 < ) A8BC0F8O :>@5==K< >1@07><
2
2
<5=O5BAO, ?>A:>;L:C C@02=5=85 (2 - (02 - )) = 0 8<55B B>;L:> >48=
2
459AB28B5;L=K9 :>@5=L 1 = 0 , ?@8G5< 2B>@0O ?@>872>4=0O 2 MB>9 B>G:5 1>;LH5 =C;O, GB>
>7=0G05B <8=8<C< 2K@065=8O 8 <0:A8<C< 0<?;8BC4K. "0:8< >1@07><, ?@8 A8;L=><
70BCE0=88 @57>=0=A0 =5 =01;N405BAO, 8 <0:A8<0;L=0O 0<?;8BC40 A>>B25BAB2C5B =C;52>9
G0AB>B5.
=0H5< A;CG05 0 = 188A-1, 0 = 479A-1 , A>>B25BAB25==> <K 8<55< 45;> A>
A;CG05< A8;L=>3> 70BCE0=8O
2
(02 < ), ?>MB><C <0:A8<C<
1,0
2
0,9
0<?;8BC4K ?@8E>48BAO =0 =C;52CN
G0AB>BC 8 ?@8<5@=K9 3@0D8:
0,8
7028A8<>AB8 4>;65= 2K3;O45BL B0:, :0:
0,7
87>1@065=> =0 @8AC=:5.
0,6
0 G0AB>B5 f0 = 30F
0,5
(0 = 188c-1) 0<?;8BC40 @02=0
0,4
D
0,3
A =
(0)
0 50 100 150 200 250 300 350 400 450 500 550 600
, c-1
A = 1,74"10-4 < = 0,174<< (12)
%0@0:B5@=0O 7028A8<>ABL 0<?;8BC4K
3. 2C:>20O <>I=>ABL.
:>;510=89 :0BCH:8 >B G0AB>BK =0?@O65=8O
(11)
0: 1K;> A:070=> 2 CA;>288, 2
A2O78 A 35=5@0F859 72C:>2KE 2>;= =0 <5<1@0=C 459AB2C5B A8;0 A>?@>B82;5=8O FA>?@ = -v .
&
-B> 7=0G8B, GB> <>I=>ABL P72 <3=>2 = vv = z2 845B =0 A>740=85 72C:0. "0:8< >1@07><,
<3=>25==0O 72C:>20O <>I=>ABL
2D2
P72 <3=>2 = v2 = [-Asin(t + )]2 = sin2(t + ) (13)
(2 -02)2 + ()2
-B> <3=>25==0O <>I=>ABL, => A ?@0:B8G5A:>9 B>G:8 7@5=8O =0A 8=B5@5AC5B A@54=OO
72C:>20O <>I=>ABL, ?>MB><C =04> CA@54=8BL <3=>25==CN <>I=>ABL ?> ?5@8>4C. 0:
1
8725AB=>, A@54=55 7=0G5=85 sin2(t + ) ?> ?5@8>4C @02=> . B>3>,
2
D2 2
P72 () = . (14)
2 (2 -02)2 + ()2
AA;54C5< P72 () =0 <0:A8<C<.
30
A, C.5.
2
Ą#ń#
d 2 2((2 -02)2 + ()2) -2(2(2 -02)2 + 2 )
== 0
ó#(
2 2
d -02)2 + ()2 Ą# 2
Ą# -02)2 + ()2 Ś#
ń#
Ł#Ś#
Ł#(
2 2 2
4 +04 - 202 + 2 - 24 + 2022 - 2] = 0
[
(04 -4) = 0
(2 -02)(02 +2) = 0
$878G5A:89 A<KA; 8<5NB @5H5=8O 1 = 0 (A>>B25BAB2C5B <8=8<C<C <>I=>AB8) 8
= 0 (A>>B25BAB2C5B <0:A8<C<C <>I=>AB8). B0:, 72C:>20O <>I=>ABL <0:A8<0;L=0 =0
G0AB>B5
= 0 =188c-1 , (15)
?@8 MB>< <>I=>ABL @02=0
D2
P72 max = = 0.0142B . (16)
2
2
>B@51;O5<0O M;5:B@8G5A:0O <>I=>ABL PM; = (t)I(t)cosĆ , 345 Ć - A4283 D07 <564C B>:><
&
(t) + gz
8 =0?@O65=85<. !8;0 B>:0 I = .
R
&
F5=8< 25;8G8=C gz =-gA sin(t + ) .
@5=51@5GL 59 <>6=>, 5A;8 0<?;8BC40
0,10
g A 1C45B <=>3> <5=LH5 0 . 0: 284=> ?>
3@0D8:C, A @>AB>< G0AB>BK 0<?;8BC40
A() C1K205B, ?@8G5< ?@8 1>;LH8E
0,05
G0AB>B0E >1@0B=> ?@>?>@F8>=0;L=>
:204@0BC G0AB>BK. @8 <0;KE G0AB>B0E A
<0;>, ?@8 1>;LH8E B>65 <0;>, 0 2 A@54=8E
G0AB>B0E 8<55B <0:A8<C<. K@065=85 0,00
0 100 200 300 400 500 600
Dg
, c-1
g A = 2545B A51O
[(2 -02)2 + ()2]1/ 2
028A8<>ABL gA >B G0AB>BK
B0:, :0: 87>1@065=> =0 @8AC=:5.
;O >F5=:8 <>6=> 2>A?>;L7>20BLAO @0AAG8B0==K< 2 ?@54K4CI5< ?C=:B5 7=0G5=85< A =0
G0AB>B5 0 = 188c-1.
g0A(0) = 0.103
>A:>;L:C 2 CA;>288 B@51C5BAO =5 =09B8, 0 >F5=8BL 25;8G8=C , <>65<
?@5=51@5GL 25;8G8=>9 g0 A(0) = 0.103 ?> A@02=5=8N A 0 = 1 , GB> ?@82545B : B><C,
GB> 7=0G5=85 ?>;CG8BAO GCBL 702KH5==K<.
02 cos2(t)
">340 ?>B@51;O5<0O <3=>25==0O <>I=>ABL PM; <3=>2 = , 0 A@54=55 7=0G5=85
R
?>B@51;O5<>9 <>I=>AB8
02
PM; = . (17)
2R
>B@51;O5<0O M;5:B@8G5A:0O <>I=>ABL @02=0 PM; = 0,125B .
0:A8<0;L=K9 @025=
P72 max = 0,113
max = (18)
PM;
;O =0E>645=8O 3@0=8G=KE G0AB>B @01>G53> 480?07>=0 =04> @5H8BL C@02=5=85
31
g
A(
)
P72 max
P72 (=,2 ) = (19)
2
D2 2 1 D2
=
2
2 (2 -02)2 + ()2 2 2
2
4 - (202 + )2 +04 = 0
22
0,016
2,= = (02 + ) ą 02 + (20)
24
0,014
5@E=OO 3@0=8G=0O G0AB>B0 @02=0
0,012
2 = 544c-1 8;8 f2 = 86,6F . (210)
0,010
86=OO 3@0=8G=0O G0AB>B0 @02=0
0,008
= = 65,0c-1 8;8 f= =10,3F . (211)
0,006
@0D8: 7028A8<>AB8 72C:>2>9 <>I=>AB8
0,004
>B G0AB>BK ?5@5<5==>3> =0?@O65=8O
0,002
87>1@065= =0 @8AC=:5.
0,000
0: 284=>, 8AA;54>20==K9 =0<8
0 100 200 300 400 500 600 700 800
48=0<8: E>@>H> ?>4E>48B 4;O , c-1
2>A?@>872545=8O =87:8E G0AB>B (10A>2).
028A8<>ABL 72C:>2>9 <>I=>AB8 >B G0AB>BK
(22)
32
72
P , B
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