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B> 65 2@5<O <=>65AB2> CAB@>9AB2 B5E=8:8 A2O78 4>2>;L=> A8;L=> ?@82O70=K : >?@545;5==><C B8?C =5;8=59=KE F5?59, 8 0=0;878@>20BL 8E C=825@A0;L=K<8 <5B>40<8 >G5=L B@C4=>. >;=>5 ?@54AB02;5=85 > =5;8=59=>9 F5?8 ;N1>3> 2840 <>6=> ?>;CG8BL, ;8HL 87CG82 5Q >B45;L=K5 M;5<5=BK. @8 MB><, 206=>5 7=0G5=85 8<55B E0@0:B5@8AB8:0 8;8 A5<59AB2> E0@0:B5@8AB8:, >1I8< A2>9AB2>< :>B>@KE O2;O5BAO =5;8=59=>ABL. 0 >A=>25 B0:8E E0@0:B5@8AB8: >ACI5AB2;O5BAO 0=0;87 3@0D8G5A:8< 8;8 0=0;8B8G5A:8< <5B>4><.  ?5@2>< A;CG05 =5>1E>48<> 8<5BL 3@0D8:8 E0@0:B5@8AB8:, 0 2>7459AB285 =0 F5?L (M;5<5=B) <>6=> 8<5BL ;81> 2 2845 3@0D8:>2, ;81> 0=0;8B8G5A:8. > 2B>@>< A;CG05 E0@0:B5@8AB8:8 8 2>7459AB28O 7040NBAO D>@<C;0<8. !;54>20B5;L=>, E0@0:B5@8AB8:0 0??@>:A8<8@C5BAO DC=:F859 A =5>1E>48<>9 AB5?5=LN B>G=>AB8. 0 ?@0:B8:5, :@><5 B>3>, ?>;CG8;8 H8@>:>5 @0A?@>AB@0=5=85 3@0D>-0=0;8B8G5A:85 <5B>4K, =5 B@51CNI85 1>;LH8E 70B@0B 2@5<5=8. 54>AB0B:>< <=>38E 87 B0:8E <5B>4>2 O2;O5BAO =52KA>:0O B>G=>ABL, GB> G0I5 2A53> 4>?CA:05BAO 2 B8?>2KE @0AG5B0E.  =0AB>OI55 2@5<O <0B5<0B8G5A:0O AB>@>=0 0=0;870 =5;8=59=KE F5?59 4>AB0B>G=> ?>;=> >B@065=0 2 =0CG=KE 8AB>G=8:0E.  A2O78 A MB8< 2 40==>9 G0AB8 CG51=>3> ?>A>18O <0B5<0B8G5A:89 0??0@0B @0AA<0B@8205BAO :@0B:>, ?@5A;54CO F5;L ;8HL >7=0:><;5=8O >1CG05<KE A >A=>20<8 B>3> 8;8 8=>3> <5B>40 @0AG5B0 ?@8<5=8B5;L=> : :>=:@5B=><C 87CG05<><C M;5<5=BC. >;LH>5 2=8<0=85 C45;5=> <5B>40< =0E>645=8O ?0@0<5B@>2 M;5<5=B0 ?> 53> AB0B8G5A:8< 2>;LB-0<?5@=K< E0@0:B5@8AB8:0< 8 C:070=K ?@545;K ?@8<5=8<>AB8 40==KE <5B>4>2.  1.  $'!  !" % " !" +% -" 06=CN @>;L 2 @0AG5B0E F5?59 A =5;8=59=K<8 M;5<5=B0<8 8<5NB E0@0:B5@8AB8:8 =5;8=59=>3> M;5<5=B0. A=>2=>5 2=8<0=85 C45;O5BAO 2>;LB-0<?5@=K< E0@0:B5@8AB8:0< (%), ?>A:>;L:C >=8 >B@060NB 7028A8<>ABL B>:0, ?@>B5:0NI53> 2 40==K9 <><5=B 2@5<5=8, >B =0?@O65=8O. =0;8B8G5A:>5 2K@065=85 AB0B8G5A:8E % =5;8=59=>3> M;5<5=B0 (-) 2 B>G=>< 2845 ?@0:B8G5A:8 =52>7<>6=>, B0: :0: 70G0ABCN >=0 A>AB>8B 87 =5A:>;L:8E :@82>;8=59=KE CG0AB:>2, >?8AK205<KE @07;8G=K<8 DC=:F8O<8. > <=>38E @0AG5B0E ?@8<5=ONB 3@0D8G5A:>5 >?@545;5=85 ?0@0<5B@>2. !B0B8G5A:85 % =5;8=59=KE M;5<5=B>2 1K20NB 42CE B8?>2: >4=>7=0G=K5 8 <=>3>7=0G=K5.  ?5@2>< A;CG05 >4=><C 7=0G5=8N =0?@O65=8O A>>B25BAB2C5B >4=> 7=0G5=85 B>:0, 0 2> 2B>@>< A;CG05 >4=><C 7=0G5=8N =0?@O65=8O A>>B25BAB2C5B =5A:>;L:> 7=0G5=89 B>:0 8 =0>1>@>B, >4=><C 7=0G5=8N B>:0 A>>B25BAB2C5B =5A:>;L:> 7=0G5=89 =0?@O65=8O. @0D8G5A:>5 ?@54AB02;5=85 % 45;05B ?@>F5AA @0AG5B0 ?0@0<5B@>2 =03;O4=K<. !;54C5B >B<5B8BL, GB>, 2 >B;8G85 >B ;8=59=KE M;5<5=B>2, F5?8 A - =5 ?>440NBAO B>G=><C @0AG5BC, 0 B>;L:> ?@81;865==><C. KG5@G820=85 % 2> <=>38E <5B>40E O2;O5BAO ?5@2>AB5?5==>9 7040G59, ?>A:>;L:C >B B>G=>AB8 2K?>;=5=8O 3@0D8:0 7028A8B B>G=>ABL =0E>645=8O ?0@0<5B@>2. @8 87CG5=88 3@0D8G5A:>3> ?@54AB02;5=8O % A;54C5B C45;8BL 2=8<0=85 A;54CNI8< 2>?@>A0<. 0:85 CG0AB:8 E0@0:B5@8AB8: <>6=> AG8B0BL ;8=59=K<8, 0 :0 :85 =5;8=59=K? 0: 2K18@05BAO B>G:0 ?>:>O 8 @01>G89 CG0AB>:? 0:85 87<5=5=8O 2 @01>BC F5?8 2=>A8B =5;8=59=>ABL %? . >;LB-0<?5@=K5 E0@0:B5@8AB8:8 48>4>2 >;C?@>2>4=8:>2K5 48>4K =0H;8 H8@>:>5 ?@8<5=5=85 2 =5;8=59=KE F5?OE. 0 8E >A=>25 ?>AB@>5=K B0:85 M;5:B@>==K5 ?@81>@K, :0: CA8;8B5;8 <>I=>AB8 M;5:B@8G5A:8E A83=0;>2, ?@5>1@07>20B5;8 G0AB>BK, 35=5@0B>@K 30@<>=8G5A:8E :>;510=89, AB018;870B>@K 8 <=>385 4@C385, @01>B0 :>B>@KE 2>7<>6=0 B>;L:> 1;03>40@O =5;8=59=K< A2>9AB20< 48>4>2. 4=0:>, B8? AB0B8G5A:8E =5;8=59=KE 2>;LB-0<?5@=KE E0@0:B5@8AB8: ACI5AB25==> @07;8G5= 4;O @07=KE B8?>2 48>4>2. "0:, =0?@8<5@, A8;>2K5 48>4K 8<5NB >4=>7=0G=CN %, 0 AB018;8B@>=K 8 A?5F80;L=K5 48>4K  <=>3>7=0G=K5 % @07;8G=KE 284>2.  G0AB=>AB8, AB0B8G5A:0O % =081>;55 H8@>:> 8A?>;L7C5<KE BC==5;L=KE 48>4>2 <=>3>7=0G=0O.  A2O78 A ?>4>1=K< ACI5AB25==K< @07;8G85< AB0B8G5A:8E E0@0:B5@8AB8: @568<K @01>BK 8 @0AG5B @01>G8E ?0@0<5B@>2 48>4>2 @07;8G5=. 0) %0@0:B5@8AB8:8 8 ?0@0<5B@K A8;>2>3> 48>40. !B0B8G5A:0O % A8;>2>3> 48>40 O2;O5BAO >4=>7=0G=>9. =0 2> <=>3>< 0=0;>38G=0 2E>4=K< E0@0:B5@8AB8:0< ;0<? 8 18?>;O@=KE B@0=78AB>@>2. >MB><C =0 55 ?@8<5@5 <>6=> ?>:070BL >A=>2=K5 ?@8=F8?K ?@>AB59H53> 3@0D>-0=0;8B8G5A:>3> A?>A>10 >?@545;5=8O ?0@0<5B@>2. A5 ?0@0<5B@K =5;8=59=>3> M;5<5=B0 @0745;ONBAO =0 AB0B8G5A:85, 48=0<8G5A:85 8 A@54=85. A5 >=8 <>3CB 1KBL 2KG8A;5=K ?> AB0B8G5A:8< % 8 ?@8<5=ONBAO 2 7028A8<>AB8 >B ?>AB02;5==>9 7040G8. 0 @8A.1.1. ?@54AB02;5=0 AB0B8G5A:0O % A8;>2>3> 48>40.  8A. 1.1. >;LB-0<?5@=0O E0@0:B5@8AB8:0 A8;>2>3> 48>40 7 @8AC=:0 284=>, GB> MB> :@820O ;8=8O, B0: :0: 48>4  =5;8=59=K9 M;5<5=B. "5>@5B8G5A:8 % 2K@0605BAO D>@<C;>9: i = ad(ebuq/kT  1), (1. 1) 345: i - B>:; u - =0?@O65=85; q - 70@O4 =>A8B5;O M;5:B@8G5A:>3> 70@O40 (M;5:B@>=0 8;8 4K@:8); ad - :>MDD8F85=B 48DDC78>==>3> B>:0; b - ?>AB>O==0O 48DDC78>==>3> B>:0; k - ?>AB>O==0O >;LF<0=0; " - 01A>;NB=0O B5<?5@0BC@0 (2 3@04CA0E 5;L28=0). @0:B8G5A:8 MB0 D>@<C;0 E>@>H> 2K?>;=O5BAO B>;L:> 2 A@54=59 G0AB8 %. 1;878 =C;52>3> =0?@O65=8O 8 ?@8 >G5=L 1>;LH8E =0?@O65=8OE (@568< =0AKI5=8O) =01;N40NBAO ACI5AB25==K5 >BABC?;5=8O >B D>@<C;K (1.1). 0 @8A. 1.1. MB8 CG0AB:8 ?>:070=K ?C=:B8@><.  AB0B8G5A:>< @568<5 ?> 48>4C ?@>B5:05B ?>AB>O==K9 B>:. 0?@O65=85 8 A8;0 B>:0 =5 87<5=ONBAO 2> 2@5<5=8. -B><C @568<C A>>B25BAB2C5B =5:>B>@0O B>G:0 =0 % (u0,i0).  MB>< A;CG05 <>6=> 2KG8A;8BL AB0B8G5A:>5 2=CB@5==55 A>?@>B82;5=85 48>40 8, A>>B25BAB25==>, AB0B8G5A:CN :@CB87=C (A<. G.1 AB@.26-27).  48=0<8G5A:>< @568<5 G5@57 48>4 ?@>B5:05B ?5@5<5==K9 B>:, 25;8G8=0 =0?@O65=8O 8 B>:0 87<5=ONBAO 2> 2@5<5=8. B ?5@5?04>2 =0?@O65=8O 1C45B 7028A5BL A?>A>1 2KG8A;5=8O :@CB87=K. @CB87=0 2 40==>< A;CG05 O2;O5BAO 48=0<8G5A:8< ?0@0<5B@>< 8 55 25;8G8=0 =5 1C45B >48=0:>2>9 4;O 2A5E B>G5: E0@0:B5@8AB8:8. ;O ?>;=>3> @0AG5B0 48=0<8G5A:>3> @568<0 2AN % CA;>2=> @071820NB =0 CG0AB:8: >B 0 4> B>G:8   =5;8=59=0O G0ABL; >B B>G:8  4> B>G:8   ?@O<>;8=59=0O (;8=59=0O) G0ABL; 2KH5 B>G:8   =5;8=59=0O G0ABL. B  4>  - 4;8=0 >B@57:0, :>B>@K9 AG8B0NB (0??@>:A8<8@CNB) ?@O<>9 ;8=859. =0 2 1>;LH>9 AB5?5=8 7028A8B >B B@51C5<>9 B>G=>AB8 @0AG5B>2. A;8 =08<5=LH0O 8 =081>;LH0O 25;8G8=0 B>:0 2 48=0<8G5A:>< @568<5 A>>B25BAB2CNB ;8=59=><C CG0AB:C, B> 2KG8A;5==0O :@CB87=0 =5 >B;8G05BAO >B AB0B8G5A:>9, ?>A:>;L:C >=0 @02=0 B0=35=AC C3;0 =0:;>=0 ?@O<>9 : >A8 01AF8AA. A;8 65 E>BO 1K >48= 87 C@>2=59 =0?@O65=8O (B>:0) 2KE>48B 70 ?@545;K CG0AB:0 -, B> =5>1E>48<> 2KG8A;OBL A>1AB25==> 48=0<8G5A:CN :@CB87=C. '0AB> 2 M;5:B@8G5A:8E F5?OE G5@57 - ?@>B5:0NB 425 A>AB02;ONI85 B>:0: ?>AB>O==0O 8 ?5@5<5==0O, ?@8G5< ?5@5<5==K9 B>: ?@54AB02;O5B A>1>9 30@<>=8G5A:>5 :>;510=85 (B0: =07K205<K9 <>=>30@<>=8G5A:89 @568< 8;8 >4=>G0AB>B=>5 CAB@>9AB2>).  MB>< A;CG05 AB0B8G5A:0O :@CB87=0, A>>B25BAB2CNI0O ?>AB>O==>9 A>AB02;ONI59 B>:0 >B=>A8BAO : B>G:5 "0, =07K205<>9 B>G:>9 ?>:>O. S0 =  EMBED Equation.3  - AB0B8G5A:0O :@CB87=0, 345 48DD5@5=F80;K B>:0 8 =0?@O65=8O:  EMBED Equation.3  8  EMBED Equation.3  70<5=5=K ?@8@0I5=8O<8 ( EMBED Equation.3 ), 27OBK<8 >B=>A8B5;L=> B>G:8 ?>:>O; i0 8 u0  B>: 8 =0?@O65=85 ?>AB>O==>9 A>AB02;ONI59. ">G:0 %, A>>B25BAB2CNI0O @07=>AB8 ?>AB>O==>3> B>:0 8 0<?;8BC4K ?5@5<5==>3> B>:0 - =86=OO B>G:0 @01>G53> CG0AB:0 ("1). u1 = u0 - Umaxcos( EMBED Equation.3 t + EMBED Equation.3 u) , i1= i0 - Imaxcos( EMBED Equation.3 t + EMBED Equation.3 i). ">G:0, A>>B25BAB2CNI0O AC<<5 ?>AB>O==>9 A>AB02;ONI59 8 0<?;8BC4K ?5@5<5==>9 A>AB02;ONI59 B>:0  25@E=OO B>G:0 @01>G53> CG0AB:0 ("2). u2 = u0 + Umaxcos( EMBED Equation.3 t + EMBED Equation.3 u), i2= i0 + Imaxcos( EMBED Equation.3 t + EMBED Equation.3 i). "0:8< >1@07><, @01>G89 CG0AB>: @0A?>;>65= =0 AB0B8G5A:>9 % >B B>G:8 "1 4> B>G:8 "2.  ?@81>@0E :;0AA0  B>G:C ?>:>O "0 2K18@0NB 2 A5@548=5 @01>G53> CG0AB:0. ;O B5E @568<>2, @01>G89 CG0AB>: :>B>@KE =0E>48BAO 2 ?@545;0E ;8=59=>9 G0AB8 %, :@CB87=0 =587<5==0 8 @02=0 AB0B8G5A:>9 :@CB87=5. > 2A5E 4@C38E A;CG0OE A;54C5B CG8BK20BL 87<5=5=85 :@CB87=K. ".5. ?@8<5=OBL 48=0<8G5A:CN :@CB87=C. !B@>3> 3>2>@O, =0 ?@0:B8:5, <564C B>:>< 8 =0?@O65=85< 2 =5;8=59=KE F5?OE ACI5AB2C5B @07=>ABL D07, B0: :0: A>?@>B82;5=85 F5?8 25;8G8=0 :><?;5:A=0O:  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3 u -  EMBED Equation.3 i -B> 7=0G8B, GB> <0:A8<C< B>:0 4>AB8305BAO =5 2 B>B 65 <><5=B, GB> 8 <0:A8<C< =0?@O65=8O, 87-70 G53> B>G=>5 2KG8A;5=85 48=0<8G5A:8E ?0@0<5B@>2, ?@5645 2A53> :@CB87=K, 25AL<0 70B@C4=5=>.  ?@0:B8G5A:8E @0AG5B0E G0AB> 8A?>;L7CNBAO A@54=85 ?0@0<5B@K, B0: :0: 1>;55 B>G=> >B@060NB ?@>B5:0NI85 ?@>F5AAK. 0AA<>B@8< ?>=OB85 A@54=59 :@CB87=K. 5@5<5==0O A>AB02;ONI0O =0?@O65=8O 2 30@<>=8G5A:>< 8 A8<2>;8G5A:>< (:><?;5:A=><) 2845: Umaxcos( EMBED Equation.3 t + EMBED Equation.3 u) = Umax 5j EMBED Equation.3 u 5j EMBED Equation.3 t, 345: Umax  <3=>25==0O 0<?;8BC40 =0?@O65=8O; Umax 5j EMBED Equation.3 u  :><?;5:A=0O 0<?;8BC40 =0?@O65=8O. =0;>38G=>, ?5@5<5==0O A>AB02;ONI0O B>:0 2 30@<>=8G5A:>< 8 A8<2>;8G5A:>< 2845: Imaxcos( EMBED Equation.3 t + EMBED Equation.3 i) = Imax 5j EMBED Equation.3 i 5j EMBED Equation.3 t, 345: Imax  <3=>25==0O 0<?;8BC40 B>:0; Imax 5j EMBED Equation.3 i  :><?;5:A=0O 0<?;8BC40 B>:0. 8=0<8G5A:0O :@CB87=0, ?> >?@545;5=8N, @02=0 >B=>H5=8N: Si =  EMBED Equation.3 [Umaxcos( EMBED Equation.3 t + EMBED Equation.3 u)]/ EMBED Equation.3 Imaxcos( EMBED Equation.3 t + EMBED Equation.3 i). !@54=OO :@CB87=0, A>3;0A=> >?@545;5=8N, 2KG8A;O5BAO 4;O >B45;L=>9 30@<>=8:8 :0: >B=>H5=85 A@54=59 :><?;5:A=>9 0<?;8BC4K B>:0 40==>9 30@<>=8:8 : :><?;5:A=>9 0<?;8BC45 =0?@O65=8O. 284C B>3>, GB> =081>;LH89 2:;04 2> <=>38E CAB@>9AB20E 2=>A8B ?5@20O 30@<>=8:0, ...>170@52 ?@54;>68; A@54=59 :@CB87=>9 AG8B0BL >B=>H5=85 A@54=59 :><?;5:A=>9 0<?;8BC4K B>:0 >A=>2=>9 (?5@2>9) 30@<>=8:8 : :><?;5:A=>9 0<?;8BC45 55 =0?@O65=8O:  EMBED Equation.3  EMBED Equation.3 SA@ = Imax.A@ 5j EMBED Equation.3 u /Umax 5j EMBED Equation.3 i (1.2)  @0AG5B0E 8<5==> MB0 25;8G8=0 8 1C45B >?@545;OBLAO 3@0D8G5A:8 ?> AB0B8G5A:8< %. A;8 @50:B82=K5 A>AB02;ONI85 - B0:>2K, GB> @07=>ABL D07  EMBED Equation.3  EMBED Equation.3  1;87:0 : G8A;C :@0B=><C 180 EMBED Equation.3  (B.5. n EMBED Equation.3 , 345 n  F5;>5 G8A;>), B> A@54=OO :@CB87=0 ?@0:B8G5A:8 @02=0 >B=>H5=8N <3=>25==KE 0<?;8BC4. > <=>38E A;CG0OE 8<5==> <3=>25==K5 0<?;8BC4K B>:0 8 =0?@O65=8O ?>440NBAO 87<5@5=8N. !>>B25BAB25==>, 2 MB8E A;CG0OE B>G=>ABL 2KG8A;5=89 1C45B B5< 1>;LH5, G5< <5=LH5  EMBED Equation.3  EMBED Equation.3  >B;8G05BAO >B n EMBED Equation.3 . !@02=820O ?>=OB8O ?0@0<5B@>2, <>6=> A45;0BL 2K2>4, GB> ?@8 C<5=LH5=88 0<?;8BC4K ?5@5<5==>9 A>AB02;ONI59 A@54=OO :@CB87=0 AB@5<8BAO : 48DD5@5=F80;L=>9. !CI5AB2CNB A;54CNI85 ?@028;0 8A?>;L7>20=8O ?0@0<5B@>2: 5A;8 @01>G89 CG0AB>: =5A:>;L:> 1>;LH5 ;8=59=>9 G0AB8 AB0B8G5A:>9 % (u<0,2u0), B> ?@8<5=ONB 48=0<8G5A:85 ?0@0<5B@K; 5A;8 65 0<?;8BC40 ?5@5<5==>3> =0?@O65=8O 25;8:0 (>4=>3> ?>@O4:0 A u0), B> ?@8<5=ONB A@54=85 ?0@0<5B@K. 8=0<8G5A:0O :@CB87=0 >B;8G0NBAO ?> 25;8G8=5 >B AB0B8G5A:>9 8 >B A@54=59 2 =5;8=59=>< @568<5 @01>BK. !CI5AB2C5B =5A:>;L:> A?>A>1>2 CG5B0 MB>3> @07;8G8O. 081>;55 B8?8G=K< O2;O5BAO @07;>65=85 2 @O4 "59;>@0 48=0<8G5A:>9 :@CB87=K >B=>A8B5;L=> AB0B8G5A:>9 :@CB87=K: Si = S0 + S/ (u u0) +  EMBED Equation.3  S//(u u0)2 + EMBED Equation.3  S///(u u0)3 + . . . + +  EMBED Equation.3 S(n) (u u0)n (1.3) 345 S/, S//, . . ., S(n)  ?@>872>4=K5 :@CB87=K ?> =0?@O65=8N, A>>B25BAB25==>, ?5@2>3>, 2B>@>3> 8 1>;55 2KA>:8E ?>@O4:>2;  EMBED Equation.3 u = (u  u0)  ?@8@0I5=85 =0?@O65=8O 2 B>G:5 ?>:>O. ! CG5B>< @07;>65=8O (1.3) ?>;CG8< 2K@065=85, A2O7K20NI55 B>: A =0?@O65=85<: i = (S0 +  EMBED Equation.3 1S/ +  EMBED Equation.3 2 S// + . . . +  EMBED Equation.3 nS(n)) u (1.4)  ?@>AB59H5< A;CG05 4>AB0B>G=> >3@0=8G8BLAO ?5@2>9 ?@>872>4=>9, 2 1>;55 B>G=KE @0AG5B0E CG8BK20NBAO 8 2KAH85 ?@>872>4=K5. i = (S0 +  EMBED Equation.3 1S/ ) u. (1.5) 1) %0@0:B5@8AB8:8 8 ?0@0<5B@K BC==5;L=>3> 48>40. % <=>38E A?5F80;L=KE 48>4>2 >B=>AOBAO : <=>3>7=0G=>9. "0:85 E0@0:B5@8AB8:8 1K20NB 42CE B8?>2: N->1@07=K5 8 S->1@07=K5, GB> ?>:070=> =0 @8A. 1.2.  8A.1.2 . =>3>7=0G=K5 %: 0) N->1@07=0O; 1) S->1@07=0O. N->1@07=>9 E0@0:B5@8AB8:5 >4=><C =0?@O65=8N A>>B25BAB2C5B =5A:>;L:> B>:>2.  S  >1@07=>9 E0@0:B5@8AB8:5 =5A:>;L:8< =0?@O65=8O< A>>B25BAB2C5B >48= B>:. % BC==5;L=>3> 48>40 N->1@07=0O. =0 8<55B ?5@5381, :>B>@><C A>>B25BAB2C5B >B@8F0B5;L=>5 2=CB@5==55 A>?@>B82;5=85. <5==> MB>B CG0AB>: % 8 8A?>;L7C5BAO 2 @01>B5, ?>B><C GB> 2 B0:>< @568<5 BC==5;L=K9 48>4 >1;0405B CA8;8B5;L=K<8 A2>9AB20<8. 0 @8A.1.3. ?@54AB02;5=0 % BC==5;L=>3> 48>40, =0 :>B>@>9 ?>:070=K B>G:8 ;8=59=>3> @01>G53> CG0AB:0.  8A. 1.3. >;LB-0<?5@=0O E0@0:B5@8AB8:0 BC==5;L=>3> 48>40 !B0B8G5A:0O % BC==5;L=>3> 48>40 B5>@5B8G5A:8 2K@0605BAO AC<<>9 42CE M:A?>=5=B: i = atue-u + ad(ebqu/kT  1) (1.6) 345: u - =0?@O65=85; at - :>MDD8F85=B BC==5;L=>9 A>AB02;ONI59 B>:0;  - :>=AB0=B0, E0@0:B5@87CNI0O BC==5;L=K5 A2>9AB20 40==>3> 48>40; ad - :>MDD8F85=B 48DDC78>==>9 A>AB02;ONI59; b - :>=AB0=B0, E0@0:B5@87CNI0O 40==K9 48>4; q - 70@O4 =>A8B5;O; k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u<0,1u0), 345 u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i = a;u3/2, (1.7) 345 a; - :>MDD8F85=B, 7028AOI89 >B M:75<?;O@0 ;0<?K. -;5:B@>20:CC<=K5 AB018;8B@>=K 8 B8@0B@>=K 8<5NB S >1@07=K5 E0@0:B5@8AB8:8. 0<?K A A5B:0<8 8<5NB =5A:>;L:> AB0B8G5A:8E %, 87 :>B>@KE 2 @0AG5B0E, G0I5 2A53>, 8A?>;L7CNBAO 2KE>4=K5 8 ?@>E>4=K5. 0 @8A.1.4 ?>:070=K: 0) C?@02;ONI85 (0=>4=>-A5B>G=K5) % 8 1) 2KE>4=K5 (0=>4=K5) % M;5:B@>20:CC<=>3> B@8>40, 8<5NI85 =5;8=59=K9 E0@0:B5@. =0;>38G=> ?>;C?@>2>4=8:>2><C 48>4C =0 % (@8A.1.4) >B<5G5=K: ;8=59=K9 CG0AB>: (-), 2 :>B>@>< <>6=> AG8B0BL :@CB87=C =587<5==>9 (AB0B8G5A:>9) 8 CG0AB:8, 345 =5>1E>48<> CG8BK20BL =5;8=59=>ABL, B.5. 2KG8A;OBL 48=0<8G5A:CN 8;8 A@54=NN :@CB87=C. "1-"2  @01>G89 CG0AB>:, 4;O :>B>@>3> ?>:070=> 3@0D8G5A:>5 >?@545;5=85 ?@8@0I5=89 B>:0 8 =0?@O65=8O.  8A.1.4. !B0B8G5A:85 % M;5:B@>20:CC<=>3> B@8>40. ++: 1. !B0B8G5A:85 2>;LB-0<?5@=K5 E0@0:B5@8AB8:8 @048>;0<? 8<5NB =5;8=59=K9 284. "5>@5B8G5A:0O 7028A8<>ABL B>:0 >B =0?@O65=8O 2K-?>;=O5BAO ;8HL ?@8 A@54=8E =0?@O65=8OE. 2. ;O ?@81;865==>3> @0AG5B0 @568<>2 @01>BK - 3@0D8G5A:8 >?@545;ONB :@CB87=C ?> C?@02;ONI8< 8 2KE>4=K< AB0B8G5A:8< % ;0<?K.A CG5B>< ?5@5?04>2 =0?@O65=8O.  @0AG5B=KE D>@<C;0E 48DD5@5=F80;K 70<5=5=K ?@8@0I5=8O- <8, >?@545;O5<K<8 3@0D8G5A:8. % 18?>;O@=KE B@0=78AB>@>2 0@0<5B@K 18?>;O@=>3> B@0=78AB>@0 (") 2 @568<5 ?5@5<5==>3> A83=0;0 =07K20NBAO 48DD5@5=F80;L=K<8. =8 2K2545=K 87 B5>@88 G5BK@5E?>;NA=8:>2 2 ?@54?>;>65=88 A;01KE ?> 0<?;8BC45 A83=0;>2 =52KA>:>9 G0AB>BK (MB8 25;8G8= 8=48284C0;L=K 4;O :064>3> "). 8?>;O@=K9 B@0=78AB>@ 8<55B A5<59AB20 AB0B8G5A:8E E0@0:B5@8AB8:: 2E>4=K5 8 2KE>4=K5. ;O @0AG5B0 48=0<8G5A:>3> @568<0 8A?>;L7CNBAO 48DD5@5=F80;L=K5 ?@>2>48<>AB8, C@02=5=8O :>B>@KE 4;O M:2820;5=B=>9 AE5<K A 35=5@0B>@0<8 B>:0, 8<5NB 284: Im1=g11Um1+ g12Um2; Im2=g21 Um1+g22Um2, (1.8) 345: Im1 8 Um1 - 0<?;8BC4K 2E>4=>3> B>:0 8 2E>4=>3> =0?@O65=8O, A>>B25BAB25==>; Im2 8 Um2 - 0<?;8BC4K 2KE>4=>3> B>:0 8 2KE>4=>3> =0?@O65=8O, A>>B25BAB25==>; g11 - 2E>4=0O ?@>2>48<>ABL; g22 - 2KE>4=0O ?@>2>48<>ABL; g12  ?@>2>48<>ABL >1@0B=>9 A2O78; g21  ?@>2>48<>ABL C?@02;5=8O (:@CB87=0). #@02=5=8O (1.8) ?>:07K20NB, GB> 2E>4=>9 B>: Im1 A:;04K205BAO 87 B>:0, A>740==>3> 2E>4=K< =0?@O65=85< Um1 =0 M;5<5=B5 AE5<K g11, 8 B>:0, 2>7=8:0NI53> 2> 2E>4=>9 F5?8 >B =0?@O65=8O Um2 70 AG5B >1@0B=>9 A2O78. KE>4=>9 65 B>: Im2 A:;04K205BAO 87 CA8;5==>3> B>:0: g21 Um1 8 B>:0, A>740205<>3> =0 M;5<5=B5 AE5<K g22 =0?@O65=85< Um2. "0:8< >1@07><, g21 Um1 - B>:, CG8BK20NI89 CA8;5=85 "; 0 g12 Um2 - B>:, CG8BK20NI89 >1@0B=CN A2O7L 2 ". @0D8G5A:8< <5B>4>< <>6=> 2KG8A;8BL 48DD5@5=F80;L=K5 ?@>2>48<>AB8: g11 8 g12 ?> 2E>4=K< %, 0 g21 8 g22 - ?> 2KE>4=K< %.  7028A8<>AB8 >B @568<0 @01>BK 8 B@51C5<>9 B>G=>AB8 :0640O 87 ?@>2>48<>AB59 <>65B 1KBL 48=0<8G5A:>9 8;8 A@54=59. 8=0<8G5A:0O ?@>2>48<>ABL: gi = EMBED Equation.3 [Umaxcos( EMBED Equation.3 t + EMBED Equation.3 u)]/ EMBED Equation.3  Imaxcos( EMBED Equation.3 t + EMBED Equation.3 i). !@54=OO ?@>2>48<>ABL: gA@ = Imax. A@ 5j EMBED Equation.3 u /Umax 5j EMBED Equation.3 i 07;>65=85 2 @O4 "59;>@0 4> B@5BL53> G;5=0: gi = g0 + g/ (u u0) +  EMBED Equation.3 g//(u u0)2, 345: g0  AB0B8G5A:0O ?@>2>48<>ABL (2 B>G:5 ?>:>O); g/, g//  ?5@20O 8 2B>@0O ?@>872>4=K5.  MB>< ?@81;865=88 B>: 2K@0605BAO G5@57 =0?@O65=85 2 2845: i = (g0 +  EMBED Equation.3 1g/ +  EMBED Equation.3 2 g//)u.  ?@>AB59H5< A;CG05 >3@0=8G820NBAO ?5@2>9 ?@>872>4=>9 @07;>65=8O: i = (g0 + EMBED Equation.3 1g/ )u. ;O ?5@5<5==>9 A>AB02;ONI59 2E>4=>3> =0?@O65=8O u EMBED Equation.3 =u u0, ?>;CG8< ?@81;865==> :204@0B8G=CN 7028A8<>ABL B>:0 >B =0?@O65=8O: i = g0u EMBED Equation.3  + g/ (u EMBED Equation.3 )2. 0 @8A.1.5 ?@54AB02;5=K 2E>4=K5 AB0B8G5A:85 % 18?>;O@=>3> B@0=78AB>@0 8 >B<5G5=K ;8=59=K5 (-) 8 =5;8=59=K5 CG0AB:8.  8A.1.5. E>4=0O % 18?>;O@=>3> B@0=78AB>@0. ++: A5 ?0@0<5B@K 18?>;O@=KE B@0=78AB>@>2 >?@545;ONBAO ?> AB0- B8G5A:8< 2E>4=K<, 2KE>4=K< 8 C?@02;ONI8< 2>;LB-0<?5@=K< E0- @0:B5@8AB8:0< 0=0;>38G=> 48>40< !B8G5A:89 8 48=0<8G5A:89 @568<K 7040NBAO 0<?;8BC4>9 ?5@5- <5==>3> =0?@O65=8O =0 2E>45 18?>;O@=>3> B@0=78AB>@0: <0;K9 A83=0;  AB0B8G5A:89 @568<; 1>;LH>9 A83=0;  48=0<8G5A:89 @568<. A=>2=K<8 ?0@0<5B@0<8 O2;ONBAO 48DD5@5=F80;L=K5 ?@>2>- 48<>AB8, 0 B0:65 A>>B25BAB2CNI85 8< A>?@>B82;5=8O.  2.  !&/ % " !" +% -" @5645, G5< ?@>2>48BL @0AG5B A;>6=KE M;5:B@8G5A:8E F5?59 =5>1E>48<> >?@545;8BL ?0@0<5B@K A0<>3> -.  ?@54K4CI59 3;025 1K;8 >?8A0=K >1I85 A2>9AB20 AB0B8G5A:8E % 8 ?@>AB59H89 <5B>4 >?@545;5=8O 25;8G8= >A=>2=KE ?0@0<5B@>2 A 8A?>;L7>20=85< 3@0D8G5A:8 ?@54AB02;5==KE 2>;LB-0<?5@=KE E0@0:B5@8AB8:. '0AB> B@51C5BAO ?>;CG8BL @5H5=85 7040G8 2 >1I5< 2845, =5 A2O70==>< A :>=:@5B=K<8 G8A;5==K<8 7=0G5=8O<8 ?0@0<5B@>2, B> 5ABL, 2 2845 D>@<C;, >B@060NI8E 7028A8<>ABL B>:0, ?@>B5:0NI53> G5@57 -, >B =0?@O65=8O, 2>7459AB2CNI53> =0 =53>.  B0:>< A;CG05 8A?>;L7C5BAO 0=0;8B8G5A:89 <5B>4 =0 >A=>25 >?@545;5=8O <0B5<0B8G5A:>9 D>@<C;K, >?8AK20NI59 E0@0:B5@8AB8:C A 4>AB0B>G=>9 AB5?5=LN B>G=>AB8. "0:85 <5B>4K =07K20NBAO 0??@>:A8<0F859. (8@>:>5 ?@8<5=5=85 =0 ?@0:B8:5 =0H;8 <5B>4K ?>41>@0 ?>4E>4OI8E ?@81;865==KE (0??@>:A8<8@CNI8E) DC=:F89: AB5?5==>3> <=>3>G;5=0, M:A?>=5=BK, >B@57:>2 ?@O<KE ;8=89 8 4@. ??@>:A8<8@CNI85 DC=:F88 ?>72>;ONB 0=0;8B8G5A:8 >?@545;OBL ?0@0<5B@K -. AN % CA;>2=> @071820NB =0 CG0AB:8, A>>B25BAB25==> @568<C @01>BK. 064K9 CG0AB>: <>65B 1KBL 8A?>;L7>20= =5 B>;L:> 4;O 3@0D8G5A:>3>, => 8 4;O 0=0;8B8G5A:>3> >?@545;5=8O ?0@0<5B@>2. @8 87CG5=88 MB8E <5B>4>2 A;54C5B C45;8BL 2=8<0=85 A;54CNI8< 2>?@>A0<: 0: 2K1@0BL 0??@>:A8<8@CNICN DC=:F8N ?@8 0=0;8B8G5A- :>< @0AG5B5? 0: >?@545;8BL ?>;>65=85 @01>G8E B>G5: 8 2K?>;=8BL ?>AB- @>5=85 4;O 2KG8A;5=8O :>MDD8F85=B>2 3@0D>-0=0;8B8G5A:>< @0AG5B5? 0:>9 <5B>4 @0AG5B0 2K1@0BL 4;O 48=0<8G5A:>3> @568<0 1>;LH>3> A83=0;0? 1I85 A2545=8O > 2K1>@5 <5B>40 @0AG5B0 8 0??@>:A8<0F88  >1I5< A;CG05 : =5;8=59=><C M;5<5=BC, @01>B0NI5<C 2 0??0@0- BC@5, ?@8;>65=> =0?@O65=85 (2>7459AB285), 8<5NI55 425 A>AB02;ONI85: ?>AB>O==CN 8 ?5@5<5==CN. -B> =0?@O65=85 8 >?@545;O5B @01>GCN >1;0ABL -. 0AG5B ;N1>3> @568<0 @01>BK - =0G8=0NB A =0E>645=8O =0 AB0B8G5A:8E % B>G5: @01>G53> CG0AB:0, A>>B25BAB2CNI53> B>9 G0AB8 E0@0:B5@8AB8:8, :>B>@0O >E20G5=0 ?@8;>65==K< 2>7459AB285<. ;O @0AG5B0 =5 >1O70B5;L=> AB@>8BL B>G:8 3@0D8G5A:8, => <>6=> 7=0BL 8E :>>@48=0BK. @8 @0AA<>B@5=88 ;8=59=KE M;5<5=B>2 >B<5G0;0AL =57028A8- <>ABL 8E A>1AB25==KE ?0@0<5B@>2 >B 2=5H=53> 2>7459AB28O: =0?@O65=8O, B>:0 B5<?5@0BC@K 8 4@. A>15==>ABLN 2A5E - O2;O5BAO 7028A8<>ABL ?0@0<5B@>2 >B 25;8G8=K ?@8;>65==>3> =0?@O65=8O 8 ?@>B5:0NI8E B>:>2. -B0 7028A8<>ABL, 2 1>;LH8=AB25 48=0<8G5A:8E @568<>2, :0: ?@028;>, =5 2K@0605BAO ?@O<>9 ;8=859. 0?@8<5@: CABL - 8<55B :204@0B8G=CN 7028A8<>ABL <564C B>:>< 8 =0?@O65=85<: i = a2u 2, u >0 (2.1) 345 a2  ?>AB>O==K9 :>MDD8F85=B. ">340 ?@>2>48<>ABL F5?8 @02=0: G = EMBED Equation.3 = a2u, 0 :@CB87=0 55 E0@0:B5@8AB8:8 S= EMBED Equation.3 =2 a2u. 10 MB8 ?0@0<5B@0 7028AOB >B =0?@O65=8O, 0 : F5?8 2 F5;>< =5 ?@8<5=8< ?@8=F8? AC?5@?>78F88 (A<.'0ABL1, AB@.24-25). 0 @8A.2.1 ?@82545=0 2E>4=0O % 18?>;O@=>3> B@0=78AB>@0, =0 :>B>@>9 ?>:070=K 87<5=5=8O D>@<K A83=0;>2 ?@8 =5;8=59=>< 48=0<8G5A:>< @568<5 @01>BK. >7459AB285 u(t)=U0+Umaxcos EMBED Equation.3 t.  8A.2.1. 568<K @01>BK 18?>;O@=>3> B@0=78AB>@0 0: 284=> 87 @8AC=:0, 1>;LH>9 A83=0; ?>425@3AO A8;L=K< =5;8=59=K< 87<5=5=8O<. 7<5=5=85 D>@<K <0;>3> A83=0;0 =525;8:>, B0: :0: <0;0 0<?;8BC40 2E>4=>3> ?5@5<5==>3> =0?@O65=8O. 0: 8725AB=>, ;N1>9 A;>6=K9 A83=0; <>6=> ?@54AB028BL 2 2845 @O40 $C@L5, ?5@2K9 G;5= :>B>@>3> 8 5ABL >A=>2=0O 30@<>=8:0. =0 2K@0605B 30@<>=8G5A:>5 2>7459AB285 =0 -.  48=0<8G5A:>< @568<5 2 2KE>4=>9 A83=0; 4>1028;8AL 1>;55 2KA>:85 G0AB>BK  2KAH85 30@<>=8:8. 0 ?@0:B8:5 >1KG=> >BA5820NB 2KAH85 30@<>=8:8 A ?><>ILN D8;LB@>2. !B@>3> 3>2>@O, 2 2KE>4=>< A83=0;5 2A5340 4>102;ONBAO 2KAH85 30@<>=8:8 87-70 =5;8=59=>AB8 %, 8 8E 2:;04 1C45B B5< 1>;LH5, G5< 1>;LH5 ?@8;>65==>5 2>7459AB285.  ?@0:B8G5A:8E @0AG5B0E @07;8G0NB 48=0<8G5A:85 @568<K ?> 25;8G8=5 ?@8;>65==>3> : - =0?@O65=8O. 0) 8=0<8G5A:89 @568< <0;>3> A83=0;0. 0;K<8 AG8B0NBAO A83=0;K, ?@8 :>B>@KE 2KE>4=>9 A83=0; >4=>3> ?>@O4:0 A 25;8G8=>9 AB0B8G5A:>3> 2E>4=>3> A83=0;0 (u0).  MB>< A;CG05 <>6=> 8A?>;L7>20BL AB0B8G5A:85 ?0@0<5B@K. 1) 8=0<8G5A:89 @568< 1>;LH>3> A83=0;0 405B A8;L=K5 =5;8=59=K5 8A:065=8O, B@51CNI85 CG5B0 2KAH8E 30@<>=8:.  MB>< @568<5 8A?>;L7C5BAO A@54=OO :@CB87=0 SA@, A>3;0A=> 2K- @065=8N (1.2). 5B>4K @0AG5B0 ?0@0<5B@>2 -, ?@54AB02;5=K =0 @8A.2.2.  _ 8A.2.2 ;0AA8D8:0F8O <5B>4>2 @0AG5B0 @8 3@0D8G5A:>< <5B>45 0=0;870 E0@0:B5@8AB8:0  8 2>7459AB285 7040NBAO 2 2845 3@0D8:>2. $>@<0 2KE>4=>3> A83=0;0 >?@545;O5BAO ?CB5< ?>AB@>5=8O B@5BL59 ?@>5:F88 ?> 42C< 40==K< (@8A.2.3).  8A.2.3. @0D8G5A:>5 >?@545;5=85 @50:F88 - 54>AB0B>: MB>3> <5B>40 2 B><, GB> =52>7<>6=> ?>;CG8BL B>G=>9 :>;8G5AB25==>9 >F5=:8 @8 0=0;8B8G5A:>< <5B>45 2>7459AB285 8 E0@0:B5@8AB8:0 7040=K 2 2845 D>@<C;. ?@545;5=85 @50:F88 - 45;05BAO <0B5<0B8G5A:8<8 <5B>40<8. 5B>4 405B B>G=K9 :>;8G5AB25==K9 @57C;LB0B 8 ?@8<5=O5BAO 2 =0AB>OI55 2@5<O 4;O <0H8==>3> @0AG5B0. @8 3@0D>-0=0;8B8G5A:>< <5B>45 E0@0:B5@8AB8:0 - 70405BAO 3@0D8G5A:8, 0 2>7459AB285 0=0;8B8G5A:8. "0: :0: 2>7459AB285 O2;O5BAO DC=:F859 2@5<5=8, B> 4;O :064>3> <><5=B0 2@5<5=8 ?0@0<5B@K @0AAG8BK20NBAO A ?><>ILN 3@0D8:0. ++:  7028A8<>AB8 >B 0<?;8BC4K ?5@5<5==>3> 2>7459AB2CNI53> =0- =0?@O65=8O 2 48=0<8=0<8G5A:>< @568<5 2KE>4=>9 A83=0; >1>30I0- 5BAO 2KAH8<8 30@<>=8:0<8. !CI5AB2C5B =5A:>;L:> B8?>2 48=0<8G5A:>3> @568<0 @01>BK -, 8, A>B25BAB25==> MB><C, =5A:>;L:> <5B>4>2 @0AG5B0 ?0@0<5B@>2. 8=0<8G5A:89 @568< <0;>3> 2>7459AB2CNI53> A83=0;0 8<55B ?0@0<5B@K AB0B8G5A:>3> @568<0. 0AG5B 48=0<8G5A:>3> @568<0 1>;L-H>3> A83=0;0 =0 ?@0:B8:5 ?@>872>48BAO 0=0;8B8G5A:8 8;8 3@0D>-0=0;8- B8G5A:8. ??@>:A8<0F8O E0@0:B5@8AB8: <=>3>G;5=>< ;O >?@545;5=8O @50:F88 -, 2K@0605<>9 A8;>9 B>:0, =0 2>7459AB285, 2K@0605<>5 2E>4=K< =0?@O65=85<, G0AB> 2K18@0NB 0??@>:A8<8@CNICN DC=:F8N 2 2845 <=>3>G;5=0. ".5. AB0B8G5A:CN % 70<5=ONB AB5?5==K< <=>3>G;5=>< A =5>1E>48<>9 AB5?5=LN B>G=>AB8. i(t) = f (u), (2.2) 345: i(t) - B>: (@50:F8O); u(t) - =0?@O65=85 (2>7459AB285); f (u) = f (u,t) - DC=:F8O, 2 >1I5< A;CG05, =5;8=59=0O. 0 ?@0:B8:5 4;O 0=0;8B8G5A:>3> >?8A0=8O MB>9 7028A8<>AB8 AB@5<OBAO @50;L=CN E0@0:B5@8AB8:C - 70<5=8BL ?@81;865==>9 1>;55 ?@>AB>9.  MB>< 8 A>AB>8B 7040G0 0??@>:A8<0F88. 7 :C@A0 <0B5<0B8:8 8725AB=>, GB> ;N10O DC=:F8O <>65B 1KBL >?8A0=0 A:>;L C3>4=> B>G=> 15A:>=5G=K< AB5?5==K< @O4>< 2840: i(t) = 00 + 01u + 02u2 + 03u3 + . . . + 0nun + . . . . (2.3) 040G0 0??@>:A8<0F88 A2>48BAO : >?@545;5=8N 7=0G5=89 :>MDD8F85=B>2 ?>;8=><0, :>B>@K5 O2;ONBAO ?@>872>4=K<8 :@CB87=K. 00 =S0 ; 01=S/ ; 02 = EMBED Equation.3 S // ; 03 = EMBED Equation.3 S/// ; . . . ; 0n =  EMBED Equation.3 S(n).  @07;8G=KE ?@0:B8G5A:8E 7040G0E B@51C5BAO @07=0O B>G=>ABL @0AG5B>2. '0AB> <=>3>G;5= 4>AB0B>G=> >3@0=8G8BL 42C<O-B@5<O ?5@2K<8 G;5=0<8. i(t) = 00 + 01u + 02u2 . (2.4) h< J L T ` d f | ( 4  0 J L \ v , . 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.68:<>ƿ֫shhTBsh"jchCJ EHH*UmH sH 'j@ hCJH*UVmHnHuhCJ H*mH sH jhCJ H*UmH sH hCJ$H*mH sH hK#hCJ H*hCJ mH sH hK#hCJ hK#hCJ H*hCJ$mH sH  hK#h hhhhCJ hCJ H*mH sH jhCJ UmH sH j*bhCJ EHUmH sH >@BDJLNPRTV`brz|~ Ǽ{wndYdYNddYhK#hmH sH hCJ$H*mH sH hCJ$mH sH hK#hCJ$h"jehCJ EHH*UmH sH 'j@ hCJH*UVmHnHuhCJ H*mH sH jhCJ H*UmH sH hCJ$H*mH sH hK#hCJ hK#hCJ H*hCJ H*mH sH hCJ mH sH  hCJ,hK#hCJ H*>@BDFJRTVX\jptvxԲֲزܲ޲"(ܳxoӚhK#hCJ$ hK#h hCJ$H*hK#hCJ$H*hK#hCJ$H*hhCJ$H*mH sH hCJ$H*mH sH j=ghCJ$EHU$j7 @ hCJUVmHnHujhCJ$UhK#hCJ$H*mH sH hK#hmH sH hCJ$mH sH %ܳ޳ "&(*,´ƴȴʴjlnpƼ׸Ƽƣƣƌ׸ƼƣpjhCJ$H*UmH sH jlhCJ$EHUhK#hCJ$H* hK#h hNHhK#hCJ$jjhCJ$EHUhhK#hCJ$H*hCJ$mH sH  hCJ$H*jhCJ$UjihCJ$EHU$j@ hCJUVmHnHu*pz|*,.68:<>@JL\dfhjlnprvxz|~Ĵꚫ꡴ֈꄫzhK#hCJ$H*h"johCJ$EHH*UmH sH  hK#hhCJ$mH sH hK#hCJ$jhCJ$H*UmH sH "j8nhCJ$EHH*UmH sH 'j @ hCJH*UVmHnHuhK#hCJ$H*hCJ$H*mH sH ,">@8:<>@B ,.0F<>NPhCJNHhmH sH  hNHhK#hCJhCJmH sH jEshqhCJU hCJh hK#hhK#hCJ$H*hK#hCJ$jhCJ$H*UmH sH "jqhCJ$EHH*UmH sH 4"\йȺ<>@2gd $^a$gdgdgdgd & F gdp^pgd$a$gd $^a$gd $`a$gd0JFTV|~`gd @gd $^a$gd $ & Fa$gd$a$gdgd$a$gd $ & FKa$gd~|&2<>@BDFPRbjlnprtv˺Տjy3hCJ$EHUmH sH $jD1@ hCJUVmHnHujhCJ$UmH sH hCJ mH sH  hK#hhK#hCJ$H*hK#hCJ$H*hK#hCJ$hCJ$mH sH  hNHh hCJhCJNH2vxz$(*,.2468:@FHJTحؖ΍΍΃xjjhCJ UmH sH hCJ H*mH sH hK#hCJ H*hK#hCJ hK#hjW5hCJ$EHUmH sH $j^ @ hCJ$UVmHnHujhCJ$UmH sH hCJ mH sH hK#hCJ$ hNHhhCJ$mH sH hK#hCJ$H*)TVfnprtvx|~LHJFH, nprtʼrgh{hmH sH jOh{hUmH sH hmH sH  hCJ$H* hCJ$ hCJ H* hCJ hNH hK#hj8hUhjhCJ UmH sH j97hCJ EHUmH sH $jT@ hCJUVmHnHuhCJ mH sH hK#hCJ (z|L*bptx~HJNPZ*`gdgdt~TVFJLN8:vx $(02468@^`bdl ǿǿǿǿǵ׵ǵǵhK#hCJ$hCJ$mH sH hCJNH hCJhCJmH sH hK#hCJjhU hh hNH hK#hhD*p&  (2^$a$gd ^`gd $`a$gd $^a$gd $ & Fa$gd & Fgdgdgd`gdhZ\24uu2v>xxZz^zzz}}}} ^`gdgd`gdZ\^`blprvxz|PR&(ܶ塓$j7 @ hCJUVmHnHuhK#hCJ$H*mH sH jhCJ$UhCJ$H*mH sH hCJ$H*mH sH hK#hCJ$H* hCJ$H*hK#hCJ$H*hK#hCJ$hCJ$mH sH  hK#h hNHh1(*,.6>BDFlnprt|߻ߨɻ߅uqkqbbXhK#hCJ$H*hK#hCJ$ hNHhjuthCJ$EHUmH sH $j8 @ hCJUVmHnHujrhCJ$EHUmH sH $j 9 @ hCJUVmHnHujhCJ$UmH sH hCJ$H*mH sH hCJ$H*mH sH hCJ$mH sH jhCJ$UjphCJ$EHU" &4tttnupuuuuuuuuuuuuuuuuuuvvv v vvvvvvvv&v2v4v"w$www&x(xxxxxxxxxxxxxxxxxzzz hNHUhK#hCJ$H* hCJ$H*hCJ$mH sH hK#hCJ$H*hK#hCJ$hJ 7028A8<>AB8 >B F5;8 8AA;54>20=8O F5?8 <>6=> ?>;L7>20BLAO C:>@>G5==K< @O4><, 8A?>;L7CO B>;L:> :204@0B8G=K9 G;5=, >B@060NI89 =08<5=LH85 =5;8=59=K5 MDD5:BK.  =5:>B>@KE 7040G0E 8A?>;L7CNB C:>@>G5==K9 <=>3>G;5= B@5BL59 AB5?5=8: i(t) = 00 + 03u3; i(t) = 00 + 01u + 03u3. (2.5) >@O4>: 2KG8A;5=89. ??@>:A8<8@CNI0O 8 0??@>:A8<8@C5<0O 7028A8<>AB8 ?@8@02=820NBAO 2 >B45;L=KE, >?@545;5==K< >1@07>< A?5F80;L=> 2K1@0==KE B>G:0E, :>B>@K5 ?@8=OB> =07K20BL C7;>2K<8.  >AB0;L=KE B>G:0E A>2?045=85 0??@>:A8<8@CNI59 DC=:F88 A % ;8HL ?@81;865==>5. 0) ??@>:A8<0F8O ?>;8=><>< ?5@2>9 AB5?5=8. >;8=>< ?5@2>9 AB5?5=8 i0?(t)=00 + 01(u-u0) - 8A?>;L7C5BAO 4;O 0??@>:A8<0F88 =5;8=59=KE CG0AB:>2, GB> ?>:070=> =0 @8A.2.4 4;O ?@>E>4=>9 AB0B8G5A:>9 % ?>;52>3> B@0=78AB>@0 (") A C?@02;ONI8< p-n ?5@5E>4>< 8 :0=0;>< n-B8?0.  8A.2.4. ??@>:A8<0F8O % ?>;8=><>< ?5@2>9 AB5?5=8.  C@02=5=88 420 =58725AB=KE :>MDD8F85=B0 00 8 01. ;O 8E >?@545;5=8O =5>1E>48<>, GB>1K 0??@>:A8<8@CNI0O ?@O<0O i0?(t)=00 + 01u (0=0) ?@>E>48;0 G5@57 B>G:C A :>>@48=0B0<8 (i=i0 8 u=u0) 8 8<5;0 :@CB87=C ;8=59=>3> CG0AB:0 E0@0:B5@8AB8:8. -B>, A>3;0A=> >?@545;5=8N, AB0B8G5A:0O :@CB87=0 S=S0. ;O >?@545;5=8O :>MDD8F85=B>2 0??@>:A8<8@CNI59 DC=:F88 ?>;CG8< 420 C@02=5=8O: i0=00 + 010,  EMBED Equation.3  = S =  EMBED Equation.3 (a0 + a1u). BAN40 8<55<: 00 =i0 ; 01= S 8 B>: @025=: i0?(t) = i0 + S0 u. 1) ??@>:A8<0F8O ?>;8=><>< 2B>@>9 AB5?5=8 >;8=>< 2B>@>9 AB5?5=8 i0?(t) = 00 + 01(u-u0) + 02(u-u0)2. -B>B ?>;8=>< ?>72>;O5B CG5ABL =5;8=59=>ABL =0G0;L=>3> CG0AB:0 %., GB> ?>:070=> =0 @8A.2.5 4;O ?>;52>3> B@0=78AB>@0 (") A> AB@C:BC@>9 <5B0;;-48M;5:B@8:-?>;C?@>2>4=8: (-AB@C:BC@0) A A>1AB25==K< :0=0;>< n-B8?0.  8A.2.5. ??@>:A8<0F8O =0G0;L=>3> CG0AB:0 % ;O >?@545;5=8O 0??@>:A8<8@CNI8E :>MDD8F85=B>2 00, 01 8 02 =5>1E>48<>, GB>1K i0?(t) A>2?0;0 A i 2 B@5E C7;>2KE B>G:0E: N1 (u=u1; i=i1); N2 (u=u2; i=i2); N3 (u=u3; i=i3). ">340 8<55< A8AB5<C C@02=5=89: i1 = 00 + 01(u1-u0) + 02(u2-u0)2 i2 = i0 = 00 i3 = 00 + 01(u3-u0) + 02(u3-u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i0?(t) >B i(t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i0?(t) = i0?(u) = eu . ;O 42CE ?@>872>;L=KE C7;>2KE B>G5: =0 % ?>;CG8<: i1 = i0?1(u1) =  eu1 , i2= i0?2(u2) = eu2 . >7L<5< >B=>H5=85 MB8E B>:>2:  EMBED Equation.3  = e(u1-u2), >B:C40 =( EMBED Equation.3 n i1 -  EMBED Equation.3 ni2) /( u1- u2). 0B5< ?> D>@<C;5 4;O ;N1>3> 87 B>:>2 i1 8;8 i2 2KG8A;ONB :>MDD8F85=B . =0;>38G=> <>6=> ?@>25AB8 0??@>:A8<0F8N DC=:F859 i0? = ( ue-u  1), (2.6) 345 :>MDD8F85=B  2K@0605B ?>AB>O==CN A>AB02;ONICN B>:0. 0 3@0D8:5 MB> A>>B25BAB2C5B ?0@0;;5;L=><C ?5@5=>AC M:A?>=5=BK 24>;L >A8 >@48=0B. 1) ??@>:A8<0F8O >B@57:0<8 ?@O<KE ;8=89 @8<5=OO 4;O 0??@>:A8<0F88 % <=>3>G;5= ?5@2>9 AB5?5=8, D0:B8G5A:8 ?>;=>ABLN 70<5=ONB 2AN E0@0:B5@8AB8:C ?@O<>9 ;8=859, =5 ?@8=8<0O 2> 2=8<0=85 =5:>B>@K5 =5;8=59=K5 CG0AB:8 ?@8 >G5=L <0;KE 8 >G5=L 1>;LH8E =0?@O65=8OE (B>:0E). -B>B <5B>4  ;8=59=0O 0??@>:A8<0F8O  405B =54>AB0B>G=> B>G=K5 @57C;LB0BK 4;O 48=0<8G5A:>3> 480?07>=0, ?@8 :>B>@>< 25;8:8 B>:8 8 =0?@O65=8O. 5A<>B@O =0 C:070==K9 =54>AB0B>:, <5B>4 ;8=50@870F88 % H8@>:> 8A?>;L7C5BAO 4;O @0AG5B0 ?0@0<5B@>2 CA8;8B5;59 :;0AA0 , 2 :>B>@KE @01>G89 CG0AB>: - =5 2KE>48B 70 ?@545;K B>G5:  8  @8A.1.1. 4=0:>, 2 ACI5AB25==> =5;8=59=KE ?@81>@0E =5 4>?CAB8<> AB>;L 3@C1>5 ?@81;865=85 0??@>:A8<8@CNI59 DC=:F88 : @50;L=>9 %. >;LHCN B>G=>ABL <>6=> ?>;CG8BL, 5A;8 0??@>:A8<8@>20BL E0@0:B5@8AB8:C =5;8=59=>3> M;5<5=B0 =5A:>;L:8<8 >B@57:0<8 ?@O<KE ;8=89. 0: ?@028;>, 0??@>:A8<0F8N >4=>3> 8 B>3> 65 B8?0 =5>1E>48<> ?@>8725AB8 :0: 4;O 2E>4=KE, B0: 8 4;O 2KE>4=KE 8 C?@02;ONI8E E0@0:B5@8AB8: =5;8=59=>3> M;5<5=B0. '0AB> 0??@>:A8<8@CNB B>;L:> @01>G89 CG0AB>:, 0 =5 E0@0:B5@8AB8:C ?>;=>ABLN.  ?@>AB59H5< A;CG05 <5B>4>< ;8=50@870F88 >?@545;ONB ?0@0<5B@K B>;L:> 42CE CG0AB:>2 %, 0 2 >1I5< A;CG05 <>6=> >?@545;OBL ?0@0<5B@K ;N1>3> :>;8G5AB20 CG0AB:>2 (;8=59=>-:CA>G=0O 0??@>:A8<0F8O). '5< 1>;LH5 :>;8G5AB2> CG0AB:>2, B5< 2KH5 B>G=>ABL 0??@>:A8<0F88. 0AA<>B@8< MB> =0 ?@>AB59H5< ?@8<5@5 0??@>:A8<0F88 42C<O >B@57:0<8 ?@O<KE C?@02;ONI59 % ?>;52>3> B@0=78AB>@0 A 8=4CF8@>20==K< :0=0;>< n-B8?0  @8A.2.6.  8A.2.6. ??@>:A8<0F8O 42C<O >B@57:0<8 ?@O<KE 0: A;54C5B 87 @8A.2.6 0??@>:A8<8@CNI0O DC=:F8O 8<55B 284: i0? = 0 ?@8 U<E EMBED Equation.3  8 U=E EMBED Equation.3 ;  EMBED Equation.3  i0? = S (U-E EMBED Equation.3 ) ?@8 U>E EMBED Equation.3 , 345: S - :@CB87=0 =0:;>==>3> (?@O<>;8=59=>3>) CG0AB:0; E EMBED Equation.3  - =0?@O65=85 >BA5G:8 8;8 70?8@0=8O. >MDD8F85=BK 0??@>:A8<0F88 >?@545;ONBAO =5?>A@54AB25==> ?> E0@0:B5@8AB8:5. =0;>38G=> 4;O 2KE>4=KE % B>3> 65 " (@8A.2.7.)  8A.2.7. ??@>:A8<0F8O 2KE>4=KE % "  >1;0AB8 <0;KE =0?@O65=89 =0 AB>:5 (U! >B 0 4> U!.min) AB>:>2K9 B>: i! ?>GB8 =5 7028A8B >B =0?@O65=8O =0 70B2>@5. >MB><C 2 MB>9 >1;0AB8 % 0??@>:A8<8@CNB ?@O<>9 =07K205<>9 ;8=859 3@0=8G=>3> @568<0 ( ). =0 ?@>2>48BAO 87 =0G0;0 :>>@48=0B 2 B>G:C <0:A8<0;L=>3> ?5@5H810 E0@0:B5@8AB8:8, A>>B25BAB2CNI59 <0:A8<0;L=><C B>:C, ?@8 :>B>@>< 1C45B @01>B0BL ". #@02=5=85 MB>9 ?@O<>9: i0?.! = S3@ U!, 345 S3@= EMBED Equation.3  - :@CB87=0   (3@0=8G=0O :@CB87=0) >?@545;O5BAO ?> 3@0D8:C. @0255   2KE>4=>9 B>: " (i! ) O2;O5BAO ;8=59=>9 DC=:F859 42CE ?5@5<5==KE: =0?@O65=8O AB>:0 U! 8 =0?@O65=8O 70B2>@0 U. =0 8<55B 284: i0?.! = U + U!+!, (2.7) 345: ,  - :>MDD8F85=BK, A @07<5@=>ABLN :@CB87=K; ! - :>MDD8F85=B, 8<5NI89 @07<5@=>ABL B>:0. 50;L=K5 E0@0:B5@8AB8:8 70<5=ONBAO A5<59AB2>< ?0@0;;5;L=KE ?@O<KE, >BAB>OI8E =0 >4=0 >B 4@C3>9 =0 @02=KE @0AAB>O=8OE ?@8 >48=0:>2KE ?@8@0I5=8OE =0?@O65=8O =0 70B2>@5. 0:;>= 8E >?@545;O5BAO =0:;>=>< 107>2>9 E0@0:B5@8AB8:8, A>>B25BAB2CNI59 (0,7-0,8) i!.max. 07>20O E0@0:B5@8AB8:0 A?@O<;O5BAO 4> ;8=88 3@0=8G=>3> @568<0, 0 2A5 >AB0;L=K5 ?@>2>4OBAO ?0@0;;5;L=> 59. 0AAB>O=85 <564C =8<8 2K45@68205BAO @02=K<8 8=B5@20;C <564C 107>2>9 8 A>A54=8<8 =59 E0@0:B5@8AB8:0<8. ?@545;8< :>MDD8F85=BK , , ! 2 2K@065=88 (2.7). "0: :0: :@CB87=0 S =  EMBED Equation.3  (?@8 U! = const) 8 2=CB@5==55 A>?@>B82;5=85 R EMBED Equation.3 = EMBED Equation.3 ( ?@8 U = const), B> 87 (2.7) ?>;CG8<: = S ,  = 1/ R EMBED Equation.3 . ">340 2K@065=85 4;O B>:0 AB>:0: i0?.! = S U + U! /R EMBED Equation.3  + !. (2.8) A;8 U =  EMBED Equation.3 , 345  EMBED Equation.3  - =0?@O65=85 70?8@0=8O, B>, A>3;0A=> @8A.2.7 i.!= 0; U!=!. B:C40 ?>;CG8<: ! = -( S  EMBED Equation.3  +  /R EMBED Equation.3 ). (2.9) >4AB028< (2.8) 8 (2.9) 2 (2.7) 8 ?>;CG8<: i0?.! = S [U   EMBED Equation.3  +  EMBED Equation.3  (U!-E!)]. (2.10) -B> 8 5ABL 0??@>:A8<8@CNI0O DC=:F8O ?@0255  , 2 :>B>@>9: S - :@CB87=0 C?@02;ONI59 E0@0:B5@8AB8:8;  EMBED Equation.3  - =0?@O65=85 >BA5G:8 AB>:>2>3> B>:0, A>>B25BAB2CNI55 E0@0:B5- @8AB8:5 ?5@5A5:0NI59 >AL 01AF8AA 2 B>G:5 U!=! (i.!= 0);  EMBED Equation.3  = SR EMBED Equation.3  - AB0B8G5A:89 :>MDD8F85=B CA8;5=8O. 8DD5@5=F80;L=K5 ?0@0<5B@K: S,  EMBED Equation.3 , R EMBED Equation.3  8  EMBED Equation.3  - >?@545;ONBAO 3@0D8G5A:8. >;=>5 2K@065=85 0??@>:A8<8@>20==KE 2KE>4=KE E0@0:B5@8AB8: 8<55B 284: 0 ?@8 U!<0 i0?.! S3@ U! =0   S[U -  EMBED Equation.3 +  EMBED Equation.3  (U!-E!)] ?@0255  . ++: ??@>:A8<0F8O =5;8=59=>3> @568<0 @01>BK 4>AB0B>G=> B>G=> >ACI5I5AB2;O5BAO >4=>9 M:A?>=5=F80;L=>9 DC=:F859, AC<<>9 42CE M:A?>=5=B 8;8 AC<<>9 ?@O<>9 ;8=88 A M:A?>=5=B>9. 8=50@870F8O % ?@8<5=O5BAO 2 42CE 20@80=B0E: 70<5=0 E0- @0:B5@8AB8:8 42C<O >B@57:0<8 ?@O<KE ;8=89 8 ;8=59=>-:CA>G=0O 0?-@>:A8<0F8O =5A:>;L:8<8 >B@57:0<8 ?@O<KE. 3. >MDD8F85=BK 0??@>:A8<8@CNI8E ?@O<KE 8 ?0@0<5B@K - >?@545;ONBAO 3@0D8G5A:8.  3.   "+  -" . &5;K9 @O4 @048>B5E=8G5A:8E :0A:04>2 8, 2 ?5@2CN >G5@54L, CA8;8B5;8 <>I=>AB8 @01>B0NB 2 48=0<8G5A:>< =5;8=59=>< @568<5 1>;LH>3> A83=0;0 8;8 65 2 @568<5 A >BA5G:>9 B>:0. >A;54=89 ?>72>;O5B ACI5AB25==> ?>2KA8BL  CAB@>9AB20, >4=0:> ?@82>48B 8 : 2>7@0AB0=8N 0<?;8BC4 2KAH8E 30@<>=8: 2KE>4=>3> A83=0;0, B.5. : 2>7@0AB0=8N =5;8=59=KE 8A:065=89 ?> A@02=5=8N A @568<>< 157 >BA5G:8 B>:0. K40NI89AO A>25BA:89 CG5=K9 ..5@3 2 30-E 3>40E %% 25:0 @07@01>B0; <5B>4 @0AG5B0 :0A:04>2, @01>B0NI8E 2 @568<5 A >BA5G:>9 B>:0. -B>B <5B>4 H8@>:> ?@8<5=O5BAO 8 2 =0AB>OI55 2@5<O :0: 4;O @0AG5B>2 ;0<?>2KE, B0: 8 B@0=78AB>@=KE :0A:04>2. 0AA<>B@5=85 C3;0 >BA5G:8 ?>72>;O5B CA2>8BL >A=>2K MB>3> <5B>40. @><5 B>3>, 2 ?@>F5AA5 2KG8A;5=8O ?0@0<5B@>2 ?> <5B>4C ..5@30 0=0;878@CNBAO >A>15==>AB8 @01>BK -, 8<5NI85 1>;LH>5 7=0G5=85 ?@8 2K1>@5 >1;0AB8 ?@8<5=5=8O 40==>3> =5;8=59=>3> M;5<5=B0. @8 87CG5=88 <5B>4>2 @0AG5B0 @568<0, 8A?>;L7CNI8E B>B 8;8 8=>9 284 0??@>:A8<0F88 AB0B8G5A:8E 2>;LB-0<?5@=KE E0@0:B5@8AB8:, A;54C5B C45;8BL 2=8<0=85 A;54CNI8< 2>?@>A0<: 0: A =5>1E>48<>9 B>G=>ABLN 2K1@0BL 0??@>:A8<8@CNICN DC=:F8N? 0: 3@0D8G5A:8 2K?>;=8BL ?>AB@>5=85 @01>G53> CG0AB:0 8 2K- 1@0BL B>G:C ?>:>O 4;O >ACI5AB2;5=8O @0AG5B=>3> @568<0 @01>BK -? 0:8< A?>A>1>< >?@545;ONBAO :>MDD8F85=BK 0??@>:A8<8- @CNI59 DC=:F88 ?> 7040==K< 2>;LB-0<?5@=K< E0@0:B5@A8B8:0<? 0: ?@>25@8BL AB5?5=L B>G=>AB8 0??@>:A8<0F88? 3.1. 01>B0 157 >BA5G:8 B>:0  MB>< 48=0<8G5A:>< @568<5 @01>BK 2E>4=>5 =0?@O65=85 8<55B 1>;LH85 0<?;8BC4K, 2A;54AB285 G53> @01>G89 CG0AB>: >E20BK205B ACI5AB25==> =5;8=59=K5 >1;0AB8 %. $>@<0 A83=0;>2 ?@8 MB>< 8A:0605BAO  @8A.2.1. 568< @01>BK =5 <>=>30@<>=8G5:89. >7459AB285 A >4=>9 (>A=>2=>9) 30@<>=8:>9 =0 2E>45 ?@82>48B : ?>O2;5=8N 2KAH8E 30@<>=8: =0 2KE>45. 040G0 @0AG5B0 70:;NG05BAO 2 >?@545;5=88 0<?;8BC4 30@<>=8:, 8<5NI8E 7=0G5=85 4;O @01>BK ?@81>@0. "0:>9 @0AG5B =07K205BAO 30@<>=8G5A:8< 0=0;87><, 8A?>;L7CNI8< @07;>65=85 2 @O4 $C@L5 :0: 2>7459AB28O, B0: 8 @50:F88 -. 7 <0B5<0B8:8 8725AB=>, GB> 5A;8 ?5@8>48G5A:0O DC=:F8O C4>2;5B2>@O5B CA;>28O< 8@8E;5, B.5 >=0 =5?@5@K2=0, <>=>B>==0 8 8<55B ?@>872>4=K5 2KAH8E ?>@O4:>2 2 40==>< 8=B5@20;5, B> 5Q <>6=> ?@54AB028BL DC=:F8>=0;L=K< @O4>< $C@L5. ;O ?5@5<5==>9 A>AB02;ONI59 2KE>4=>3> B>:0 - >= 8<55B 284:  EMBED Equation.3 , 345:  EMBED Equation.3  - 0<?;8BC40 k-B>9 30@<>=8:8; k EMBED Equation.3  - G0AB>B0 k-B>9 30@<>=8:8;  EMBED Equation.3  - =0G0;L=0O D070 k-B>9 30@<>=8:8.  48=0<8G5A:>< @568<5 @01>G89 CG0AB>: 8<55B =5;8=59=K5 G0AB8, B>:, ?@>B5:0NI89 G5@57 -, 2K@0605BAO AC<<>9 @O40 8 ?>AB>O==>9 A>AB02;ONI59: i(t) = I0 + Imax.1 cos EMBED Equation.3 t+Imax.2cos2 EMBED Equation.3 t +Imax.3cos3 EMBED Equation.3 t +. . .+, 345 4;O ?@>AB>BK >?CI5=K =0G0;L=K5 D07K 30@<>=8:. =0;>38G=> <>6=> ?@54AB028BL <=>3>G;5= 0??@>:A8<8@CNI59 DC=:F88 B>:0, 5A;8 =0 2E>4 2>7459AB2C5B :>A8=CA>840;L=>5 =0?@O65=85: i0?(t) = 00 + 01(u-u0) + 02(u-u0)2 + 03(u-u0)3 +. . . 0<5=8< @07=>ABL =0?@O65=89 53> 30@<>=8G5A:8< 2K@065=85<: i0? (t) = 00 + 01 Umaxcos EMBED Equation.3 t + 02(Umaxcos EMBED Equation.3 t )2 + 03(Umaxcos EMBED Equation.3 t)3 +. . . >72545< 2 =5>1E>48<K5 AB5?5=8, ?>A;5 G53> 2>A?>;L7C5<AO B@83>=>-<5B@8G5A:8<8 A>>B=>H5=8O<8: cos2 EMBED Equation.3 t =  EMBED Equation.3  (1+ cos2 EMBED Equation.3 t); cos3 EMBED Equation.3 t =  EMBED Equation.3  cos EMBED Equation.3 t + EMBED Equation.3  cos3 EMBED Equation.3 t; cos4 EMBED Equation.3 t =  EMBED Equation.3  + EMBED Equation.3  cos2 EMBED Equation.3 t + EMBED Equation.3  cos4 EMBED Equation.3 t 8 B0: 40;55. >A;5 2A5E ?@5>1@07>20=89 ?>;CG8<: i0? (t) = (00 +  EMBED Equation.3 02 U EMBED Equation.3  +  EMBED Equation.3 04 U EMBED Equation.3 + & ) + + (01U EMBED Equation.3  +  EMBED Equation.3  03U EMBED Equation.3 +  EMBED Equation.3  05 U EMBED Equation.3 + & ) cos EMBED Equation.3 t + + ( EMBED Equation.3 02U EMBED Equation.3  +  EMBED Equation.3 04 U EMBED Equation.3  + & ) cos2 EMBED Equation.3 t + + ( EMBED Equation.3 01U EMBED Equation.3  +  EMBED Equation.3 05 U EMBED Equation.3 + & ) cos3 EMBED Equation.3 t + & + = = I0 +  EMBED Equation.3 , 345 I0=00 + EMBED Equation.3 02 U EMBED Equation.3 + EMBED Equation.3 04 U EMBED Equation.3 + & - ?>AB>O==0O A>AB02;ONI0O; Imax.1=01U EMBED Equation.3  +  EMBED Equation.3  03U EMBED Equation.3 +  EMBED Equation.3  05 U EMBED Equation.3 + & ; Imax.2 = EMBED Equation.3 02U EMBED Equation.3  +  EMBED Equation.3 04 U EMBED Equation.3  + & ; Imax.3 = EMBED Equation.3 01U EMBED Equation.3  +  EMBED Equation.3 05 U EMBED Equation.3 + & 8 B.4. A53> 2 40==>< @O4C (@07;>65=88) 1C45B AB>;L:> 30@<>=8: :@0B=KE  EMBED Equation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u1 = u0 + Umax cos EMBED Equation.3 t  8A.3.1. 0AG5B @568<0 <5B>4>< B@5E >@48=0B CABL B@51C5BAO >?@545;8BL ?>AB>O==CN A>AB02;ONICN I0 8 0<?;8BC4C ?5@2KE 42CE 30@<>=8: Imax.1 8 Imax.2.  MB>< A;CG05 @O4 >3@0=8G8205BAO ?5@2K<8 B@5<O G;5=0<8: i(t) = I0 + Imax.1cos EMBED Equation.3 t + Imax.2cos2 EMBED Equation.3 t . (3.1) ;O >?@545;5=8O B@5E =58725AB=KE 2K15@5< =0 3@0D8:5 B@8 >@48=0BK: i1 ?@8  EMBED Equation.3 =0 U = U 0+ Um i2 ?@8  EMBED Equation.3 = EMBED Equation.3  U = U 0 i3 ?@8  EMBED Equation.3 =  EMBED Equation.3  U = U 0- Um >4AB028< 7=0G5=8O MB8E B@5E B>:>2 8 A>>B25BAB2CNI85 8< 7=0G5=8O 0@3C<5=B0 2 D>@<C;C (3.1) i1 I 0+ Im1 + Im2 i2 I 0- Im2 i3 I 0- Im1 + Im2 5H0O A8AB5<C >B=>A8B5;L=> I 0, Im1, Im2 ?>;CG8< D>@<C;K, 8725AB=K5 ?>4 =0720=85< D>@<C;K B@5E >@48=0B: I 0=(i1+i3+2i2)/4; Im1 =(i1- i3)/2; Im2=(i1+i3-2i2)/4. !;54C5B >B<5B8BL, GB> >3@0=8G5=85 @O40 $C@L5 ?5@2K<8 42C<O G;5=0<8 ?@82>48B : >H81:5 2 >?@545;5=88 0<?;8BC4 30@<>=8:, B0: :0: ?>;8=>< 2B>@>9 AB5?5=8 =5B>G=> >B>1@0605B E0@0:B5@8AB8:C 2 7040==>< 8=B5@20;5. A;8 2>7=8:05B =5>1E>48<>ABL >?@545;8BL 30@<>=8:8 1>;55 2KA>:8E ?>@O4:>2, B> ?@81530NB : 1>;55 B>G=K< <5B>40<. 1) 5B>4 ?OB8 >@48=0B 0 @8A.3.2. >B<5G5=K ?OBL >@48=0B MB>3> <5B>40, ?>72>;ONI85 2KG8A;8BL I 0, Im1, Im2, Im3, Im4. ;O :064>9 B>G:8 A>AB02;O5BAO C@02=5=85 B>:0, 0 70B5< @5H05BAO ?>;CG5==0O A8AB5<0 87 ?OB8 C@02=5=89 A ?OBLN =58725AB=K<8.  8A.3.2. 5B>4 ?OB8 >@48=0B. i1 ?@8  EMBED Equation.3 =0 U = U 0+ Um i2 ?@8  EMBED Equation.3 = EMBED Equation.3  U = U 0 +  EMBED Equation.3 Um i3 ?@8  EMBED Equation.3 = EMBED Equation.3  U = U 0 i4 ?@8  EMBED Equation.3 =2 EMBED Equation.3  U = U 0 -  EMBED Equation.3 Um i5 ?@8  EMBED Equation.3 =  EMBED Equation.3  U = U 0- Um 5H5=85 A8AB5<K 405B D>@<C;K ?OB8 >@48=0B: I 0=[i1+i5+2(i2+ i4)]/6; Im1 =(i1- i3 + i2 - i4)/3; I m2=(i1+i5-2i3)/4; I m3=[i1-i5-2(i2-i4)]/6; Im4 =[i1+ i5 - 4(i2+ i4) +6i3]/12. 5B>4K: B@5E 8 ?OB8 >@48=0B >G5=L ?@>ABK 8 ?@8<5=ONBAO G0AB> 4;O >F5=:8 =5;8=59=>AB8 2 CA8;8B5;OE. 4=0:>, ?@8 A8;L=>9 =5;8=59=>AB8 B>G=>AB8 MB8E <5B>4>2 =5 4>AB0B>G=>. ;O 1>;LH59 B>G=>AB8 8A?>;L7CNB :>MDD8F85=B 30@<>=8::  =  EMBED Equation.3 . -B>B :>MDD8F85=B ?>:07K205B M=5@35B8G5A:>5 A>>B=>H5=85 2:;04>2 2KAH8E 30@<>=8: >B=>A8B5;L=> ?5@2>9 (>A=>2=>9) 30@<>=8:8. A;8 87<5@5=8O ?>72>;ONB >?@545;8BL B>;L:> 0<?;8BC4K =0?@O65=89 30@<>=8:, 8;8 <>I=>AB8 >B45;L=KE 30@<>=8:, B> 2 MB>< A;CG05 ?@8<5=ONBAO A;54CNI85 D>@<C;K 4;O 2KG8A;5=8O :>MDD8F85=B0 30@<>=8::  =  EMBED Equation.3 ;  =  EMBED Equation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min>E EMBED Equation.3 ). @8 MB>< B>: G5@57 =5;8=59=K9 M;5<5=B ?@>B5:0; 2 B5G5=85 2A53> ?5@8>40 =0?@O65=8O 2>7459AB28O. A;8 65 <8=8<0;L=0O 25;8G8=0 2E>4=>3> =0?@O65=8O <5=LH5 =0?@O65=8O 70?8@0=8O (Umin< E EMBED Equation.3 ), B> ?>O2;ONBAO ?@><56CB:8 2@5<5=8, :>340 B>: G5@57 - =5 ?@>B5:05B, GB> ?>:070=> =0 @8A.3.3.  8A.3.3. $>@<0 A83=0;0 2 @568<5 A >BA5G:>9 B>:0 0: 284=> 87 MB>3> @8AC=:0, B>: ?@54AB02;O5B A>1>9 ?>A;54>20B5;L=>ABL 8<?C;LA>2. 0AA<>B@8< @01>BC - 2 @568<5 1>;LH>3> A83=0;0, :>340 AB0B8G5A:85 % <>6=> 0??@>:A8<8@>20BL >B@57:0<8 ?@O<KE. =0;8B8G5A:>5 2K@065=85 0??@>:A8<8@CNI59 DC=:F88 8<55B 284: i0? = 0 ?@8 U<E EMBED Equation.3  8 U=E EMBED Equation.3 ;  EMBED Equation.3  i0? = S (U- E EMBED Equation.3 ) ?@8 U>E EMBED Equation.3 , 0 2>7459AB2CNI55 =0?@O65=85: u = u0 + Umax cos EMBED Equation.3 t . "0: :0: 2>7459AB285 ?5@8>48G5A:>5, B> 8 2KE>4=>9 B>: (A83=0;)  ?5@8>48G5A:0O DC=:F8O, ?@54AB02;ONI0O A>1>9 ?>A;54>20B5;L=>ABL :>A8=CA>840;L=KE 8<?C;LA>2. <?C;LAK B>:0 E0@0:B5@87CNBAO 42C<O >A=>2=K<8 ?0@0<5B@0<8: 0<?;8BC4>9 Imax 8 C3;>< >BA5G:8  EMBED Equation.3 . #3>; >BA5G:8  MB> ?>;>28=0 2@5<5=8 70 ?5@8>4, 2 B5G5=85 :>B>@>3> G5@57 - ?@>B5:0; B>:, 2K@065==0O 2 @0480=0E (4>;OE  EMBED Equation.3 ) 8;8 2 3@04CA0E. 0: 284=> 87 @8A.3.3, C3>; >BA5G:8  MB> 7=0G5=85 0@3C<5=B0 2KE>4=>3> A83=0;0, >BAG8BK205<>5 >B <><5=B0, :>340 2KE>4=>9 B>: 4>AB8305B <0:A8<C<0 4> <><5=B0, :>340 B>: AB0=>28BAO @02=K< =C;N (>BA5:05BAO ?@8 2>7459AB2CNI5< =0?@O65=88 @02=>< =0?@O65=8N 70?8@0=8O). @8  EMBED Equation.3 = EMBED Equation.3  u = E EMBED Equation.3 . @8 MB>< =0?@O65=85 ?@8=8<05B 284: E EMBED Equation.3  = u0 + Umax cos  EMBED Equation.3 , >B:C40 cos  EMBED Equation.3  = EMBED Equation.3 , (3.2) ?@8G5<, 2A5 =0?@O65=8O 15@CBAO 0;351@08G5A:8 (A> A2>8<8 7=0:0<8). -B0 D>@<C;0 ?>72>;O5B >?@545;8BL 25;8G8=C C3;0 >BA5G:8 ?> 8725AB=K<: E EMBED Equation.3 , u0 8 Umax. 7 D>@<C;K (3.2) 284=>, GB> ?@8 D8:A8@>20==>< =0?@O65=88 >BA5G:8 (E EMBED Equation.3 ) 25;8G8=0 C3;0 >BA5G:8 7028A8B >B @568<0 @01>BK -, B.5. >B 25;8G8=K =0?@O65=8O A<5I5=8O u0 8 0<?;8BC4K ?5@5<5==>3> =0?@O65=8O Umax. 028A8<>ABL  EMBED Equation.3 = EMBED Equation.3 (u0,Umax), ?@8 CA;>288 E EMBED Equation.3 = Aonst, 8;;NAB@8@C5BAO =0 @8A.3.4 4;O @07;8G=KE @568<>2 @01>BK =5;8=59=>3> M;5<5=B0.  8A.3.4. #3>; >BA5G:8 @07;8G=KE @568<>2  :0G5AB25 ?@8<5@0 =0 MB>< @8AC=:5 @0AA<>B@5=0 AB0B8G5A:0O % A8;>2>3> 48>40, @01>B0NI53> 2 48=0<8G5A:>< @568<5.  7028A8<>AB8 >B 25;8G8=K C3;0 >BA5G:8 @07;8G0NB A;54CNI85 @568<K @01>BK -, 0 A>>B25BAB25==> 8 @048>B5E=8G5A:8E CAB@>9AB2: - :;0AA  ?@8  EMBED Equation.3 = EMBED Equation.3  (@568< 157 >BA5G:8 B>:0); :;0AA  ?@8  EMBED Equation.3 /2< EMBED Equation.3 < EMBED Equation.3 ; :;0AA  ?@8  EMBED Equation.3 = EMBED Equation.3 /2; :;0AA ! ?@8  EMBED Equation.3 < EMBED Equation.3 /2. 0: C65 >B<5G0;>AL, B>:, ?@>B5:0NI89 G5@57 - 2 @568<5 A >BA5G:>9 B>:0, ?@54AB02;O5B A>1>9 ?5@8>48G5A:CN ?>A;54>20B5;L=>ABL 8<?C;LA>2, :>B>@CN <>6=> @07;>68BL 2 @O4 $C@L5 A CG5B>< G5B=>AB8 DC=:F88 :>A8=CA0. '0AB>B0 ?5@2>9 30@<>=8:8 AG8B05BAO >A=>2=>9, 8<5==> >=0 2E>48B 2 0@3C<5=B @8A.3.4. =0;8B8G5A:>5 2K@065=85 B>:0 8<55B 284: i = S(u0-Umaxcos  EMBED Equation.3 t - E EMBED Equation.3 ). (3.3) @8  EMBED Equation.3 = EMBED Equation.3 , B>: @025= =C;N i =0 8 ?>;CG05< 2K@065=85: 0 = S(u0-Umaxcos EMBED Equation.3  - E EMBED Equation.3 ). (3.4) KG8B0O (3.3.) 87 (3.4.), ?>;CG8<: i = SUmax(Aos  EMBED Equation.3 t - cos EMBED Equation.3 ). (3.5) @8  EMBED Equation.3 t =0 B>: ?@8=8<05B <0:A8<0;L=>5 7=0G5=85. Imax = SUmax(1 - cos EMBED Equation.3 ). (3.6) -B8 D>@<C;K 8A?>;L7CNB 4;O >?@545;5=8O 0<?;8BC4 30@<>=8: ?> <5B>4C C3;0 >BA5G:8. 07;>68< 2 @O4 $C@L5 (3.5): Imax.1 =  EMBED Equation.3  EMBED Equation.3 i(t)cos  EMBED Equation.3 t d( EMBED Equation.3 t) = =  EMBED Equation.3  EMBED Equation.3 S Umax (cos EMBED Equation.3 t.- cos EMBED Equation.3 ) cos EMBED Equation.3 t d( EMBED Equation.3 t) = = S Umax EMBED Equation.3 (  EMBED Equation.3  EMBED Equation.3 d( EMBED Equation.3 t) + EMBED Equation.3  EMBED Equation.3 cos 2 EMBED Equation.3 t d( EMBED Equation.3 t) - - cos EMBED Equation.3  EMBED Equation.3 cos  EMBED Equation.3 t d( EMBED Equation.3 t) = S Umax EMBED Equation.3 ( EMBED Equation.3  - sin EMBED Equation.3  cos  EMBED Equation.3 ). (3.7) B=>H5=85 0<?;8BC4K ?5@2>9 30@<>=8:8 : <0:A8<0;L=>9:  EMBED Equation.3 1= Imax.1/Imax =  EMBED Equation.3 . (3.8) -B> 2K@065=85 7028A8B B>;L:> >B C3;0 >BA5G:8 8 >1>7=0G05BAO  EMBED Equation.3 ( EMBED Equation.3 ). =0;>38G=> A>AB028< >B=>H5=8O 4;O 4@C38E A>AB02;ONI8E 2KE>4=>3> B>:0:  EMBED Equation.3 0 = I0/Imax =  EMBED Equation.3 ; (3.9)  EMBED Equation.3 2= Imax.2/Imax = EMBED Equation.3 . (3.10) B=>H5=85 0<?;8BC4K ;N1>9 30@<>=8:8 (k>2) : <0:A8<0;L=>9:  EMBED Equation.3 . (3.11) >MDD8F85=BK (3.11) =07K20NBAO :>MDD8F85=B0<8 @07;>65=8O :>A8=CA>840;L=>3> 8<?C;LA0 (:>MDD8F85=BK 5@30). =8 @0AAG8B0=K 8 A2545=K 2 B01;8FK 8;8 >?@545;ONBAO ?> A?5F80;L=K< 3@0D8:0<, ?>:070==K< =0 @8A.3.5.  8A.3.5. >MDD8F85=BK 5@30. 7 3@0D8:0 284=>, GB> 0<?;8BC4K 30@<>=8: <0:A8<0;L=K ?@8 2?>;=5 >?@545;5==KE (>?B8<0;L=KE) C3;0E >BA5G:8:  EMBED Equation.3 k.>?B= EMBED Equation.3 , 345 k  =><5@ 30@<>=8:8. 7 @8A.3.5 284=>, GB> Imax.1 <0:A8<0;L=0 ?@8 1200, Imax.2 <0:A8<0;L=0 ?@8 600, Imax.3 <0:A8<0;L=0 ?@8 400 8 B.4. "0:8< >1@07><, 2K18@0O @568< @01>BK - <>6=> >15A?5G8BL 2 2KE>4=>< B>:5 <0:A8<0;L=CN 0<?;8BC4C B>9 8;8 8=>9 30@<>=8:8. 0AG5B @01>BK 48>40 ?@>872>48BAO 2 A;54CNI5< ?>@O4:5. > 8725AB=K< E EMBED Equation.3 , u0 , Umax =0E>4OB C3>; >BA5G:8  EMBED Equation.3  ?> (3.2). cos  EMBED Equation.3  = EMBED Equation.3 , >B:C40  EMBED Equation.3  = arc cos EMBED Equation.3 . > D>@<C;5 (3.7) Imax = S Umax EMBED Equation.3 ( EMBED Equation.3  - sin EMBED Equation.3  cos  EMBED Equation.3 ) 8 ?> 3@0D8:C E0@0:B5@8AB8:8 (@8A.3.4) >?@545;O5BAO <0:A8<C< 2KE>4=>3> B>:0 A83=0;0. > B01;8F0< 8;8 3@0D8:C (@8A.3.5) =0E>4OBAO :>MDD8F85=BK 5@30 4;O A>>B25BAB2CNI53> C3;0 >BA5G:8. <?;8BC40 ;N1>9 30@<>=8:8 >?@545;O5BAO ?> D>@<C;5 Imax.k=  EMBED Equation.3 k Imax. (3.12) !;54C5B >B<5B8BL, GB> @0AG5B 2KE>4=>3> B>:0 B@0=78AB>@0 <5B>4>< C3;0 >BA5G:8 =5A:>;L:> >B;8G05BAO >B 2KH5?@82545==>3> @0AG5B0 48>40. @8G8=0 2 8=5@F8>==>AB8 ?@>F5AA0 ?@>B5:0=8O B>:0 2 B@0=78AB>@5: =0 ?@>BO65=88 ?5@8>40 ?5@5<5==>3> 2>7459AB28O 2 1075 " ?@>8AE>48B =0:>?;5=85 8 @0AA0AK20=85 M;5:B@8G5A:>3> 70@O40. >MB><C C3>; >BA5G:8 :>;;5:B>@=>3> B>:0 2 >1;0AB8 2KA>:8E G0AB>B  EMBED Equation.3 ' =5 @025= C3;C >BA5G:8 ?> 2E>4=>9 F5?8  EMBED Equation.3 %. 0AA<>B@5=85 48=0<8G5A:>3> @568<0 1>;LH>3> A83=0;0 4;O " ?>72>;O5B 0??@>:A8<8@>20BL 3@0D8: DC=:F88 7028A8<>AB8 70@O40 2 1075 >B 2@5<5=8 Q( EMBED Equation.3 t) 2 2845 A8=CA>840;L=>3> 8<?C;LA0 A 0<?;8BC4>9 Imax 8 C3;>< >BA5G:8  EMBED Equation.3 '. 0AG5B=K5 D>@<C;K, 2 MB>< A;CG05 ?@8=8<0NB 284:  EMBED Equation.3 ' =  EMBED Equation.3 .  2KG8A;5=85 0<?;8BC4K 30@<>=8: B>:0 :>;;5:B>@0 ?@>872>48BAO ?> A;54CNI8< D>@<C;0<: I0 =  EMBED Equation.3 0 ( EMBED Equation.3 ') I.max. Imax1 =  EMBED Equation.3 1 ( EMBED Equation.3 ') I.max.  >1I5< 2845: Imaxk =  EMBED Equation.3 k ( EMBED Equation.3 ') I.max. ++: 0AG5B 48=0<8G5A:>3> @568<0 <5B>4>< C3;0 >BA5G:8 =0H5; H8- @>:>5 ?@8<5=5=85 1;03>40@O ?@>AB>B5 @0AG5B0 8 4>AB0B>G=>9 B>G=>AB8. 0AG5B ?@>2>48BAO A ?><>ILN AB0B8G5A:>9 % , 48=0<8G5A:>9 (A@54=59) :@CB87=K, C3;0 >BA5G:8 8 :>MDD8F85=B>2 5@30. 3. @8 @0AAG5B5 F5?59 A 18?>;O@=K< B@0=78AB>@>< CG8BK205BAO 8=5@F8>==>ABL ?@>B5:0=8O B>:0 8 =0:>?;5=85 M;5:B@8G5A:>3> 70@O40 2 1075. .'  MB>9 G0AB8 ?>A>18O 1K;> ?>:070=>, GB> ?>;=>5 ?@54AB02;5=85 > =5;8=59=>9 F5?8 A:;04K205BAO 87 @0AA<>B@5=8O =5;8=59=KE A2>9AB2 5Q >B45;L=KE M;5<5=B>2. KOA=5=>, ?@8 :0:8E CA;>28OE @01>BK B>3> 8;8 8=>3> M;5:B@>==>3> M;5<5=B0 <>6=> ?@8<5=OBL ;8=59=K5 <5B>4K @0AG5B0 ?0@0<5B@>2 F5?8, 8 2 :0:8E A;CG0OE =5>1E>48<> ?>;L7>20BLAO ?>;>65=8O<8 =5;8=59=>9 B5>@88. !B@>3> 3>2>@O, 2A5 M;5<5=BK =5;8=59=K, 8 2K1>@ <5B>40 @0AG5B0 2> <=>3>< 7028A8B >B B@51C5<>9 B>G=>AB8 >?@545;5=8O ?0@0<5B@>2. 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A5 <5B>4K @0AG5B0, @0AA<>B@5==K5 2 MB>9 G0AB8 ?>A>18O, 1C4CB 2 40;L=59H5< ?@8<5=5=K : A;>6=K< F5?O<. " "#  >=>@>2A:89 .!. 048>B5E=8G5A:85 F5?8 8 A83=0;K. - .: 048> 8 A2O7L, 1986. - 512 A. >=B>@>28G .. 5;8=59=K5 :>;510=8O 2 @048>B5E=8:5 (02B>:>;510B5;L=K5 A8AB5<K).  .: !>25BA:>5 @048>, 1973.  320 A. 59:5 ., C4;0E $. 048>B5E=8G5A:89 A?@02>G=8:. ">< 2. 5@52>4 A =5<5F:>3>. - .-.: >AM=5@3>8740B, 1962. - 576 A. $@>;:8= ."., >?>2 .. <?C;LA=K5 8 F8D@>2K5 CAB@>9AB20.  .: 048> 8 A2O7L, 1992.  336 A. (8=0:>2 ..!., >;>4O6=K9 ... A=>2K @048>B5E=8:8.  .: 048> 8 A2O7L, 1983.  320 A. 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