я . . . . . - 2005 -2- 621.391.828 / . . . .- . : 2005. - 34 . " , " " , " " ". , , . . . . . . . . , 2005 , -3- . 1. Ч я 1.1. (1768 – 1830 . .), 21 1807 . , , , . , , . , (t1, t2) u1(t), u2(t),…, un(t), … , . . t 0, i j, 2 u (t )u j (t )dt k 0, (k ), t i i i 1 i j. , s(t): s(t ) c u (t ) c u (t ) ... cnu n (t ) ... ci ui (t ), 11 2 2 i1 (1.1) t 1 2 ci s(t )ui (t )dt. ki t 1 дciж ciui(t) , s(t), . -4- . дui(t)}, (t1, t2), . . : t 2 (t )dt 0, i j, u t u ( ) k , j i j. t i i 1 (1.1) : s(t ) ci ui (t ), i1 (1.2) t 1 2 * ci s(t )ui (t )dt, * ki t 1 . , , . ? , . , ( . ) , ( ) , . , , . ( ). -5- дsin nt, (t0 , t0+T) , cos ntж , t0 - T=2/ - , (t0 , t0+T) s(t) s(t ) a0 (an cos nt bn sin nt ), t0 t t0 T . n1 a0, an (1.3) : bn t T t T t T 2 0 2 0 1 0 a0 s(t ) sin ntdt, n 1,2,3,... (1.4) s(t ) cos ntdt, bn s(t )dt, an T t T t T t 0 0 0 (1.3) Ф (1.4) , : s(t ) An cos(nt n ), t0 t t0 T , n0 (1.5) An an2 bn2 , n arctg(bn / an ). (1.6) jnt дe (t0, t0+T) Ф (1.2) s (t ) t0 } (n=0, 1, 2,…), T=2/ , : jnt , t0 t t0 T , C e n n (1.7) t T . 1 0 jn t dt Cn . s (t )e T t 0 , (1.8) (1.3) (1.7) , . , , (1.4) (1.8) (1.3) . . . . . 1 Cn (an jbn ), a0 C0 , an Cn C n , bn j( Cn C n ). 2 (1.7): (1.9) -6- s(t) (t0, t0+T). s(t) . , (-, ). , , : s(t) , s(t) = s(t+mT), m=1, 2…. (1.7) ejnt = ejn(t+mT). T=2/), ( Т s(t) - Т s(t) (1.3), (1.4) (-, ), , (1.7), (1.8) . t0 1.2. (1.8) s(t) Т ( ) Ancos(nt+n) (n=1, 2, 3,…), n, =2/T), ( . . n. An Ancos(nt+n) . – , =, . 2, 3, …, . . . = , 2, 3, …, ( ) . . (1.7) . Cn e jnt (n 1,2,...) . Cn . , 2, 3, …, -7- : 2 cost e jt e jt , – , . , . Cn (1.7), , n An (1.5). . j Cn Cn e n , : , , Cn, n , , =n (n=1,2,…). (1.6) (1.9) Cn= An/2, n= n, , C-n= Cn, -n=-n, , . – , . s(t)=Acost + Bcos2t , , 2 . я 1.3. 1. , . 1. s(t) A -/2 0 /2 T T t .1 : -8- A, / 2 t / 2, s(t ) 0, / 2 t T / 2. . Cn t0 = -/2: /2 . A 1 / 2T 1 / 2 jn t Cn s(t )e jn t dt dt e jn t | Ae T / 2 T / 2 jn T / 2 n A sinn / 2 2A 2A (e jn / 2 e jn / 2 ) sin nT 2 T n / 2 2 jn T A sinn / T A sinn / q , T n / T q n / q q=T/ , , =2/T . ( s (t ) ) : A sin(n / T ) jnt e . T n n / T . 1 Cn (an jbn ) 2 (1.10) : . A sin(n / T ) A bn 0, an 2 C n 2 , a0 C0 . T n / T T : s (t ) sin(n / T ) A [1 2 cos nt ]. T n1 n / T . Cn ( .2). (1.11) -9- Cn C1 C2 C3 -4/ 0 -2/ 2 =2/T 4/ 2/ .2 . 3 3 A sin(n / 2) , n 1,2,... An 2 T n / 2 n() . sin(n / T ) n / T e j 2k 1, e j (2k 1) 1 , k = 0, 1, 2, … . , T, , 1. An ) A1 =2C1 A2 A A0 T A3 0 2 2/ 4/ n() ) 0 2 - -2 .3 - 10 - « 2. s(t ) A sin 1t , t , .4. S(t) A -2/1 -/1 2/1 /1 0 t .4 T=/1 , =21 . : t0 =0, A1 / 1 . j 2 nt 1T jn t 1 dt t e dt Cn s ( t ) e sin( ) 1 T 0 0 j t j t 1 j 2 nt A1 / 1 j (2n1)t A1 / 1 e 1 e 1 1 dt e dt e j j 2 2 0 0 A1 / 1 j (2n1)t j (2n1)t / 1 A 1 1 e dt | e 2 j 0 2 (2n 1) 0 j (2n1)t / 1 A A A 2A 1 e . | (2n 1) (2n 1) 2 (2n 1) 0 (4n 2 1) : s (t ) 2A 1 e j 21nt , n 4n21 .5. 2A/ 21 41 61 -61 -41 -21 0 -2A/3 -2A/15 -2A/3 .5 » - 11 - . 3. , Umcost, U0 (U0<Um) T=2 . .6. , Umcos = U0, . . =arccos(U0/Um). , 2 . s(t) 2 Um U0 - 0 2 .6 .6 : s(t) = Umcos t – U0, - < t <. s(t) 1 bn s(t ) sin ntdt 0, n 1,2,..., 2 : s(t ) a0 an cos nt . n1 : Um 1 (sin cos ). a0 (U m cos t U 0 )dt 2 t - 12 - : Um 1 a1 ( sin cos ). (U m cos t U 0 ) cos tdt 2 n =2, 3, 4, …: an an 2U m sin n cos n cos n sin . n(n 2 1) : a0=Um0(),…,an=Umn(), 0 ( ) 0(), 1(),…, n(), …- : 1 (sin cos ) , 1( ) ( sin cos ) , …, 1 n ( ) 2 sin n cos n cos n sin , n 2,3,... . n(n 2 1) ( ) , , , . , Д2Ж. 1.4. я я 1. : A=1 , = 0,05 , ) = 0,1 , ) = 0,25 , ) = 0,5 , ) =1,0 . 2. , t ( s (t ) .7) : A sin(n / T ) jnt jnt e e . T n n / T . . - 13 - s(t) A t t T .7 3. , « » . 8, , : s (t ) A A 2 A cos 2t cos 4t cos 2nt sin t ... ... 2 3 2 15 4n 1 S(t) A -2/ -/ / 0 2/ t .8 , ( 2). , 4. . 9, . 2 sin(n / 2) 2 A A sin(n / 4) n 2n A cos t ; cos t ; s2 (t ) : s1(t ) A 4 2 n1 n / 4 T 2 T n1 n / 2 1 A 2n s (t ) A t. sin 3 T 2 n1 n - 14 - S1(t) A -T/2 0 T/2 t S2(t) 2T A 0 -T/2 T/2 t S3(t) A 0 -T t T .9 5. s(t) ( . . 10). S(t) A 0 .10 . s(t+) , . . . t - 15 - 2. . Ч я 2.1. , , , , , . (-, ), s(t), . S ( ) , : . jt S ( ) s(t )e dt, Ф s(t ) (2.1) . 1 . jt d , S ( )e 2 (2.2) ejt . . (-< <) . . S ( ) 1 . S ( )d , 2 ( ) . (2.1) , (2.1) (2.2) , (2.2) – . s(t) S ( ) Ф . . : s(t) S ( ) . , . S ( ) s(t) . . - 16 - (1.8) (2.1) , . , , . j ( ) S ( ) A( ) jB ( ) S ( )e , (2.3) A( ) s(t ) costdt, B( ) s(t ) sin tdt, S ( ) A2 ( ) B 2 ( ) , B( ) ( ) arctg . A( ) S() . () S ( ) ( Ч ) ( s(t). , (2.4) Ч ( Ч ) - ) , Ч () - (2.4) S() - . (2.2) : 1 1 j[t (t )] ( ) S e d S ( ) cos[t ( )]d 2 2 j 1 ( ) sin[ ( )] S t d S ( ) cos[t ( )]d , 2 0 s(t ) S() (2.5) (). . Ф . Д1Ж . 1. . . s (t ) S ( ) , s (t ) S ( ) , 1 1 2 2 s2 (t ) s1(t t ) 0 . . jt 0. S ( ) S ( )e 1 2 S2 () = S1(), 2() = 1() - t0. (2.6) - 17 - , - t0 , (-t0), - . j t s (t ) s (t )e 0 , 2 1 . . S ( ) S ( ) , 2 1 0 , (2.7) 0 . . e 2. j t 0 . s (t ) s (kt) , . . k>1 2 1 , 0<k<1 - . . 1 . S ( ) S ( / k ). 2 k 1 , , (2.8) k . s (t ) 2 3. ds1(t ) , dt . . S ( ) j S ( ) 2 1 (2.9) . 1 . S ( ) S ( ) . 2 j 1 (2.10) t s (t ) s ( x)dx , 2 1 4. s(t)= s1(t)s2(t). . . . 1 . 1 . ( ) ( ) S ( ) S x S x dx S ( x ) S 2 1( x)dx . 2 1 2 2 (2.11) , 1/2 . - 18 - s(t ) s ( x)s (t x)dx , . . 2 1 5. s1(t) s(t) s2(t). . . . S ( ) S ( ) S ( ) . 1 2 (2.12) s(t) . 6. – . , s(t ) ri si (t ) , i , , ri - . si(t) Si ( ) , . . S ( ) ri S i ( ) . i (2.13) 7. t , . ., . s(t) S ( ) , . S (t ) 2s( ) . (2.14) , , . , E s 2 (t )dt . . (2.11) s1(t) = s2(t) = s(t) : s(t) 1 2 1 2 E s 2 (t )dt S ( ) d S ( )d . 2 0 (2.15) ( - S2() (2.15) (2.15) ) . , - 19 - , . S2() . , K ( ) s(t )s(t )dt , S2() : jt 2 S ( ) K (t )e dt , K (t ) (2.16) 1 2 jt d . S ( )e 2 , (2.17) (2.1) , , s(t ) dt . , , - (t) , . , , . 2.2. 1. я ( .11 ): A, / 2 t / 2, s(t ) 0, / 2 t / 2. (2.1): / 2 jt . A jt / 2 2 A j / 2 j / 2 S ( ) A e dt e e | (e ) j j 2 / 2 / 2 sin / 2 sin / 2 . 2 A S (0) , sin A / 2 / 2 2 (2.18) - 20 - . S (0) s(t )dt A . . 11 S() S(t) ) ) A A -/2 0 t /2 -4/ -2/ 0 2/ . 11 1 .1.3 - . : s(t)=Ae-t1(t), 2. >0, 1, t 0, 1(t) : 1(t ) 0, t 0. s(t) A A/e t =1/ 0 t .12 ( j )t . A A jt jarctg( / ) S ( ) A e t e dt A e dt e . (2.19) j 2 2 0 0 (2.19) Ч S ( ) A 2 2 Ч : , ( ) arctg( / ), - 21 - . 13 ) . ) S() A/ () /2 /4 A/2 - 0 - 0 -/4 -/2 .13 3. ( ) ( s(t ) Ae t 2 .14 ): , 0. 2 t j t 2 j t ( t 2 j t ) . 2 2 4 S ( ) A e e dt A e dt Ae dt e A 2 4 x2 dx A e 2 4 , e e x2 dx . : e ( ) .14 ). s(t) S() A ) A A e A/e t -1/ 0 1/ -2 .14 0 2 - 22 - 4. ( ). - (t) t=0, , 1. , (t )dt 1. 0 , t 0 , , t 0, (t ) , (t) – ( , . 15 ). , . . (t) , ( , ) . : s ( t ) ( t t ) dt s ( t ) (t t0 )dt s(t0 ), 0 0 (2.20) . . j t S ( ) (t )e dt e 0 1. (2.20) (t) ) S() ) 1,0 1,0 t 0 0 . 15 ( .15 ). (2.6) j t 0. (t t0 ) e , (2.21) : (t ) , j 1 . 1 j t 1 1 j t ( ) cos sin S e d e d td td costd . 2 2 2 2 0 : - 23 - (t ) 1 j t 1 e d costd , 0 2 (2.22) (t). (2.22) , , : s(t)=A. jt . 1 jt S ( ) A e dt A2 dt 2A ( ). e 2 (2.23) 5. , . 16 . (2.9), s(t). s(t) ) A -b -a 0 b a t ds(t)/dt ) A/(b-a) -b -a 0 a b t d2s(t)/dt2 ) -b -a 0 a b t .16 s(t), : d 2s A [ (t b) (t a) (t a) (t b)]. dt 2 b a - 24 - . d 2s ( j ) 2 S ( ) , dt 2 . ( j ) 2 S ( ) . S ( ) (2.21) A j b j a j a j b e e e e , b a e j a e j a e j b e j b 2 A cosa cosb . b a 2 2 2 (b a) 2 2A (2.24) b =/2, (2.24) a = 0, : 2 . 4 A 1 cos( / 2) 8 A sin2 ( / 4) A sin( / 4) S ( ) , 2 / 4 2 2 . 17 . ) S() ) s(t) A -/2 A/2 0 -8/ -4/ /2 t 0 4/ 8/ .17 . 6. S ( ) : 1, t 0, 1(t ) 0, t 0. 1(t) : 1(t ) 1/ 2 (1/ 2)sign(t ) , 1, t 0, sign(t ) 1, t 0, . S1( ) , sign(t), ( .18). – - 25 - sign(t) 1 0 t -1 2(t)=dsign(t)/dt 2 0 t .18 dsign(t ) 2 (t ) dt , (2.9) (2.21), . . ( j ) S1( ) 2, S1( ) 2 / j. (2.25) (2.23) (2.25), 1(t): . S ( ) ( ) 1 / j. , (2.26) 1(t), (2.26) . - 7. Т. s(t) (1.7) s(t ) s(t): jn t , C e n n . (1.8). =2/T, Cn . . j t j t S ( ) s(t )e dt C n e jn t e dt n . 1 j( n)t . e dt 2 C n 2 C n ( n). 2 n n (2.27) , (n=0, 1, 2, …), ( =n , ) . 2 C n . - 26 - ( (1.10), . 1), : . 2A sin(n / 2) S ( ) ( n) . T n n / 2 (2.28) (2.28) .19. S() 2A T -4/ -2/ 0 2 4/ 2/ .19 j ( t ) A j j t A j j t A j (0t ) 0 0. e e 0 e e s(t ) A cos( t ) e e 2 0 2 2 , . A j A j ( 0 ) S ( ) 2 e ( ) 2 e 0 2 2 j j Ae ( ) Ae ( 0 ). 0 . 8. S ( ) (2.29) s(t), : x(t) s(t ) x(t ) cos( t ). 0 . X ( ) . x(t) (2.11), . X ( ) (2.29), (2.20), : . 1 . j j ( 0 ) d S ( ) X ( ) e ( 0 ) e 2 . . 1 j 1 j e X ( ) e X ( ). 0 2 0 2 (2.30) - 27 - , (2.30) , x(t) . 0 . . , , - . 20 , ) . X() x(t) -/2 0. X ( ) 0 /2 -2/ t 0 2/ S()=1/2[X(-0)+X(+0)] ) s(t)=x(t)cos(0t+) -0 /2 -/2 0 0 .20 . X ( ) 9. x (t), Т x(t) T(t): x (t ) x(t ) T (t ) x(t ) (t nT ) x(nT ) (t nT ). n n , x (t) ( , Т ( . . x(t ) X ( ) . T (t ) (t nT ) . n (2.27) ), ) ( ) x(t) - 28 - . T (t ) 2 Cn ( n), n . 1 T /2 jn t dt 1 , Cn (t )e T T / 2 T =2/T. , T (t ) 2 ( n) ( ). T n (2.31) (2.11) x(t)T(t) . X ( ) (): . . . X ( ) X ( ) ( n)d X ( ) ( )d 2 2 n 1 . X ( n). T n . 21 , (2.32) , . X() x(t) ) 1,0 0 ) 1,0 0 T T(t) t x (t) t - =2/T 0 () 0 X () ) 1/T T - t 0 0 =2 .21 (2.32) . X ( ) ( . X ( ) , Т . =2/T) - 29 - 2 , . - X ( ) , . X ( ) , , . X ( ) . X ( ) , ( .21 ). , x(t) x (t). В.А.К ( : ), F = /2, x(t) x(t) , T(1/2F )=(/ ) , 2.3. я . я 1. , .22. s(t) A 0 t t .22 - 2. s(t), .23. - . - 30 - S() A - ()=- t0 0 0 .23 3. , .24, ()=0. S() A 2 -0 2 0 0 .24 4. , . , s(t) S ( ) , . 1 [s(t T ) s(t T )] S ( ) cosT . 2 , 8, . , t= 0. s(t ) A(1 e t )1(t ). 5. . 6. , : A cos t, t / 2, 0 s1(t ) t / 2. 0, ) s (t ) s (t nT ), n 1 ) s (t ) x (t nT ) cos0t, n 1 , A, t / 2, x1(t ) 0, t / 2, 0 >>2/T=, <T. - 31 - 7. , 1: 1, t / 2, x1(t ) 0, t / 2, s(t ) A(1 m cos1t ) x1(t nT ), n 1</T = /2, <T , m1. , 8. , , , ( ) , . 9. A cos t, 0 0, t / 2, s (t ) : ) 0 =20, t /2 ) 0=2 ( ). 10. ( (n+1) , . 25. s(t) 0 T nT t .25 , : j (n1)T . . 1 e S ( ) S ( ) 0 jT , 1 e . S0 ( ) - 11. . , . 26: ), - 32 - s(t) A - 0 t -A .26 12. : s(t ) A[1(t / 2) 1(t / 2)]. А . 13. A cos t, t / 2, 0 s (t ) t / 2. 0, . 14. « , » . 27: s (t) A 0 T T .27 . t - 33 - 1. . . 2. . . . . 3. . . ., , 1986. ., , 1988. . . . ., , 1982. 4. . ., . ., « . . ». 5. ., . . 6. . . . . , 1972. ., , 1971. . ., . , 1987. 7. . . . . ., . , 1989. . 1. . Ч ………………………………………………………………………..……...3 1.1. ………………………………………………….3 1.2. …………………………………………………….6 1.3. ………………………...7 1.4. ………………………………………………12 2. . Ч …………………………………………………..15 2.1. ………………………………………………...15 2.2. …………19 2.3. ……………………………………………….29 …………………………………………………………………………...33 - 34 - Уч б ич к - би . . . 603950, , . . 60 84 1/16. . . , 23. . . . 2,1. № .- . . . . 2,3. 300 . . . . 603600, . , . № 18-0099 , 37 14.05.01 .