. 13.
x(t) = A cos (ω0t + α), x(t) = A sin(ω0t + β),
x(t) = A1 cosω0t + A2 sinω0t,
β = α + π/2, A1 = A cos α, A2 = - A sin α, 2 = A21 + A22.
, - , , .
(.. , ), , , :
(2) (4).
(1), .
, , ; .. , ( ), , , , - . , (7).
, , π, ; .. , , , . , (.. ) (10) (11).
(24) (25), , , .
,
- . . .
, ,
,
A, α - .
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. :
- , ;
- . ;
- (14), (12).
;
) , , , . . , . , ;
) . , , , ( ). , ;
) , (, ) ;
) = ,
) Fx = (, , , ). , Fx - Fx = - kx, k - . Fx (13), ,
max + kx = 0, ax + ω02 x = 0,
ω0 = - .
, ω0 , . :
) - , ;
) , ( , );
)
2 1 = (F) (67)
, . (, ), , .. 1 = const. , U 2= + U. , mgx, + , - - . k, U , 1 - . U = mgx+ . , . ( ), (, - ). (F)= (F,1+ F,2), F,1 F,2 - , F,2 ;
|
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)
mgx+ (68)
,
, , , .
, + ω02 = 0, ω0 - .
, , . :
) - , ;
) , , , Fx = , , ), - , (.. );
) Fx c ;
) Fx Fx = - kx, , k (14) - (15). Fx ≠ - kx, , Fx ;
) .
. , .
, ( , ) (15) (36) .
( ) , , (38) (39). , , Fk, ( ) (44) (45),
. ( ), ( ), ( , ) . a0 (, , ..), (44) (45), g (47). , g , .
, , - .
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, .
, : ) X0 ; ) v0 ; ) , .. x0 v0. x0 v0 .
x0 v0 ( ), A (17). (17), , , x0 = 0, v0 ≠ 0, . , (17) , . , , .
, , , , .
, , , , . , (63). , 0 , (64). , - , 0, . (61) - (65).
, 11 + ω02 = 0, . , ( ), , , ,
sin α ≈ tg α ≈ α. (69)
. , ( ), , - , . , , , .
. ( ) ( ). () , , . , , . , .
, , , , , . , , , , . . , .
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, - . , ( 9, 18). , : ( ) g ( ). :
x(t) = v0cos α0 t; y(t) = v0sin α0 t 0,5gt2, (70)
v0 , α0 .
t x(t) y(t).
y(x) = - (1 + tg2α0)x2 + tgα0 x. (71)
, ,.
:
1 = 1 cos (ω1 t + α1); 2 = 2 cos (ω1 t + α2).
= 1 + 2
= cos (ω1 t + α)
2 = 12 + 22 + 21 2 cos(α1 - α2),
α (. (73)).
.
1 = 1 cos (ω1 t + α1); 2 = 2 cos (ω2 t + α2).
= (t) cos ((1 cos α1 + A2 cosy) + α(t))
w2 - w1 = W > 0, a2 + W t = y(t),
2 2 = 2 cos (ω1 t + y(t)).
(t)
2(t) = 12 + 22 + 21 2 cos(α1 - y(t)), (72)
α , .
:
(1 cos α1 + A2 cosy)cos ω1 t (1 cos α1 + A2 cosy) sin ω1 t =
cos (ω1 t + ),
( ) :
cos = (1 cos α1 + A2 cosy) /,
sin = (1 sin α1 + A2 siny) /. (73)
tg , .
, / 1/.
:
, . , . .
ω, ω + Δ ω. Δ ω ≤ ω. . , , . , , , .
. 16.
:
x1 = a cos ω t, x2 = a cos (ω + Δ ω) t.
, :
= x1+ x2 = (2 a cos ((Dw/2) t)) cos(w + Δ ω /2) t (74)
( Δ ω < ω, Δ ω /2 ω). (74) . 16. ω/Δ ω =10. . 28.
(74) , . D ω < ω , s wt , , , . (74) w, .
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,
x(t) = (t) cos (ω1 t + α1),
(t) - , .
, , (72).
x(t) = x1 (t) + x2 (t) = 1 cos (ω1 t + α1) +2 cos (ω2 t + α2) = 1 cos (ω1 t + α1) +1 cos (ω2 t + α2)+ (A2 - 1)cos (ω2 t + α2) = 2 A1cos()cos() +
+ (A2 - 1)cos (ω2 t + α2).
. 16 . . 16 .
. 16 . ,
. 16 .
. .16 , , .
11.
, :
; x2(t) = 2 sin (ωt + 300);
.
, . , x2(t).
; ;
; x2(t) = 2cos (ωt+ - ) = 2cos(ωt - ).
x1(t). , π:
x1 = 5 cos(ωt + π + ) = 5 cos (ωt + ).
, , sin ωt cos ωt:
δ :
, ,
2310,802 = (2310,802 /1800).π = 4,046
: x = 5,38 cos(ωt + 4,046 ).
. 17 .
. 17. ( - )
. , , , , . (70,71).
, , (x = 1coswt, y = A2 cos (wt + j) ),
( ):
a) = (2/1) x ( j = 0) .
b) = - (2/1) x ( j = p) .
c) x2/A12 + y2/A22 = 1 ( j = p/2) .
d) x2/A12 + y2/A22 (2xy/ 1 2)cos(j2 -j1) = sin2(j2 - j1) (75)
= 1cosw1t = 2cos(w2t +j)
( 21 22) . w1 w2 . w1/w2 , .
, , .
. 18 19.
, Dt. , , ( ), , , - t . x,y z , .
.
1 2
3 4
5 6
7 8
. 19.
1. - x = A cos(2ωt + φ), y = B cos(ωt); 2. - x = A cos(2ωt), y = B cos(ωt + φ); 3. - x = A sin(2ωt + φ), y = B sin(ωt); 4. - x = A sin(2ωt), y = B sin(ωt + φ); 5. - x = A cos(ωt), y = B cos(2ωt + φ); 6. - x = A cos(ωt + φ), y = B cos(2ωt); 7. - x = A sin(ωt), y = B sin(2ωt + φ);
8. - x = A sin(ωt + φ), y = B sin(2ωt), ( = 1, = 4, φ = 1350)
, 1,
11 + 2β1 + ω02 = 0, (76)
β , ω0 . (.20)
x = A0e-βtsin(ωt + j0). (77)
β << ω0 .
12
a B , k. . R, J, . φ, , . .
.
, ,
= S sin φ.
Ɛ :
Ɛ = - = - BS cos φ ; I = Ɛ /R.
, . F, ,
F = I a B sin α (α = 900).
.20. y = A e bX sin (wx + j0)
: (0, sin φ0), 1, 2, 3...((kπ φ0) /ω, 0). : 1 , 2, 3, : (((k + ½)p - j0)/w: (- 1)k A e-bX). : D1, D2, D3, . (k p - j0 - a)/w. : 1, 2, 3, (kp - j0 + 2a)/w, tg α = ω/b
.
φ: = 2 F h. h F. . h = (a/2) cos φ. , .
, .
, , () , ():
1 = - k φ (k = , J = 0,5 πr4, G = , J r, L - , G , μ , G 0,4 , μ = 0 - 0,5).
= 2, Ɛ = -- B 2 , I = - .
. 21. ,
J φ11 + φ1 + kc φ = 0,
(J ).
2β = , ω0 2 = .
, ω0 > β.
. , , .
, . , , : F = η. grad v.dS, grad v = dV/d - , dS - , η . , , ( ).
:
A R, , OO. J, h, h << R. , A, T, λ.
dS r .
dS = π(r+dr)2 - πr2 = 2πrdr, (78)
2- , dr.
v = ω.r,
ω - R. v
grad v = dv/d = v/h = ωr/h, (79)
.. , , .
dF = η. grad v.dS = η.(ωr/h) 2πrdr.
dF
dN = dF.r = (2π ωr/ h) r2 dr.
, , r.
.
. , () . () kφ.
Jε =- Nb- N= -bω - kφ,
b = (πωηR4)/2h
:
φ″ + 2βφ′ + ω20φ = 0,
2β = b/J; ω20 = k/J,
J . 2- , .
.
.
= + = 0,5 (kx2 + mv2).
x = A0e-βtcos(ωt + j0).
(. . 171): sin α = ω /ω0, cos α = β /ω0, ω02 = k/m, ω 2 = ω 02 + β2.
v(t) = - A0ω0 e-βtcos(ωt + j0 - α).
E (t) = 0,5 (k A02 e-2βtcos2(ωt + j0) + m A02 ω 02 e-2βt cos2(ωt +
j0 - α)) = 0.5 k A02 e-2βt(0.5(1 + cos2(ωt + j0)+1+ cos2(ωt +
j0 - α))= 0.5 k A02 e-2βt(1+cos((2 ωt+2j0+2 ωt+2j0-2
α)/2)cos((2 ωt+2j0 -2 ωt-2j0 +2 α)/2)= 0.5k A02 e-2βt(1+cosα
cos(2ωt+2j0 -α))= 0.5k A02 e-2βt(1+(β/ω0) cos(2ωt+2j0 -α))
(80)
E (t) .
() R, L . . () dt I Ɛ dt, I - , Ɛ - .
dQ dW. dQ = I2Rdt.
W = q2/2C WL = LI2/2. :
I Ɛ dt = I2Rdt + dW, (81)
W = , .
, I = ,
Ɛ, (82)
, :
2β = , ω02 = , f = .
β2 > ω02, (76) () () (. 21).
x = C1exp(-β - )t + C2 exp(-β + )t;
; . (83)
β2 = ω02,
x(t) = e-βt(x0 + (n0 + βx0)t). (84)
. 0 v0 . , (77) (83) (76) , .
. 21. (0 , t1 , , t , )