,
(ω01 = ω02 = ω0),
x 1 = A 1os (ω0 t+ a1)
x 2 = A 2os (ω0 t + a2) (1.13)
:
x = x 1 + x 2 = A 1 cos ( ω0 t + a1 ) + A 2 cos ( ω0 t + a2 ) = A cos ( ω0 t + a ).
A a (. 1.8), , 1 2, a1 a2 . ω0,, (1.13).
(. 1.8):
(1.14)
(1.15)
5. ω0 1 = 3 , 2 = 5 = 7 . .
. (1.14),
:
6. , . .
.
(1.14):
α = - 0,93 .
:
:
:
, .. ω01 ≈ ω02 = ω, (. 1.9, ).
, p . , p p pp p : T = 2 p/D ω.
, p p
x = A 1os(ω0 t + a1), y = A 2os(ω0 t + a2).
p t , p pp :
. (1.16)
(1.16) (. 1.10), Ox Oy (. 1.10, ). (. 1.10, ) (. 1.11).
, (1.16).
1. (a2 - a1) = (2 k + 1)p/2, k = 0,1,2...
|
|
os(2 k + 1) p /2 = 0, sin(2 k + 1)p /2 =
p - p .
: A 1 = A 2 = R, pp p (p. 1.10, ).
2. (a2 - a1) = k p, os(k p) = , sin(k p) = 0.
y = A 2 x / A 1.
, , (. 1.11).
p, pp
p p p (p. 1.12), p p . , pp . pp p : p p p Ox Oy p p .
O x O y (p. 1.12) 1:2 (p p Ox O y: x p Oy, p).
7. XOY y . , . t = 0,5 ?
. , t .
.
= 10 , b = 5 . (.1.13).
. t = 0 = 0 y = 5 . , . t , y . . ,
:
p p. , .
, , . .
F = - r υ = - r dx / dt,
r - . , F u .
. (1.17)
b = r / ( 2 m) (1.18)
.
(1.17) .
x = A0 e- b t os(ω t + a). (1.19)
p ,
- , ω0 - , . . p (r = 0).
(. 1.14)
A = A 0 e - b t . (1.20)
:
.
, (b << ω0) T = T 0 = 2p/ω0.
|
|
b > ω0 , () (. 1.15).
- , .. , (. 1.14):
(1.21)
b d pp p. , p p p , , p p p p, pp p .
p b d. p, p t p (e p), p p p N e ( N e = t / T). p (1.20),
b = 1 / t, .. , p p, p e p. p (1.21) ,
, p p p pp , p e p.
β δ p c-1, p p pp.
8.
.
.
. (1.17):
9. 10 6 . 0,2 c-1. .
. (1.19) .
10
(t 2 t 1) = 10 T, .
ω =2π / T = 2π∙10β / ln1,67 = 7,8 π, -1.
, , :
, .
, , F = F 0os(ω t), F 0 - ; ω - . :
(1.22)
b - ω0 - .
p (1.22) p p p (1.17). (1.22)
x = A cos ( ω t - j ), (1.23)
A - ; j - ,
, (1.24)
. (1.25)
p (1.17) (1.19). p p p p, (. 1.16). p p (1.24), p (1.23).
p, (. 1.16).
(1.24) (1.25) , b ( ω02 - ω2). b = 0 ω = ω0 . b . . , - ω.
|
|
b ω .
(1.24) ω:
, (1.26)
(1.26) (1.24), p :
. (1.27)
(. 1.17) , b p .
11.
. ?
.
ω = 3 -1, ω02 = 4 -2, β = 0,6 -1,