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Ужгородский национальный университет

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§5. ( )

( ) ,

( ,

). ?

, . ,

dA	~	dA	= −βA , β – ,

dt		dt

“ ”, dA<0. :

dA
dA	= −βdt	ln A − ln A	= −βt	A = A e−βt	.	(17)
	= −βdt	ln A − ln A	= −βt	A = A e−βt	.	(17)
A		0		0
A

: .

(17) t , β =1,

, , ≈2,7 .

, (

). β = 1τ

, ( , ,

	−	t
), (17) :			.
	A = A e	τ
	0

, ,

(

, “ ”),

F = −α dx = −αv (α – ), dt

(2) (ma = F + F ):

m	d2x	+ α	dx	+ kx = 0	.	(18)
	dt2		dt

, ,

( )

x = A e−βt sin ωt	.	(19)
0

: ω – ,

, α – .

(19) (18).

α β ( k – “ ” ,

141

k = mw20 , w0 – ),

(18) , (19) . 1- 2-

v = dx = A0we−βt cos wt - A0be−βt sin wt , dt

a= d2x = -A0w2e−βt sin wt - A0bwe−βt cos wt - A0bwe−βt cos wt + A0b2e−βt sin wt dt2

(18), , : [m(b2–w2) – ab + k]sinwt + (aw–2mwb)coswt = 0.

sin i cos 0, 0

(1) m(b2–w2) – ab + k = 0

| k= mw2

–

(2)

aw–2mwb = 0

b =

a = 2mb –

(1)

m(b2–w2) – 2mb2 + m w2

= 0

w =

w2 - b2 =

w2 -

. (20)

4m2

(20)

w0 b (

a). w w0.

1) : b<<w0 (b 0)

w»w0.

T =

» T0

w2 -b2

w0 1 -b2

1 - b2

ç1

2 ÷

/ w2

2w0

> 0 –
.
2) : b£w0 (b»w0 , b>w0)			x
.			x
.

b=3, w0=3,14	:
) t=0	A=A0 v = 0,
x(t)=A0e-3tcos(3,14t);
) t=0	=0 v = v =tga.	0	α	t

( ) –

1 2,

(t2–t1=T ® t2=t1+T),

( . 17):

142

A = A e−βt1

A º A

= A e−βt2

= A e−β(t1 +T )

= A e−βt1 e−βT ;

A e−βt1

= eβT

= const , Þ

= bT =

q = ln

, (21)

A e−βt1

× A e−βT

–

b , b = 1t ,

q = bT =	T	Þ	1	=	t	= Ne	q =	1	.	(22)
	t		q
					T			Ne

: , ,

Ne ,

. ,

, ,

(

, ,

) (

A	x(t)
A
Acosα	Aexp(-βt)
Acosα
0	t
	t
-A

143

18.

. . . -

. . .

. . )

§1.

, ,

. ,

, .

, , ,

, – .

, , ,

( ).

, ( )

, .

. .

F= F0×cosw1t,

( ),

“ ” – .

2- :

d2x

+ a

+ kx = F cos w t

(1)

dt2

a=2b k= mw2

d2x

+ w2x =

cos w t .

(1 )

dt2

2- .

+ ,

144

= 0 -βtcos(wt) = A e−βt cos(	w2	- b2 t)	( )
0	0

– ,

. 17, (18) (19) ); x – (

) .

w1: x =Acos(w1t+j), ( )

j –

(1 ),

( ., ,

. “ ”, .1, .: , 1973 ., .254-261.):

A =

(2)

(w02 - w12 )2 + 4b2w12

m2 (w02 - w12 )2 + a2w12

2bw1

aw1

2bw1

tg j = -

= -

;

j = -arctgç

÷.

(3)

w0 - w1

m(w0 - w1 )

è w0

- w1

( ) ( (2) (3)) ( )

(1 ), . ( )

( ). ,

e-βt, ( ) ,

. ( ).

§2. .

, ,

w1.

F0,

b, .

, j

w1 w0 ,

( ):

w1 w0 , ,

, – .

(b 0), w , (2),

w0. ,

: ( w1=0 = ).

( w1) 0:

145

[(w2

- w2 )2

+ 4b2w2

= 2(w2

- w2 ) ×(-2w ) + 8b2w ,

dw1

2(w2

- w2 ) ×(-2w ) + 8b2w

= 0 - w2

+ w2

+ 2b2 = 0

: w

= ±

w2 - 2b2

;

“ ”

, ,

w =

- 2b2

(4)

, b,

b=a/2 , w w0, b=0

	w1=w (					w2	= w2	- 2b2 )
(2) :						1	0
(2) :

		Ap =	F0		.			(5)

			2mb w2	- b2
			2mb w2	- b2
			0
(4) (5) , b=0 w =w0,				.
	( 2, .						1)

( –

A(ω)
			β1
(β )		β	2
2			2
f0 2		β3
mω0
ω (β	)	ω (β )
3			1
		ω (β2) ω0		ω

. 2b2>w02,

–

(b<<w0)

. 1	Ap »	F0	.	(5 )
		2mbw0
ω0=1,41;
β1=0,18, β2=0,35, β3=0,55.	“ ” (
	) f0,	F0

x			f	0	2	k		f0
		=			= { f0=kx, w =		} =		.
		=	k		= { f0=kx, w =	m	} =	mw2	.
	0		k		0	m		mw2
								0

(4) (5): w =0 ( , 1- ) w20 = 2b2 ,

A	º A	=	F0	=	F0	. (5 )
			2b2m		mw2
p	0
				0

0, , f0 = F0:

146

mw2

1 2p

= Q

(6)

2mbw

2b T

, =b 0

–

). Q

,	(
!). , , 1/ = N ,
Q = N .	(6 )
(N – ). –

. ( ) ,

§3.

, ,

j 0 . j(w)

(3), . 2)

). w0 j /2 (90 ). b 0, w <w0

j /2; w »w0 j /2.

, , ,

(v = w)

(F=F0coswt), ( ):

= cos(wt+j),

v =

= -Awsin(wt + j) ,

–sin(wt +

) = cos wt ,

v = wcoswt = v coswt.

(3) (4),

( b>0)

- 2b2

æ w

tg j

= -

;

= arctgç

÷.

- 2b

)

- (w0

ϕ(ω)
π/2	β3
	β3
	β1	ω0
		ω0	ω
		β1	ω
		β1	β2
			β2
-π/2	+π		β3
-π/2	+π

. 2 .

tgϕ(ω)

β3
β1
ω0	ω
	β1
	β2
	β3

147

. 2 . ω0=3,14

β1=0,5 , β2=1 β3=2.

, L,

R ( ). ,

, +UR+UC=0.

= L

, U

I =

UR=RI, (q –

d2q

+ R

q = 0 – :

L m, R α, 1/C k;

, ,

( R=0)

ω2 =

T = 2π

148

19.

( ). . , ,

. .

. . .

§1. ,

– ,

, ,

, ( ),

, ,

, .

, “ ” ,

sinωt , ,

= sinωt,

–

( ). ,

r	r	r	r
, ( x	v ) ( x		\|\| v ),
		.		,
		,		( ,

) , –

. ,

, ,

, .

, (

) .

x=Acosωt, t=0 , , =0

149

( , ;

Y . t=t1

=0 ( ) ( )

, t1 ( )

t1 :

, “ ” , t–t1.

, :

x	= Acos w(t - t ) = Acos w(t -	y	)	(1)

1	1	c

, ).

, – t .

( , ) .

) , t=0		( , t=const,
)	y		2p	y		2p
x = Acos w		= Acos			= Acos		y = Acos ky ,	(1 )

1	c		T c			l

l= × – ( , – ), k=2 /l – ( 2 ).

(1 ) , : 1=f( ).

) , =0

x1 = Acos wt – .


x = Acos w(t -	y	) = Acos(wt - w		y	) = Acos(wt - ky) ,	(1 )

	c			c
– ( ) ,						–
, ( ) –						, –
			–

, .

2pn

c =

= ln =

;

k =

(1) :

	2p		y	æ t		y ö
x(y,t) = A cos		(t -		) = Acos 2pç	-		÷	,
	T		c
				è T		l ø

– , l – .

, (±Dr), .

, , , ,

. ( )

150

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