1

2

1.1 1. - ( ) . , , . 1.2 1.1 , u , . m , k , c f f(t )= .

1.1

dAlembert . f , if , sf

df . 2

1 () .

f

ufd

fs

fic

k m

3

, ( ), Hooke, () ( ), , , . , . . :

i d sf (t ) f (t ) f (t ) f(t ) m u(t ) c u(t ) k u(t ) f(t )+ + = + + = (.1) 1.3

, (.. ) ( 1.2)

1.2

, , . , . 1.2 tu

ug(t) us(t)

ut(t)=ug(t)+us(t)

4

gu -

su , :

t g su u u= + (.2.1)

t g su u u= + (.2.2) t g su u u= + (.2.3)

[.2.2] [.2.3] [.2.1] . , , . , :

i d s t s sf (t ) f (t ) f (t ) 0 m u (t) c u (t ) k u (t ) 0+ + = + + = ( )g s s sm u (t ) u (t ) c u (t ) k u (t ) 0 + + + =

s s s gm u (t ) c u (t ) k u (t ) m u (t) + + = , eff gf (t ) m u (t)= s, , , :

effm u(t ) c u(t ) k u(t ) f (t ) + + = gm u(t) c u(t ) k u(t ) m u (t) + + = (.3)

1.4 [.1] [.3] . , ( )0t 0= . ( ) ( ), . 0u(t ) 0u

0u(t ) 0u ,

5

0 0 00 0

f c u k uu(t ) um

= = . , , . , m u(t) c u(t ) k u(t ) 0 + + = 0 0u(t ) u= , 0 0u(t ) u= . c c 0= c 0 . , , , crc .

cr 0c 2 m= (.4)

o 2 o km =

oo

2T = 3 .

() ( ). :

cr o

c cc 2 m

= = (.5)

2 o 2 . 3 oT .

6

.1.

0=

o0 o 0

o

o

sin( t )u(t ) u cos( t ) u

pcos( t )

= + = +

(.6)

1

0t0 0 00 d d

d

u uu(t ) u cosh( t ) sinh( t ) e + = +

(.9)

d 2d 0 1 = , 2d 0 1 = .

[.6] ( )2

2 00

0

up u = + 1 0

0 0

utan

u =

. [.7]

( )2

2 0 0 00

0

u up u + = +

1 0 0 00 0

u utanu

+ = .

n , n

2 1i in n2

i d i n

u(t ) u 2ln ln n n2u(t nT ) u 1

dT

dd

2T = . , . . , . 1.5 of(t ) f sin( t )= of(t ) f cos( t )= . ( ) :

0mu(t) cu(t ) ku(t ) f s in( t )+ + = (.11) 0 0u(t ) u= , 0 0u(t ) u= . c pu(t ) u (t ) u (t )= + , cu (t )

pu (t ) .

0

= .

8

.2. ( of(t ) f sin( t )= )

c pu(t ) u (t ) u (t )= +

(.12)

( ) 0tc 1 d 2 du (t ) C sin( t ) C cos( t ) e= + (.13) * 1 2C ,C

( ) ( )( ) ( )2op 2 22

f 1u (t ) 1 sin( t ) 2 cos( t )k 1 2

= +

( ) ( )o

2 22

f 1 sin( t )k 1 2

p sin( t )

= + =

(.14)

( ) ( )

o2 22

f 1pk 1 2

= + ,

12

2tan1

=

( [.14]) D , ( ) .

( ) ( )2 22stmax(u) 1D( , )

u 1 2 = =

+ (.15)

0= 1= , D( , ) D(1,0) = = . ,

9

00

1 1= = = . 0

21 2=

max 21D

2 1=

.

2 2res res 00

1 2 1 2 = = . 1.6 ( Duhamel) f(t )

0t 0= . , : mu(t) cu(t ) ku(t ) f(t )+ + = 0 0 0 0u(t ) u , u(t ) u= = ( ) - ( ) .

( )( ) ( )

t0 00 d d

d

tt

dd 0

u uu(t ) u cos( t ) sin( t ) e

1 f( )sin t e dm

+ = + +

(.16)

[.16] Duhamel, . [.16] 0= d o = Duhamel [.16] . .

10

1.7 . , , . . Duhamel ( ). , . , t=t, . max stD u /u= stu

of

, ostfu k= .

.3.

Tt 2 max Tt 2 < max

0 t t ( )ofu(t ) 1 cos( t )k

= omax 2fu k= t t ( ) [ ]{

[ ]}

o

u(t )f 1 cos( t ) cos (t t )k

sin( t )sin (t t )

=

+

omax

2f tu sin( )k 2

=

tf0

f(t)

11

.4.

t 0.37101 T max t 0.37101 T < max

0 t t o

u(t )

f sin( t ) tcos( t ) 1k t t

= +

max max

21

2

max

u u(t )

1 (t )cos1 (t )

t

= + =

t t

o

o

u(t )f

[sin( t ) sin( (t t ))]kt

fcos( t )

k

=

max max

max

1

u u(t )t

cos( t ) 1tant sin( t )

==

.5.

t 0.5 T > max t 0.5 T < max , t 0.5 T = max ou f 2k=

0 t t

[ ]o 2fu(t ) sin( t ) sin( t ) ,

k(1 )

,t

= = =

max max

max

u u(t )2t(1 )

== +

t t

u(t )u(t ) sin( (t t ))

u(t )cos( (t t ))

= +

max

o2

u

f 2 cos( )k 1 2

=

f(t)

f0 t

f(t)

f0 t

12

1.8 () , 0T ( 0 ) . 1.8.1 , ( 1.3), , ,

0 00

(T ) ( ) = = = .

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5

D(,)

=0

=0.05

=0.1

=0.2

=0.3

=0.4

=0.5

=1

1.3

( ) , D

13

=0, t tT 2 = ( 1.4 1t t = )

.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t/

D

()

1.4 =0

1.8.2 () ( ) [.3] ( ) , . , , , . , [.3] :

g gc kmu(t) cu(t ) ku(t ) mu (t ) u(t ) u(t ) u(t ) u (t )m m+ + = + + =

2

2g g2

4 4u(t ) 2 u(t ) u(t ) u (t ) u(t ) u(t ) u(t ) u (t )T T + + = + + =

14

(.. iT T

iT i T= i 0,1, 2,...= ) . ii 0 T 0 T 0= = = , , ( ), ( - ). , . dS ,

vS , aS (

maxu , maxu t ,maxu ). - -. .

v dPS S= (.17.1)

2

vmPSmax E2

= (.17.2)

2

a dPS S= (.18.1) amax V mPS= (.18.2)

(PSv PSa) [.17.2, .18.2].

15

1.8.3 sf , . max du S= , :

s d a aWf k S m PS PS Wg

= = = = (.19)

aPSg

= , - g ( ). , ( , ) ( ) , . [.20.1-3]. : b sV f W= = (.20.1) : b sM h f h W= = (.20.2) : ac d2 2 2

o

PSv E I v E IM Sh h = = (.20.3)

v=3 , v=6 h .

1.8.4 , . ,

16

, . - , sf . , - [..6].

.6. ( )

0 T

.

18