nblib.library.kz - /elib/library.kz/jurnal/v_2006_2/

(1)

,

[1–3]:

0 /

, 1 /

2 2 2 2

t d

u t

u t u t u t

v

x , (1)

x E v

t ,

d – , ; – ;v₀ – -

; – ; – ,

sign( ) , – (-1;1),

.

, -

« », ,

, .

. -

:

n

k

v t x k l l l v t x l k l l l

t EF x F

0

3 2 1 1

0 3

2 1

0

4 4

,

3 2 1 2

1 0

3 2 1 1

0

t x 2 l k 4 l l l v t x 2 l l k 4 l l l

v

3 2 1 2

1 0

3 2 1 2

1

0

t x 3 l l k 4 l l l v t x 4 l l k 4 l l l

v

, (2)

k – , ;l₁, l₂, l₃– -

; – ; – ;F – ; (z) –

;v₀ – .

,

, (1) (2) :

n

k k m

i

v t x k l l l

EF a

t x x t t u

u a x

u

0

3 2 1 0

1 2 , 2 2 2 2

1 4

3 2 1 2

1 0

3 2 1 1

0

t x 2 l k 4 l l l v t x 2 l l k 4 l l l

v

3 2 1 2

1 0

3 2 1 2

1

0

t x 3 l l k 4 l l l v t x 4 l l k 4 l l l

v

, (3)

u(t,x_i) – i .

(3) « –

»;m – .

(3)

t=0: u(0,x)=0; (0, ) 0 x

x

u (4)

(2)

n

k v

l l l t k x t

t E u t x

1 0

3 2 1 0

0

0 4 0 ,

, , 0

0

3 2 1 2 1 0

3 2 1 1 0

3 2 1

1 4 2 4 2 4

v

l l l k l t l

v

l l l k t l

v l l l k t l

0

3 2 1 2 1 0

3 2 1 2

1 4 4 4

3

v

l l l k l t l

v

l l l k l

t l ,

x=L; t,L 0 (5)

, ₀ – =0,L – .

, (3)–(5)

:

n

k

v l l l k px a k

px m

i

e

v p e EF

x p a u

x p p a u p dx

x p u

d

ⁱ

0

4

1 0 2

2 2

2

0 3 2 1

, , ,

0

3 2 1 2 1 0

3 2 1 1 0

3 2 1

1 4 2 4 2 4

v

l l l k l l px v

l l l k l px v

l l l k l px

e e

e

0

3 2 1 2 1 0

3 2 1 2

1 4 4 4

3

v

l l l k l l px v

l l l k l l px

e

, (6)

t=0; u 0,p 0; ( , ) 0 dx

x p u

d , (7)

; 0

x ⁰

3 2 1 1 0

3 2

1 4

1

4 0

0

,

v

l l l k pl n

k

v l l l pk

e e

E p E p

x p u

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1

1 4 2 4 3 4 4 4

2

v l l l k l p l v

l l l k l p l v

l l l k p l

e e

e

, (8)

L

x ; ( , ) 0

x L p

u . (9) :

x p u e c e c x p

u

^a^p

p x a x

~ ,

,

₁ ₂ . (10)

, (10) -

, :

0 3 2 1 1 0

3 2

1 4

4

1 2 0 2

0 2 2

1

, 1

^v

l l l k l px v

l l l k px n

k p k

a p x

a x

e p e

v a EF

v e a

c e

c x p u

0

3 2 1 2 1 0

3 2 1

1 4 2 4 3 4 4 4

2

v

l l l k l l px v

l l l k l px

e e

xi

a m p

i

e

x p p u

a 1

,

. (11)

(3)

0 0

1 2 0 2

3

1 ,

^v

px

a pL a pL n

k

e

a v e

e

e e

e p v

a EF x a p u

0 3 2 1 2 1 0

3 2 1 1 0

3 2

1 4 2 4 2 4

4

v l l l k l p l v

l l l k p l v

l l l k pl v

l l l pk

e e

a pL a pL

a L px a

L px v

l l l k l p l v

l l l k l p l

e e

E e a

e

⁰

4 4 4

3

0 3 2 1 2 1 0

3 2 1 2 1

0 3 2 1 2 1 0

3 2 1 1 0

3 2

1 2 4 2 4

1

4 4

1

^v

l l l k l p l v

l l l k p l n

k

v l l l k pl v

l l l pk

e e

xi

a m p

i

i v

l l l k l p l v

l l l k l p l

p e x p u a e

e 1

,

1 4

4 4

3

0 3 2 1 2 1 0

3 2 1 2 1

. (12)

, :

1

, 1

,

^a^x

m p

i

e

x p p u a x

p

u

ⁱ

0 0

1

1 2

0 2

3

1

v

L a p x v

L a p x a

L px n

k

a L px a p

L s r

r

k

e e e e e

v p a EF

a

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1 1 0

3 2

1 4 2 4 2 4 3 4

4

v l l l k l p l v

l l l k l p l v

l l l k p l v

l l l k pl v

l l l pk

e e

e

0 3 2 1 1 0

3 2 1 0

3 2 1 2

1 4 4

1 2 0 2

0 4 2

4

1

v

l l l k l px v

l l l k px n

k v k

l l l k l p l

e p e

v a EF

v e a

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1

1 4 2 4 3 4 4 4

2

v l l l k l l px v

l l l k l l px v

l l l k l px

e e

n

k s

r

a L r px a

rL px s

r

a L r px a

rL px

E e e a

E e a

1 0

1 2 2

0 0

1 2 2

0

(4)

0 3 2 1 2 1 0

3 2 1 1 0

3 2

1 4 2 4 2 4

4

v l l l k l p l v

l l l k p l v

l l l k pl v

l l l pk

e e

0 3 2 1 2 1 0

3 2 1 2

1 4 4 4

3

v l l l k l p l v

l l l k l p l

e

. (13)

:

0 3 2 1 1 0

3 2

1 2 4

1

4 2

0 2

3

1 ,

^v

l l l k l a

rL p x n

k

v l l l k a

rL p x s

r

k

e e

v p a EF x a

p f

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1

1 4 2 2 4 2 3 4

2 2

v l l l k l l a

rL p x v

l l l k l l a

rL p x v

l l l k l a

rL p x

e e

e

0 3 2 1 1 0

3 2 1 0

3 2 1 2

1 4 2 2 4 2 2 4

4 2

v l l l k l a

L rL p x v

l l l k a

L rL p x v

l l l k l l a

rL p x

e e

e

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1

1 4 2 2 2 4 2 2 3 4

2 2 2

v l l l k l l a

L rL p x v

l l l k l l a

L rL p x v

l l l k l a

L rL p x

e e

e

0 3 2 1 1 0

3 2 1 0

3 2 1 2

1 4 2 1 4 2 1 4

4 2 2

v l l l k l L a

L r p x v

l l l k L a

L r p x v

l l l k l l a

L rL p x

e e

e

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1

1 4 2 1 2 4 2 1 3 4

2 1 2

v l l l k l l L a

L r p x v

l l l k l l L a

L r p x v

l l l k l L a

L r p x

e e

e

0 3 2 1 1 0

3 2 1 0

3 2 1 2

1 4 2 1 4 2 1 4

4 1 2

v l l l k l L a

L r p x v

l l l k L a

L r p x v

l l l k l l L a

L r p x

e e

e

0 3 2 1 2 1 0

3 2 1

1 4 2 1 2 4

2 1 2

v l l l k l l L a

L r p x v

l l l k l L a

L r p x

e e

0 3 2 1 2

1 4

3 1 2

v l l l k l l L a

L r p x

e

⁰

3 2 1 2

1 4

4 1 2

v l l l k l l L a

L r p x

e

0 2 1 0

1 0

2 2

1 2 0 2

0

2 ⁿ

1

^p^x_v^k ^p^x _v^l ^k ^p^x _v^l ^k ^p^x ^l_v ^l ^k

k

e e e e

p v

a EF

v

a

^p^x ^l_v ^l ^k

e

3

0 2 1

n

k s

r

v l l l k a

rL p x s

r

a L r px a

rL px v

k l l px

E e e a

1 0

4 2 0

0

1 2 2

0 4

0 3 2 1 0

2 1

(5)

0 3 2 1 2 1 0

3 2 1 2

1 4 2 4 4

3 2

v l l l k l l a

rL p x v

l l l k l l a

rL p x

e e

0 3 2 1 1 0

3 2 1 1 0

3 2

1 2 1 4 2 1 2 4

1 4 2

v l l l k l a

L r p x v

l l l k l a

L r p x v

l l l k a

L r p x

e e

e

(14)

0 3 2 1 2 1 0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 3 4 2 1 4 4

2 1 2

v l l l k l l a

L r p x v

l l l k l l a

L r p x v

l l l k l l a

L r p x

e e

e

_.

(13) :

m

i

x a p i

e

i

x p p u a x

p f x p u

1

, 1 ,

,

. (15)

(t,x) f(p,x) :

n

k s

r k

v l l l k a

rl t x

v H l l l k a

rl t x

v a EF x a t

1 0 0

3 2 1 0

3 2 1 2

0 2

3 2 4 2 4

,

0 3 2 1 1 0

3 2 1

1 4 2 4

2

v l l l k l a rL a t x v H

l l l k l a rL a t x

0

3 2 1 1 0

3 2 1

1 4 2 2 4

2 2

v

l l l k l a

rL t x

v H l l l k l a

rL t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 2 4

2 2

v

l l l k l l a

rL t x

v H

l l l k l l a

rL t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 3 4

2 3

v

l l l k l l a

rL t x

v H

l l l k l l a

rL t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 4 4

4 2

v

l l l k l l a

rL t x

v H

l l l k l l a

rL t x

0 3 2 1 0

3 2

1 2 1 4

1 4 2

v l l l k a

L r a t x v H

l l l k a

L r a t x

0 3 2 1 1 0

3 2 1

1 4 2 1 4

1 2

v l l l k l a

L r a t x v H

l l l k l a

L r a t x

0

3 2 1 1 0

3 2 1

1 4 2 1 2 4

2 1 2

v

l l l k l a

L r a t x v H

l l l k l a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 2 4

1 2 2

v

l l l k l l a

L r a t x v H

l l l k l l a

L r a t x

(6)

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 3 4

1 3 2

v

l l l k l l a

L r a t x v H

l l l k l l a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 4 4

4 1 2

v

l l l k l l a

L r a t x v H

l l l k l l a

L r a t x

0 3 2 1 0

3 2

1 2 1 4

4 1

2

v l l l k L a

L r a t x v H

l l l k L a

L r a t x

0

3 2 1 1 0

3 2 1

1 4 2 1 4

1 2

v

l l l k l L a

L r a t x v H

l l l k l L a

L r a t x

0

3 2 1 1 0

3 2 1

1 4 2 1 2 4

1 2 2

v

l l l k l L a

L r a t x v H

l l l k l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 2 4

1 2 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 3 4

1 3 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 4 4

1 4 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

0 3 2 1 0

3 2

1

2 1 4

1 4 2

v l l l k L a

L r a t x v H

l l l k L a

L r a t x

0

3 2 1 1 0

3 2 1

1 4 2 1 4

1 2

v

l l l k l L a

L r a t x v H

l l l k l L a

L r a t x

0

3 2 1 1 0

3 2 1

1 4 2 1 2 4

1 2 2

v

l l l k l L a

L r a t x v H

l l l k l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 2 4

1 2 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 3 4

1 3 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 4 4

1 4 2

v

l l l k l l L a

L r a t x v H

l l l k l l L a

L r a t x

s

r a

L t r

a H t rL E H

a

0

0 2 2 1

3 2 1 1 1 0

3 2 1

0 2 4 2 4

v l l l k l a rL a t x v H

l l l k a rL a t x E H

a ⁿ

k s

z

(7)

0

3 2 1 2 1 0

3 2 1 2

1 4 2 4 4

3 2

v

l l l k l l a rL a t x v H

l l l k l l a rL a t x H

0 3 2 1 1 0

3 2

1 2 1 4

1 4 2

v l l l k l a

L r a t x v H

l l l k a

L r a t x

H (16)

0

3 2 1 2 1 0

3 2 1

1 4 2 1 2 4

1 2 2

v

l l l k l l a

L r a t x v H

l l l k l a

L r a t x H

0

3 2 1 2 1 0

3 2 1 2

1 4 2 1 4 4

1 3 2

v

l l l k l l a

L r a t x v H

l l l k l l a

L r a t x

H .

xi

p u ,

1

1 ,

,x t x

t

u , (17)

0

2 1

1 2

2 , , 1

! , 1

,

j

i

x j x j t

x a t x

t

u , (18)

0

3 2 2 3

1 1 1

3

3 , , , ,

! , 1

,

j

i

jx x x t mx

x x j t

x a t x

t u

3 2 2 1 1 0

1 2

,

! ,

1 t x x x j x x

j a

j

i

. (19)

x

i

t u ,

2

2 0

1 1

0 0

1

! , 1

! , , 1

,

i

g j

j g g

g g i

g j

j i

i

j

jx a x x j t

x a t x

t u

g i

g j

ij i i

g

t x j x

j x a

j x t

1

1 1 0

1 1 2

1

, , 1

! 1 1

,

, (20)

x

i

t,

–

a / p

^j

exp jx

_i

p / a f p , x

_i , -

[4],

x

ⁱ_g² – g

2 2

1

, x ,..., x

_i

x

.

, :

m

i

x

i

t u x

t x t u

1

, ,

,

, (21)

x

i

t

u ,

(23).

(8)

1.Nikitin L.V., Tyurehodgaev A.N. Wave Propagation and Vibration of Elastic Rods with Interfacial Frictional Slip. Wave Motion 12 (1990) 513-526 North-Holland.

2.Nikitin L.V., Tyurehodgaev A.N. Defor d’un pipeline souterrain sous l’action de l’onde sismique. Deformation of the Underground Pipeline under Action of Seismic Wave. XII^th European Conference of Soil Mechanics and Geotechnical Engineering. 7-10 June 1999 Amsterdam, the Netherlands.

3. ., . -

//

. . 1987. 1.

. 98-106.

4. ., . -

. ., 1965.

, -

i ,

,

i .

Summary

Solution of the problem about longitudinal vibration of the rail in railway motion which consists of six-axis van and lies on ties taking into account dry friction on the “wheel-rail” contact are provided.

. . . ,

. 2.04.06 .

. .

2.

. -

, -

( -

, ) -

,

( ) -

.

- - -

, [1].

- ,

- - .

, -

- [2–4].

-

, -

- . - -

, -

. -

, ,

, -

, ,

. ,

- ,

. -

, ;

[5–7].

-

, «

»