.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

.., .., .. -

. . . . . . . . . . . . 11

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

.., ..

-

, 20

.., .., .. -

- ,

. . . . . . . . . . . . . . . . . . . . . 24

.., .., .., .. -

. . . . . . . . . . . . . . . . . 29

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

..

. . . . . . . . 39

.. ,

. . . . . . . . . . . . . . . . 44

.., .., .., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

.., .. -

, -

. . . . . . . . . . . . . . . . 54

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

.., ..

. . . . . . . . . . . . . . . . . . 60

.. , -

. . . . . . . . . . 65

.., ..

. . . . . . . . . . . . . . . . . . 70

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

.. -

, . . . . . . . . . . . . . 82

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

.., ..

. . . . . . . . . . . . . . . . . . . . . . . . 92

4 .. -

. . . . . . . . . . . . . . . 97

.., .., ..

. . . . . . . . . . . . . . . . . . . . . . 102

.. - -

. . . . . . . . . . . . . . . . . . 106

.., .. .. -

. . . . . . . . . . . . . . . . 110

.. - -

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

.., .., .. -

-

. . . . . . . . . . . . . . . . . . . 117

..

. . 123

.. -

. . . . . . . . . . . . 128

.., .., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

..

. . . . . . . . . . . . . . . . . . . 147

.., ..

152

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

.., .., .. -

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

.. -

. . . . . . . . . . . . . . . . . 167

.., .., .., .. -

. . . . . . 172

.. -

. . . . . . . . . . . . . . . . . 180

.., .. , -

. . . . . . . . . . . . . . . . . . . . . . 184

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . 188

5 ..

. . . . . . . . . . . . . . . . . . . . . . . . . . 193

.., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

.., ..

. . . . . . . . . . . . . 203

.., .. -

. . . . . . . . . . . . . . . . . . . . 208

.. -

. . . . . . . . . . . . . . . . . . . . . . 213

.. -

. . . . . . . . . . . . . . . . . . . . 218

.., .., .., .. -

. . . . . . . 223

..

. .., ,

--

-

. -

, .

, -

.

, -

. ,

, -

. .

-. -

:

- [1-4. -

. -

(), (

,

).

,

.

, -

() .

, ,

-

. -

, ,

, -

. -

.

, . -

, ,

.

- [3-

4. -

, .

. .

7

() -

, -

() .

,

. -

( ). -

: ,

, , , -

, . : ,

, .

,

. , ,

, .

, -

.

[3-4 , -

, -

.

. , , -

.

, . ,

, ,

, .

-

.

, : ,

, .

: /,

( ).

-

: [5.

( - -

),

.

() -

[5. (

), -

[1, 2.

. V (T, P ) , - C(r)

(T, V ) =k(V 1 + 2/3V 3) + C2(T ) B -, K = B1 , -

8 ..

, CV .

dT

dt= T K C1V

dV

dt.

. -

-

P0 V0. - , ,

.

= T T0 (TCVT0

)d

dt=

(de1dt

)2.

-

de1/dt.

, -

.

,

[3

= m exp((U0 )(1 T/Tm)

R T),

, U0 , - , , Tm -, m , R .

U0 [3. -

-

. -

[3, (t) T (t) t

0

[m exp((U0 (t))(1 T (t)/Tm)/R T (t))]1 dt = 1.

t

-

(t) T (t) = T0 + (t). (t) . -

, -

.

t

= 1/f ,

9

f . I(t

), t

= N t

N = 1/I(t

).

( ) -

. U0 . - , .

. -

,

.

,

.

C(r) Nr =t

(r)/t

= 1/Ir(t

) ,C

(r) = C(r)Nr/Nm, Nm Nr. - CH(r) C(r) ( ) [5.

,

. CH(r) - , r Nr. , -

.

.

, -

.

.

( ).

( )

.

( )

( ). , -

( ).

, -

.

-

-

. -

,

, .

,

, . -

. -

, , , -

,

10 ..

. , -

.

.

. ,

.

( -

, ). -

, -

. -

, -

. -

, .. .

[1 .. / . . ..

. .. . .: , 1976. 414 .

[2 .., .. -

. .: , 1983. 248 .

[3 .., .. . -

? .: , 1992. 320 .

[4 . / . . .. . .. -

. .: , 1981. 440 .

[5 .., .., ..

// . . . . .: . II .

. - --: "". 1996. . I. . 120125.

Azarov A.D. The model of the polymer material with the varying mehanial properties.

The spei hanges of the mehanial deformations properties of the polymer material under

the long-term multiyloid ations are onsidered. The method of modeling is developed Using

the struture from the phase omponents, being haraterized by the loal density. The fatigue

proess, leading to the hanging of the phase omponents ontributions into the dynami

harateristis, is modelled.

.., .., ..

. .., ,

--

(), -

.

, ,

. -

. .

- ( -

) -

().

. -

:

x(t) = 0u(t) +

ni=1

ii(t)

i(t) = 1i (i(t) u(t)) (1)

(1) .

u(t) x(t). -. 0, i, i. i(t) ,

.

(1) , [1. -

z1(s) = u(s), z2i(s) =1

1 + sTiu(s), z2i+1(s) =

sTi1 + sTi

x(s) (2)

x(s) =ni=1

cizi(s) (3)

i(0) = 0, , [1.

(2) (1) Ti. (2) Ti - , , (2)

12 .., .., ..

- c(c1, c2, ..., cn). (2) , (1) -

, , ,

.

(2) -

.

z1(t) = u(t)

z2i(t) =1

Ti

t0

exp((t )Ti

)u()d (4)

z2i+1(t) = x(t) 1Ti

t0

exp((t )Ti

)x)d

c - .

() .

, ..

u(tj) x(tj) (4) - z(tj)

J(N) =

Nj=1

(x(tj) zT (tj) c)2 (5)

(5)

t, m = 1, 2, ..., N - :

Vz =

m1j=1

z(tj)zT (tj), xz =

m1j=1

z(tj)x(tj), x =

m1j=1

x2(tj) (6)

y = (x(tm) z(tm))T , V =

(V11V21

V12V22

), (7)

V11 = x, V12 = Txz, V21 = xz, V22 = Vz

V := V + yyT (8)

c -

c = V 1z xz, D(c) = 2V 1z ,

2 =x Txz cm n (9)

c , D(c) , 2 , n .

13

-

(4) , -

er .

I(t) =t0

erf()d . -

. I(ti) =ti

ti1

erf()d .

, f() . -

.

( -

).

.

i := i + i, i := i + i

J(N, i, i) =

Nj=1

(x(tj) x(tj))2

x(t) = 0u(t) +Nj=1

izai(t)

zai(t) =1

i

t0

exp((t )i

)u()d

i i , -

.

, -

/.

.

,

. -

, : ,

.

.

( ) -

. , -

.

14 .., .., ..

,

( , -

).

-

: , -

. , ,

,

.

:

. -

.

.

,

(

).

,

, /-

,

. , ,

-

[2. ( n = 2,0 = 5530, 1 = 2310, 0 = 0, 34, 1 = 3, 9) 102 10 . .

. 1

. 1 (

), .

15

[1 .. -

// . . 1991. 6.

. 8288.

[2 . . .: - . ,

1963. 535 .

Azarov A.D., Isaev K.V., Azarov I.D. Identiation of the linear dierential models

of the viso-elasti materials. The problem of identiation of the linear dierential models

(LDM), used for the desription of the properties of the viso-elasti materials, is solved.

Diulty of this problem is nonlinear parameters of time. Twostage algorithm of identiation

is proposed. The reommendations for appliation is given.

..

, --

-

. -

. -

. .

(,

) , -

. -

.

.

[1

, .

U(m)i (x, ), i,m =

1, 2, 3 x, , ( m ). - :

Cijkl(i)U(m)k,lj (x, ) +

2U(m)i (x, ) + im(x, ) = 0, i, j, k, l,m = 1, 2, 3, (1)

Cijkl(i) , , , (x, ) = (x1 1)(x2 2)(x3 3).

.

U(m)i (x, ), i,m = 1, 3 (1), -

i,m = 1, 3.

: 1 (i) =C11(i)

C033, 3 (i) =

C33(i)

C033,

5 (i) =C55(i)

C033, 7 (i) =

C13(i)

C033, k2 =

2

C033, C033 -

. , k , j (i) =

j (ik), j = 1, 3, 5, 7.

x1, x3 - :

U(m)j (x, ) =

1

421

C033

+

+

p(m)j (1, 3, k)

p0(1, 3, k)ei(1(1x1)+3(3x3))d1d3, (2)

p0(1, 3, k) = (1

21 +

5

23 k2)(521 + 323 k2) (7 + 5)22123,

17

p(m)1 (1, 3, k) = 1m(

5

21 +

3

23 k2) 3m(7 + 5)13,

p(m)3 (1, 3, k) = 3m(

1

21 +

5

23 k2) 1m(7 + 5)13.

p0(1, 3, k) . (2). i =ki,

p0(1, 3, k) = k4p0(1, 3, 1).

, , ,

p0(1, 3, 1) k, k j , j = 1, 3, 5, 7.

p0(1, 3, 1) = p0(1, 3, k).

1 = cos, 3 = sin,

p0( cos, sin, k) = a(, k)(2 21 (, k))(2 22(, k)),

21,2(, k) = (b(, k)d(, k)) (2a(, k))1,

a(, k) = 15 cos

4 + 53 sin

4 + (13 (7)2 275) cos2 sin2 ,

b(, k) = 5 + 1 cos

2 + 3 sin2 ,

d(, k) = ((1 5) cos2 + (5 3) sin2 )2 + 4(7 + 5)2 cos2 sin2 .

, -

, .. n(+ ) = n() = n(), n = 1, 2 [2. k

(k) = n(, k) = Re(n(, k)) +iIm(n(, k

)), n = 1, 2, - .

. 1

k = 0.2. - ,

.

. 1.

18 ..

(2) , , [2

U(m)j (x, ) =

1

221

C033

0

2n=1

Anjm(, k)

a(, k)F (tn(, k, r, ))d, (3)

Anjm(, k) = (1)n1(jm 2nGjm)(22 21 )1, n = 1, 2, j,m = 1, 3,G11 =

5 cos

2 + 3 sin2 ,

G33 = 1 cos

2 + 5 sin2 ,

G13 = G31 = (5 + 7) cos sin,F (z) = 0.5iei|z| (ci |z| cos |z| + si |z| sin |z|),tn(, k, r, ) = krn(, k) cos( ), r 1x1 = r cos, 3x3 = r sin.

U(2)2 (x, ) (1), i = m = 2.

4 (i) =C44(i)

C066, 6 (i) =

C66(i)

C066, k2 =

2

C066,

j (i) = j (ik), j = 4, 6, C

066 ,

.

,

U(2)2 (x, ) =

1

421

C066

+

+

ei(1(1x1)+3(3x3))

p0(1, 3, k)d1d3, (4)

p0(1, 3, k) = 621 + 4

23 k2.

,

p0(1, 3, k) = k2p0(1, 3, 1).

j , j = 4, 6 k,

p0(1, 3, 1) = p0(1, 3, k)

p0( cos, sin, k) = b(, k)(2 20(, k)),

20(, k) = (b(, k))1, b(, k) = 6 cos

2 + 4 sin2 .

-

, , -

(k) = 0(, k) = Re(0(, k)) + iIm(0(, k)) - k () .

. 2 , -

. 1.

(4), , (2), -

U(2)2 (x, ) =

1

221

C066

0

1

b(, k)F (t0(, k, r, ))d, (5)

19

. 2.

F , t0(, k, r, ) r . ,

. 1 . 2, ,

60 [3.

.. .

[1 . . .: . 1974. 338 .

[2 .., ..

. // . 2004. . 45. 5. . 131-139.

[3 Garnih M.R., Hansen A.C. A multiontinuum Approah to Strutural Analysis of

Linear Visoelasti Composite Materials. // J. of Applied Mehanis. Deember 1997.

Vol. 64. P. 795-803.

Azarova P.A. The fundamental solutions for viso-elasti orthotropi plane. The

fundamental solutions for viso-elasti orthotropi plane a

ording to the onformity

priniple are investigated. The polar sets of the Fourier's transforms in planar and antiplanar

ases are investigated. The integral presentations of the fundamental solutions are derived.

The omparison with the elasti problem is arried out.

,

.. ..

, --

, .

-

. . -

. -

,

1, 4, 8, 16-.

.

.

. - (x, y). P -M , 2a. . .

. 1.

21

M :

1) = 0(y),M = M0(y), H y 02) = 1 = 0(H),M = M1 = M0(H), < y < H

P M y + x. :

y = 0, (1)xy = 0,

{(1)y = 0, |x| > a

v(1) = ( + x (x)) = f(x), |x| a

+x P M , (x) . , -

:

y = H,{

(1)xy =

(2)xy ,

(1)y =

(2)y , |x| > a

u(1) = u(2), v(1) = v(2), |x| a

u v x y . (1) H y 0, (2) < y < H , (|x| ; y) . :

(1)y |y=0 = q(x), |x| a

P . q(x), P M , :

P =

aa

q()d,

[1.

: ,

. -

, ,

, . 2

-

, ,

1, 4, 8, 16 .

22 .., ..

. 2.

4- 2 -

. 3,

2 .

. 3.

, , -

. ,

1, 4, 8, 16-. 5,

4 -

.

. 4.

23

-

5, -

4 -

.

. 5.

[1 .., .., .., .., ..

. .:, 2006.

240 .

Aizikovih S.M., Andreeva J.L. Approximate analytial solution of the indentation

problem for a half-plane with alternating sign gradient in elasti properties oating .

The problem of the penetration of an indenter into a half-plane with a funtionally graded

elasti oating is onsidered. It is assumed that the law of the elasti properties variation

with oating depth is an arbitrary suiently smooth funtion depending on the distane

from the surfae of the oating. For redution this problem to the integral equations method

Fourier integral transformation is used. In the analysis, the previously obtained approximate

analytial asymptotially exat solution of the orresponding ontat problem is used [1. It

is investigated of the integral equations kernels transform for the example when alternating

sign the variation gradient of elasti properties of oating take plae 1,4,8 and 16 times.

- ,

.

.., .., ..

, --

- -

. ,

, ,

. -

, .. [1.

- -

, ,

.

1. .

P . (r, , z).

, a - R, . . ,

z = (r) = r2 (1)

.

P z., E(z)

, (z) .

1.E = E0(z), = 0(H), H z 02.E = E(H), = 0(H), z H (2)

, :

z = 0, zr = z = 0,

{z = 0, r > aw = (r) = (r), r a (3)

w - z, zr, z , z - , .

z = H .

(1)z = (2)z ,

(1)rz =

(2)rz , w

(1) = w(2), u(1) = u(2)

, 25

(r,z).

z(r, 0) = q(r), 0 < r a (4) a, P .

,

q(a) = 0

q(r).,

,

q(r) 0 r a. . -

:1

(0)

0

Q()L()J0(r)d = f(r), 0 r 1

0

Q()J0(r)d = 0, r > 1

(5)

Q() =

10

q()J0()d,

f(r) = (ra)/a, = /a, 0 r 1(6)

J0 - , (ar) = a2r2, = (2R)1, R -

, = H/a - - , L() - . L() , . :

minz(;0]

(z) c > 0, maxz(;0]

(z) c

26 .., .., ..

A = limz

(0)1(z), (9)

A,B,D - .

L() LN (), LN() =i=1

a2 + A2i2

a2 +B2i 2 (10)

(5) , L() LN () , :

qN (r) =2

{2A0L1N (0)

1 r2

Ni=1

Ciai

1r

sh(ait)t2 r2dt

}(11)

Ci :

Ni=1

Ciai sh ai + bi ch ai

b2k a2i+

L1N (0)b1k [B

0 + A0[1 + 2(bk + 1)b2k + 2

Ni=1

(a2i + b2i )]] = 0

(12)

B0 q(1) = 0. :

Ni=1

Ci ch(ai) + L1N (0)

[B0 + A0

[1 + 2

Ni=1

(a2i b2i

)]]= 0 (13)

ai = Ai1, bi = Bi1, B0 = (0), A0 = 2a(0).

:

P = 2a2 10

q(r)rdr

:

PN = 4a2

[43L1N (0)(0) +

Ni=1

Ci( ch ai + a1i sh ai)]

(14)

. -

-

- -

. , ,

,

.

, 27

. 1. ,

3.5 . 1 6

.

2 -

. -

, -

.

, ,

.

0,1 1 10 100

1

L()

1 3.5

1 1/3.5

2 3.5

2 1/3.5

3 3.5

3 1/3.5

. 2.

, -

. -

-

.

:

EW (a) = S(a) =3

4

P

a

1

1 2 (15) a , . -

a/H = 1.

28 .., .., ..

3

.

, -

. ,

- -

.

0,125 0,25 0,5 1 2 4

1

E

c

-1

1 3.5

1 1/3.5

2 3.5

2 1/3.5

3 3.5

3 1/3.5

. 3.

[1 .., .., .., .., ..

. .:, 2006.

240 .

Aizikovih S.M., Krenev L.I., Trubhik I.S. Penetration of a paraboli punh in

funtionally graded oating, whih Young modulus is hanged inmonotonially .

Problem is onsidered in this work on indentation in funtionally graded elasti half-spae

axially symmetri stamp. Expeted that stamp is a paraboloid of revolution, but ontat

between the stamp and oating a smooth. When solving the ontat problem is used two-way

asymptoti method, designed Aizikovih S.M. [1. In numerial simulations is analysed stress-

strain distribution of oating, whih Young modulus is sinusoidally varying, and number of

sine waves varies from one to three.

-

..

, ..

, ..

,

..

. .. ,

. --

- (),

.

,

. --

-

, .

, .

-

, -

. , -

, , Struture

Health Monitoring .

, ,

()

() . ,

[1,2 , ,

-

. ,

-

[3.

.., .., ..,

, -

, -

.

, -

. ,

, ,

.

,

[4, -

30 .., .., .., ..

, -

.

-

,

- 2D 3D

, -

.

- . [4 -

- -

( 250 ,

4 8 ). , -, [4.

2D , 3D . (

12 ) . 1 2 - ,

, ,

= t/t0, = fr(t)/fr(t0) 1 t , t0 , fr(t) - .

,

-

.

1.

, . -

, ,

.

,

2D 3D .

2. -

.

3. () : -

, , (.. ),

0.5 < < 0.75.4. ,

2D , .

, -

. :

, (. 1, 2) , , -

,

- 31

. 1.

2D

12 .

. 2.

3D

12 .

32 .., .., .., ..

,

(. 3 -

, );

-

, , -

, (. 4);

, -

(. 5).

, , -

,

-

.

. 3. 3- -

(),

0,875 ().

. 4. 4- -

0,875.

- 33

. 5. 10- : (),

0,875 ().

07-08-12193

08-08-90700-_.

[1 Lim Tae W, Kashangaki Thomas A.L. Strutural damage detetion of spae truss

strutures using best ahievable eigenvetors // AI FF Journ. - 1994 V. 32, 5.

P. 1049-1057.

[2 Jenkins L.S. Craked shaft detetion on large vertial nulear reator oolant pump //

Pro. of the onf. on Instability in Rotating Mahinery. - 1985 P. 253-266.

[3 A. Del Grosso, F. Lanato A ritial review of reent advanes in monitoring data analyses

and interpretation for ivil strutures// Pro. Eur. Conf. on Strutural Control (4ECSC).

S-Peterburg. 2008, V.1. P. 320-327.

[4 .., .., .., .. --

-

// . XI . . . -

, . --., , 2007, .1. .11-17.

Aopyan V.A., Soloviev A.N, Kabelkov A.N., Cherpakov A.V. FEM modal

analysis of the building onstrution element of the triangular onguration with the utting.

Features of behaviour of the spetrum of fundamental frequenies of the building onstrution

element of the triangular shape are investigated at presene of a fault as an utting in an

internal angle, depending on the size of an fault. The features of this spetrum onneted to

shapes of osillation and kinemati of motions in the damage area are analysed.

. ., . .

-

. -

..

. ,

.

. -

, -

.

. 1.

x1, x2, x3(. 1), x3 . |x3| (h1 + h2) , h1 , 2h2 . -

, , -

(1), -

(2).

Gm, m(m = 1, 2), Gm , m , m .

:

x1 = x1/R, x2 = x2/R, x3 = x3/h = x3/(R),

= h/R, h = h1 + h2, 1 = h1/h, 2 = h2/h,

ui(m) (x1, x2, x3) = ui(m) (x1, x2, x3)/R, ij(m) = ij(m)/(2G2

), i, j = 1, 3, m = 1, 2,

Gm = Gm/G2, G1 = G, G2 = 1.

223u1(m) + (D

2 + 2m/2) u1(m) + 0(m)1m = 0,

223u2(m) + (D2 + 2m/

2) u2(m) + 0(m)2m = 0,

223u3(m) + (D2 + 2m/

2) u3(m) + 10(m)3m = 0,

(1)

m = 1u1(m) + 2u2(m) + 13u3(m), i = /xi

(i = 1, 3

), D2 = 21 +

22 ,

0(m) = 1/(1 2m), 2m = h22m/Gm = h22/c22(m), c2(m) =Gm/m,

35

m , .

(x3 = 0)

ui(1) (x1, x2, 1) = 0,

ui(1) (x1, x2, 2) = ui(2) (x1, x2, 2) ,

i3(1) (x1, x2, 2) = i3(2) (x1, x2, 2)(i = 1, 3

).

(2)

. . [1

(1), (2)

ui(m) (x1, x2, x3) = ui(m)B (x1, x2, x3) + u

i(m) (x1, x2, x3)

(i = 1, 3; m = 1, 2

).

.

u1(m)B (x1, x2, x3) = p(m) (x3) 2B

(x1, x2) ,

u2(m)B = p(m) (x3) 1B (x1, x2) , u3(m)B = 0.(3)

(3) (1), (2),

B (x1, x2)

D2B (x1, x2)(/

)2B (x1, x2) = 0

p(m) (x3) :

p(m) (x3) + l2(m) p(m) (x3) = 0,

p(1) (1) = 0, p(1) (2) = p(2) (2) , Gp(1) (2) = p

(2) (2) ,

(4)

l2(m) = 2m +

2. x3.

(4)

p+(1)k (x3) = cos l+(2)k2 cos l+(1)k (2 x3) +

1

G

l+(2)k

l+(1)ksin l+(2)k2 sin l+(1)k (2 x3) ,

p+(2)k (x3) = cos l+(2)kx3,

p(1)k (x3) = sin l(2)k2 cos l(1)k (2 x3)

1

G

l(2)kl(1)k

cos l(2)k2 sin l(1)k (2 x3) ,

p+(2)k (x3) = sin l(2)kx3,

k

l+(2) sin l+(2)2 sin l+(1)1 +Gl+(1) cos l+(2)2 cos l+(1)1 = 0,l(2) cos l

(2)2 sin l(1)1 +Gl(1) sin l(2)2 cos l(1)1 = 0.

(5)

u1(m)B =k=1

p(m)k (x3) 2Bk , u

2(m)B =

k=1

p(m)k (x3) 1Bk , u

3(m)B = 0 (6)

36 .., ..

= 0 (5)

(1 + 1/G) cos +k + (1 1/G) cos (22 1) +k = 0,(1 + 1/G) sin k + (1 1/G) sin (22 1) k = 0,

[2.

1 = 2 = , G = 1 (G1 = G2 = G), 1 = 2 (5) [3

cos

2 + (+)2 = 0, sin

2 + ()2 = 0, 2 = h22/G.

uj(m) (x1, x2, x3) = n(m) (x3) jC

(x1, x2) (j = 1, 2) ,

u3(m) (x1, x2, x3) = q(m) (x3) C

(x1, x2)(7)

(1), (2), (7) , C (x1, x2)

D2C (x1, x2)(/

)2C (x1, x2) = 0,

n(m) (x3), q(m) (x3)

n(m) +[2m +

2(1 + 0(m)

)]n(m) + 0(m)q

(m) = 0,

q(m) + (2m +

2)/(1 + 0(m)

)q(m) +

120(m)/(1 + 0(m)

)n(m) = 0;

(8)

n(1) (1) = 0, q(1) (1) = 0, n(1) (2) = n(2) (2) , q(1) (2) = q(2) (2) ,

G[q(1) (2) +

1 n(1) (2)]= q(2) (2) +

1 n(2) (2) ,

G[12

(0(1) 1

)n(1) (2) +

(0(1) + 1

)q(1) (2)

]=

= 12(0(2) 1

)n(2) (2) +

(0(2) + 1

)q(2) (2) .

(9)

(8)

n1 (x3) = H1 cos

1(1)x3 +H

2 sin

1(1)x3 +H

3 cos

2(1)x3 +H

4 sin

2(1)x3,

q1 (x3) = Q1 sin

1(1)x3 +Q

2 cos

1(1)x3 +Q

3 sin

2(1)x3 +Q

4 cos

2(1)x3,

n+2 (x3) = H+5 cos

+1(2)x3 +H

+6 cos

+2(2)x3,

q+2 (x3) = Q+5 sin

+1(2)x3 +Q

+6 sin

+2(2)x3,

(10)

n2 (x3) = H5 sin

1(2)x3 +H

6 sin

2(2)x3,

q2 (x3) = Q5 cos

1(2)x3 +Q

6 cos

2(2)x3.

(1(m)

)2= 2m/

(1 + 0(m)

)+ ()2,

(2(m)

)2= 2m + (

)2.

Qi ,(i = 1, 6

) Hi :

Qi = Ai H

i , A

1 = 1(1)/, ... , A6 =

()2/(2(2)

).

37

(10) (9) -

Hi . -

F (,) = det{aij}= 0. (11)

aij = a+ij i = 1, 6, j = 1, 4 i = 1, 2, j = 5, 6,

a11 = cos 1(1), . . . , a

+66 = 2

(+)2cos +2(2)2, a

66 = 2

()2sin 2(2)2,

uj(m) (x1, x2, x3) =p=1

n(m)p (x3) jCp (x1, x2) (j = 1, 2) ,

u3(m) =p=1

q(m)p (x3) Cp .

(12)

. -

= h

cp/Gcp,

cp = 11 + 22, Gcp = 1G1 + 2G2.

21 = 2

1

11 + 22(1 + 2/G),

22 =

22

11 + 22(1G + 2).

. 2. . 3.

. 23 (5) + -

, . 45 (11) + - . , . . 2, 4 ,

, , . 3, 5

( ) . - -

:

= 2, 7 103 /3, G = 2, 61 1010 /2, = 0, 35; = 18, 7 103 /3, G = 15, 3 1010 /2, = 0, 29.

38 .., ..

. 4. . 5.

: 1/2 = 1/2 ,

[3 -

. ,

, -

, .

- (. 4, 5) -

. , -

(. 4), . 5.

[1 .., .. //

. 1966. . 30. . 5. . 963-970.

[2 .., ..

// . -. . . 2001. 1. . 314-321.

[3 .., .., .. -

// .

1996. . 26. . 13-19.

Altukhov E.V, Fomenko M.V. The steady osillations of a three-layer isotropi plate.

The three-dimensional problems of the steady osillations of a three-layer isotropi plate with

rigidly pinhed plane sides are onsidered in this paper. The homogeneous solutions of system

of the equations of tra in movings are got by a I.I. Vorovih's half-return method. The

dispersive equations originating at onstrution of potential and whirlwind onditions are

researhed.

..

. --

-

-

.

[1, 2. -

(h) (h) [3, 4

. , -

,

.

1.

:

u d2u

dt2= o, u = u +

1

1 2v( u) (1)

(1) [5:

2u = F + 4(1 ), F (r, , t) = 0(r, , t) + rr(r, , t)

= {r(r, , t),(r, , t)}, 20 1p220t2

=

(1

p2 1s2

)r2rt2{

r = 1(r, , t) cos +2(r, , t) sin

= 1(r, , t) sin +2(r, , t) cos , 21,2 + 1

s221,2t2

= 0

(2)

- :

1(r, , t) =

1 c

2

s2Imf

(t r cos

c ir sin

c

1 c

2

s2

), 0 < <

2(r, , t) = Re f

(t r cos

c ir sin

c

1 c

2

s2

)

0(r, , t) = Re f0

(t r cos

c ir sin

c

1 c

2

p2

)

r[1(r, , t) cos +2(r, , t) sin ]

(3)

40 .., ..

(3) s, p -, c , , , , , f(z), f0(z) - .

(h), (h) (1)

(3)

c c - :

Rh(x) = [(1 + 2)x 1](x 0.5) 2x

x 2x 1 = 0,

2 =1 22(1 ) , x = s

2/c2, 1 < x < 2

QCh(k) =

P1(k) 1 P2(k) k2 S1(k) k2 S2(k)

2k2P1(k) 21k

2 + S21(k) 2k2P2(k) 22k2 S22(k)k2 + 0.5K21 S1(k) k2 0.5K21 S2(k)

= 0Pn =

k2 k2n, S1,2 =

k2 K2n, 2n =

1 2n2(1 n) , = 1/2, n = 1, 2

k = /c, k1,2 = /p1,2, K1,2 = /s1,2, QCh(k) = 6QCh(s1/c) = 0

(4)

2. -

, ,

:

u2L2,T {H1()} =T

u(t)2H1()dt, T = (t1, t2) (5)

H1() . .

1. u(r, , t) (h), (h)

u|t=0 = u0,du

dt

t=0

= o, u(r, , t) = w(r, ) exp(iwt)

,

u(r, , t) exp(iwt)w(r, )H1() < C exp(wt), t (6) -

.

2. A = , B = d2

dt2

T L2,T{H1()} (u,v), - (5),

41

(0, ). :{(Au,v) ()(Bu,v) = 0, v C([u]L,v

)=([]L,v

)=((u)| ,v

)= 0

(7)

(0, ) , , () = 1, (7).

- [6

(8) -

. 0 < 6 1 R, . (7) R, {} , , , 1,2 , (1, R) < 1 < (2, R). - R > 0 R. 3.

{exp

[iwn

(t r cos

c

) nr

cM sin

]},{

exp[iwn

(t r cos

c

) nr

cm sin

]}, M =

1 c

2

p2, m =

1 c

2

s2

(n = 1, 2, . . .) L2,T{H1()}, = { (r, ) : 0 < r < ,0 < 6 }, T = [t1, t2].

{zk} K = {z : |z| 6 1}.

1, 2, 3, (h), -

- (h) -

.

, -

.

:

H(u) =

t2t1

a(u)dtt2

t1

b(u)dt, (B(u),u) = 1

a(u) =

W (u)d, b(u) =1

2

(du

dt

)2d,

u = u(r, , t), t2 > t1 > 0

(8)

W (u) . -

, [6, -

1, -

(6).

42 .., ..

, -

(h)

(Ch) -

. -

2, , 3, ,

(9),

uN =Nn=1

B(N)n un(z, )

z = exp[iw

(t r cos

c

) rcM sin

],

= exp[iw

(t r cos

c

) rcm sin

],

M =

1 c

2

p2, m =

1 c

2

s2

(9)

- (9) -

,

(8) [7, -

B(N)n :

Nn=1

B(N)n

t2t1

(un) um dl dt

undumdt

t2t1

d

= 0, m = 1, 2, . . . , N (10) (10)

, -

t1,2 1 -

.

3. [8 -

-

, -

-

,

, .

[1 : . 2- . / . ..

.: . 1990. 336 .

[2 ., . . . . 1987. (. . ..

. . .. ..)

43

[3 Budaev B.V., Bogy D.B. // Reyleigh wave sattering by a wedge. Part II. //Wave Motion.

1996. V. 24. No. 3. P. 307314.

[4 Budaev B.V., Bogy D.B. // Reyleigh wave sattering by two adhering wedges. //Pro.

R. So. Lond. 1998. A 454. No. 1979. P. 29492996.

[5 .., .. //

. . . 1976. . 227. 1.

. 7174.

[6 .., .. . .

. .: . 1964. 368 .

[7 .. . .: . 1970.

512 .

[8 .. . .: . 1962. 472 .

Berkovih V.N. The eet of formation of the wave eld in a sphenoid medium

under plane steady vibrations.

The paper is devoted to the investigation of wave eld in an elasti wedge exited by plane

steady harmoni vibrations. The problem is to study the existene of fae waves in a wedge and

interfae ones in a biwedge. Equations to estimate its phase veloity and ritial span of angle

of sphenoid omposants are obtained by virtue of speial representation of ommon solution of

dynami elastiity and generalized statement of the orresponding boundary value problem.

,

.

..

-

( )

, -

, . -

-

.

, , -

H , h, - . , . -

[1

. -

x y (). (). [2 -

,

|x| < a, |y| < b. |x| > a, |y| > b, 0 z H . , , , -

, ,

. , -

, . [3

,

.

= {x > a, |y| b, 0 z }

, U2 , 1/x, . [4

,

|x| a, |y| b,

. [5 -

. -

, ,

45

.

[6, , -

, . ,

y [b, b] :

f (x, y) = p (x) cosmy.

, -

-

, .

. -

, -

, ,

.

|x| > 0 , |y| > b . x > 0 , y > b . m , , b H . x

x > 0, |y| b

, -,

m > 0 m < 0 (0 - ), .

m = n/b , n = 0, 1, 2...

x > 0, |y| b, 0 < z < H

,

n = 1, 2... O (1/x) . [7

1 = {x > 0, |y| b}

m > 0 m < 0 . - y < b x , , - , x = 80H , .. , .

(m = /b) (m = 2/b). [8 , , , -

|x| < a , |y| < b .

F (, ) = psina

(sin (m ) b

m +sin (m+ ) b

m+

)

46 ..

- . -

, -

,

x > a , y > b ;

2 = {|x| a, y > b, 0 z }

m > 0 , m < 0 , , -

,

O (1/x) , . , ,

. -

. , -

: x2 + y2 = 3 , x2 + (y 1)2 = 1 . [1[8 ,

. [9 -

: ( ),

( ) ( ).

, , -

,

,

, . -

, -

= 0 , 0 . , -

|x| < a , |y| < b . x > a , y > b . - , -

= 0, 01 . , - , ,

0. , -

. -

:

,

, . -

, . -

, U1 U3 1/x , U2 1/x .

. U1 U3 ,

47

, ,

, .

-

, D . f (x, y) = p = P/SD, P = 1 (SD D ), - : ( = 1+cos ), ( =

sin 2)

( = cos , = 0, 5 , = 1 ). - .

.

. , -

, , ,

,

,

. , , , -

.

.

[1 .., .. -

/ /, -

. . 29 1986. 3359 56. 31 .

[2 ..

, : - : . . . /.-

. . . -. [: , 2004. - . 3-9

[3 .., .. -

: . . .. . . .

2005. N2. .31-35

[4 .. :

- : . . . / .-. . . -. -

: , 2005. - . 3-10

[5 .. -

: ,

: VI . .-. ., . , 27 .

2006.: 3 ./.-. . . - (). : , 2006. .

1. 56 .

[6 .., .., ..

:

. . .. . . . 2006. N4. .3-8

[7 .., .. -

: - :

. . . / .-. . . -, : , 2007. - . 65-71

48 ..

[8 .., .., .. , -

. . -

. 2007. - 6 (56) - . 30-45

[9 .. -

// -

. 2008. N2. .13-23.

Bolgova A.I. The waves in three-dimensional elasti layer with dierent loads on the daily

surfae.. The elasti three-dimensional layer with innite plate on the daily surfae, whih lies

without alienation and frition is examined. The wave and energy stream propagation is

studied in dependene on osillating on the plate surfae load setting in ultimate region.

.., ..,

.., ..

. ..,

. --

- -

. -

.

-

.

, -

. -

. -

; -

.

(, , -

),

.

,

-

. -

, :

. -

,

.

--

.

.

. -

. -

.

, -

, 1000

.

50 .., .., .., ..

1000

,

1700

.

, -

-

. : , -

, . ,

,

.

.

.

-

, . -

, -

. -

. ,

.

,

, -

. -

, , [1.

-

,

,

.

-

- .

-

. -

.

. -

, -

.

, -

, , . -

, .

, .

,

.

-

. -

, . -

1,51,7

, .

... 51

-

, , -

, .

,

, , -

. -

.

, ,

-

.

, ,

, -

.

.. [2 ,

-

,

- : , -

,

.

. -

-

23

.

,

- ,

. -

,

.

-

.

.

, -

. ,

, ,

6570 /

3.

.

. : -

52 .., .., .., ..

9095

3-4 ,

1416 , 40

34 . -

, ,

,

2436 . , -

,

20

. 1 2.

-

-

-

R

,

W,

%

,

/

F,

R

,

W,

%

,

/

F,

-

-

400 0,87 45,3 1,85 1,25 43,7 1,47

500 1,63 42,7 1,74 2,30 39,7 1,65

-

-

-

700 2,75 40,3 1,36 15 3,64 35,2 1,28 15

800 3,74 39,2 1,33 15 4,08 29,9 1,30 15

900 5,38 37,0 1,28 15 6,67 27,3 1,13 15

-

1000 10,80 32,6 1,07 35 10,40 23,7 1,07 35

1100 13,55 28,6 1,04 35 14,70 20,9 0,92 35

1. -

: R

; W ;F .

-

, -

0,25 , 1,0 . -

1:1.

(/) 0,280,62, 300500 /

3, -

0,40,6 /

3.

1,15-1,45 ,

- 2,76-3,54 , 9,00

14,75 , F15.

(D 500, D 600, D 700, D 1100)

: 0,1430,278 /(). -

-

,

.

... 53

-

-

wco

.

-

wco

.

-

75 97 75 97

-

-

400 0,087 0,197 9,9 16,6 0,089 0,191 9,3 16,0

500 0,093 0,172 10,3 17,0 0,097 0,176 10,7 16,8

-

-

-

700 0,146 0,140 11,0 17,2 0,144 0,142 11,0 17,4

800 0,173 0,117 12,4 20,8 0,178 0,115 12,6 20,3

900 0,198 0,109 12,9 21,3 0,198 0,103 13,0 21,0

-

1000 0,227 0,094 13,7 21,5 0,225 0,097 13,9 21,4

1100 0,254 0,098 14,6 21,8 0,254 0,090 14,2 21,6

2. -

: ,/(

); , /( ); W

-

, %

[1 .., .., .., ., ..

//

. , .-. . . . 1998. 4.

[2 .. -

( ) // . 1996.

9. 9.

Buravhuk N.I., Gurjanova O.V., Okorokov E.P., Pavlova L.N. Eet materials

from tehnogeni raw material on properts of a ellular onrete. Inuene of harms and burnt

mine breeds on physiomehanial properties of ellular onrete is investigated. Eieny of

use of suh additives in tehnology of ellular onrete is proved.

,

.

.., ..

-

( )

-

, -

k N . ,

V ,

Q (x, t) (z, t). -

.

-

q = y (z, t) (1)

qm = m[y (z, t) + 2y (z, t)V + y (z, t)V 2

](2)

: y (z, t) - , ,m - - ; y (z, t) - , 2y (z, t)V - y (z, t)V 2 - . - (1) (2)

yIV + y + y + y + cy = f (3)

: y - ; , , , c - ; f - - .

f (, ) = f () sin ,y (, ) = y () sin

(4)

, (3)

yIV + y y + cy = f. (5)

(4) . (5)

dx

d= Ax+ F, (6)

... 55

: A =

0 1 0 00 0 1 00 0 0 1

c 0 0

; x =

yy

y

y

. (, )

dx

d= Ax,

(6)

x () = (, 0)x (0) +

0

(, )F () d. (7)

A ,

.

( ) = eA() = E+A ( ) + 12A

2 ( )2 + ... .

, -

.

Voronov G.V., Kabelkov A.N.Dynami interation of in regular intervals distributed

mobile weight with the elasti beam laying on vinkler's basis with fator is "beds"k and

ompressed longitudinal fore N . It is supposed, that weights move with onstant speed V in

relation to the beam loaded by distributed fores Q (x, t) and the moments (z, t). For thedeision of a problem orresponding transitive matrixes are onstruted.

..

, --

-

,

.

, .

1. .

-

, -

. ,

-

,

.

-

-

. ,

, -

0ij . , S, Su . [1, [2:

Tij,j + 2ui = 0 (1)

Tij = ij + ui,m0mj , ij = Cijkluk,l (2)

ui|Su = 0, Tijnj |S = pi (3) - V -

.

S fi . -

0ij . [3.

2. -

.

-

l Peit .

57

(1)(3) , -

011 6= 0. u3 = w(x1) x3, u1 = x3w(x1) x1. T11 = 11 + u1,1

011, T13 = 13,

T31 = 31 + u3,1011, T33 = 33, 11 = E11 = Ex3w.

: (1) -

(3) ui. - ,

V S. [4 , -

:

(J(E + 011)w) (F011w) F2w = 0

w(0) = 0, w(0) = 0,

(J(E + 011)w)(l) = 0, ((J(E + 011)w

) F011w)(l) = P.3. .

: =max011E0

,

21 =F0l2

J0 , 42 =

0F0l42

J0E0, P0 =

P l2

J0E0, J = J0f1(), E = E0f2(), F = F0f3(),

= 0f4(), g1() = f1()f2(), g2() = f4()f3(), () =011

max011, = x

l [0; 1].

, :

w() =

0

( ) y()g1()(1 + ()/f2())

d,

y() = 1

0

K(, s)y(s)ds+ f(), f() = P0( 1),

K(, s) =1

g1(s)(1 + (s)/f2(s))

1max(,s)

(21f3()() + 42( )( s)g2())d

-

-

.

J, E, 011, , F -:

J(E + 011)wIV F011w F2w = 0

= xl,

:

(1 + )wIV 21w 42w = 0

58 ..

w(0) = 0, w(0) = 0, w(1) = 0, (1 + )w(1) 21w(1) = P0 :

(2 2) sh sin + 222 + (4 + 4) ch cos = 0, (4)

=

41 + 4(1 + )

42 +

21

2(1 + ), =

41 + 4(1 + )

42 21

2(1 + )

j() =j()j(0)

j(0), j()

(4).

. 1. j()

, -

, 104 . , , -

.

4. .

, -

, , -

, . -

(1)(3),

.

1 2: u(1)i ,

(01)ij ,

(1)ij u

(2)i ,

(02)ij ,

(2)ij .

(1), [2: S

pi(u(2)i u(1)i )ds

V

((01)mj u

(1)i,mu

(2)i,j (02)mj u(2)i,mu(1)i,j )dV (5)

(5)

-

1- , :

(01)mj = t

(n1)mj ,

(02)mj = t

(n1)mj + t

(n)mj , u

(1)i = u

(n1)i , u

(2)i = u

(n1)i + u

(n)i

59

(5) , S

pi(fi u(n1)i )ds+V

(t(n)mju

(n1)i,m u

(n1)i,j )dV (6)

-

t(n)mj

, ,

. -

011, (6) :

l0

(t(n)11 (J(w

(n1))2 + F (w(n1)

)2)dx P (f w(n1)(l)) = 0, [1;2] (7)

(7) 1-

, -

. , -

. ,

[3.

..

.

[1 ..

-

. // . 1994. . 30. 1 . 317.

[2 .. . //

. 2007. . 54. 4. . 93103.

[3 .. . .:

, 2007. 223 .

[4 .. . //

, . VI

. 2008. . 9092.

Dudarev V.V. On about improved model of bending vibrations of prestressed beam. The

equations of bending vibrations of rod are onstruted at the heterogeneous prestressed state,

redution to Fredholm's integrated equation of seond kind is realized. Examples of alulation

of resonant frequenies and its analysis are given, and also the binding equation for inverse

problem is given.

. ., . .

,

,

.

-

.

.

.

. -

.

[1-[4.

.. [5. -

.. -

[6.

.

1

-

-

h. V, . (x). XOY . OX . OY - OX, . OX. - f(x). OX [a; +a].

: , , -

, , ; , ( -

), ;

; -

, .

:

= 0, (1)

61

y

y=0

= Vf(x), |x| a, (2)

y

y=h

= 0, x, (3)

V

x+ + g = 0, (4)

y

y=0

= V

x, |x| > a, (5)

> 0 , 0. (1) h < y < 0.

(2) (3).

( y = 0) - (4)

(5).

, ( x +) , ( x ) .

(4) (5)

V 22

x2+ V

x+ g

y= 0, y = 0. (6)

y = 0 +(x)( y +0) (x) ( y 0), (x)

(x) =

[

x

]y=0

=

(+x

x

)y=0

, |x| a. (7)

(1) (5) (1) (3), (6), (7).

-

p p+ = V(x). , -

.

2

, , -

(1) (3), (6), (7)

(, y) = F [](, y) =

+

(x, y)eixdx.

(1) :

62 .., ..

(, y) = Ach(y) +Bsh(y), (8)

(8) -

A, B,

V () = F

[(x, 0)

y

]() =

+

(x, 0)

yeixdx = () K() (9)

K() = 1i

th(h)th(h) .

K() , K()

||1isgn().

3

(9) ,

(x) [a; +a] +a

a

(s)k(x s)ds = vy(x) = Vf (x), |x| a, (10)

k(x) = F1[K()](x). k(x)

. K() - 0 0 ( h > 1), 0 , k k, k = 1, 2, ... (h 1 ).

h > 1

k(x) = (1 + sign x) 12

0sh (20h)

ch2 (0h) h cos (0x) + sign x1

2

n=1

n sin (2nh) en|x|

cos2 (nh) h ,

h 1

k(x) = sign x1

2

n=0

n sin (2nh) en|x|

cos2 (nh) h .

4

(10)

[7. a = 1. :

2

N

Nj=1

(sj)k(xi sj) = vy(xi), i = 1, N, (11)

63

xi = 1 +(i 1

4

)2N, i = 1, N, sj = 1 +

(j 3

4

)2N, j = 1, N.

(11) j.5

(11)

(sj), j = 1, N , () cy cm.

cy =

+11

(s)ds; cm =

+11

s(s)ds.

Fortran. .1-2

cy(Fr) cm(Fr), Fr =12,

vy = 1, N = 50.

. 1. cy

. 2. cm

64 .., ..

, -

(h = 1) , h.

[1 .., ..

// . .: ,

1937. . 3162.

[2 .. . : , 1965.

552 .

[3 ..

// -

. : . -, 1990. . 143147.

[4 .., .. -

// .

--: , 2005. . 138144.

[5 .. . .: , 1966. 448 .

[6 .. -

// .

. 1940. 4. . 5778.

[7 .. . .:

, 1965. 242 .

Efremov I.I., Kolesnikova J.N. Gliding the thin struture on the surfae of the layer of

the powerful liquid . Hydrodynamial harateristis of a thin struture, gliding on a surfae of

a liquid of nal depth are researhed. The reeived boundary value problem of distribution of

pressure in a struture by means of Fourier transformation is redued to deision the singular

integral equation. To whih digitization the sheme of a method of disrete whirlwinds is

applied. Results of alulation of fators of elevating fore and the moment are resulted.

,

..

,

. ..

- -

.

-

,

(-

)

- .

. 1. : (a) L = 0 (b).

-

(

) (. 1)., E , , O. : R1 R2, , L1 L2, F , E , ( O) L.

66 ..

. ,

(. 1a) O - L, , .

. 2. ( -

) ( ) -

.

, , , -

(

) (L=0) (. 2). ( E P -. 2), x = const 2- . L = 0 x=0; x = A x = L ( yc(x=L) > 0, yc(x=L) - x = L), ( E ) ( P ) 1 2, :

(x A)2a2

+y2

b2= 1; 0 x A; xA

L A +y2

c2= 1; A x L; yc(x=L) = 0, (1)

: a b - ; c - - ; b = c = ycmax; L - ; A - BD (. 2) O -.

L,A, a b , F = 0, 3 E = 2, 08.105 . , L (yc(x=L) = 0) F , . -, L,A a ,

67

ycmax = b BD -

zmax:

b =

8

(LA + 2 2A

2

a2); zmax =

1

2

b

; = 1 + 2; 1,2 =

1 2E

(2)

: - . F (x, y) yc(x=L) = 0

F =

(x, y)dydx = F1+F2;F1 =

E

E(x, y)dydx = 2

A0

b1 (xA)2a2

0

E(x, y)dydx;

F2 =

P

P (x, y)dydx = 2

L0

b1 xALA

0

P (x, y)dydx.

.

,

( ) -

. -

-

( ) .

= 0...2.103

. , = 0, ( ) -

= 103...2.103 . . -

(x, y):

(x, y) = (x, 0)

1 y

2

y2c; (x, 0) =

q

; yc = 2

q

;

-

( -

) F1 F2:

F1 =

A0

qE(x)dx; F2 =

LA

qP (x)dx; qmax =b2

4

F1 =

4

A0

y2c (x)dx =Ab2

4

(1 A

2

3a2

);

F2 =

4

LA

y2c (x)dx =b2

8(LA); o =

qmax

=1

2

b

,

: q qmax - yc ycmax, .

68 ..

-

o = zmax = z(A, 0). - yc(x=L) = 0 ( . 1b DDBB) - ( . 1b DOB):

E = o

1 (xA)

2

a2 y

2

b2, 0 x A;

P = o

1 x A

LA y2

b2, A x L; (3)

:o - . . -

-

()

(), V V1 (. 2) E P z > 0:

V = V (x, y, z) = VE(x, y, z) + VP (x, y, z); V1 = V1(x, y, z) = V1E(x, y, z) + V1P (x, y, z);

VE(x, y, z) =

E

E(, )(x )2 + (y )2 + z2dd =

= 2

A0

b1 (A)2a2

0

E(, )(x )2 + (y )2 + z2dd;

V1E(x, y, z) = 2

A0

b1 (A)2a2

0

E(, ) ln[(x )2 + (y )2 + z2 + z]dd;

VP (x, y, z) =

P

P (, )(x )2 + (y )2 + z2dd =

= 2

LA

b1 ALA

0

P (, )(x )2 + (y )2 + z2dd;

V1P (x, y, z) = 2

LA

b1 ALA

0

P (, ) ln[(x )2 + (y )2 + z2 + z]dd;

: , - - d, d, (x, y). (x, y, z) :

z(x, y, z) = z2

2V

z2+

1

2

V

z; xz = z

2

2V

xz; yz = z

2

2V

yz;

x(x, y, z) = z2

2V

x2+

V

z 1 2

2

2V1x2

;

69

y(x, y, z) = z2

2V

y2+

V

z 1 2

2

2V1y2

;

xy = 1

2V

xy 1 2

2

2V1xy

. (4)

. - -

( ) -

( , -

-

,

- ). -

= 0 F qmax , , ,

- ( 0). 6= 0 - ( ) F , qmax . , Fp .

. -

-

.

. -

-

( ) , -

(, ,

)

.

,

.

, -

,

(, ),

.

Zhuravlev G.A. On Determination of the Stress Condition of Bodies Modeled by Elasti

Rollers with Crossing Axes . The analyst-experimental method of the determination of the

regularities for hanges of ontat parameters of the stress ondition of bodies modeled by

rollers rossing axes is proposed.

. ., ..

, --

-

.

, -

.

,

.

. -

,

- , . -

. ,

-

. .

,

.

.

[1[4, -

. -

, ..

[5, 6. , -

. [1, 2 -

, -

,

.

. -

, ,

-

.

[7. [3, 4 -

, [1, 2, ,

,

. [3

, [4 -

.

[8.

, ,

, -

, , -

,

71

. .

.

. ,

H1 H2. ,

. x3 = (x1, x2, t) - ,

= 0 T T k; k = 1 , k = 2 . .

, -

k = 0k(1 kT k). - , x3 , ~ = (0, 0, 1) . , ~s = (cos, 0, sin) x3 = b/f(t), f 2- . -, b . - ()

k

(~vk

t+ (~vk )~vk

)= pk + k~vk +

+k(Q0~ bf (t)~s), div ~vk = 0, (1)T k

t+ (~vk )T k = CkT k, (2)

x3 = (x1, x2, t) : ~v1 = ~v2, ~vk ~ =

t, ~ = (x1 ,x2, 1), (3)

(p1 p2)ni + ( 1ij 2ij)nj = 2Kni ()i, ~n =~

|~|, (4)

= C MT k, kij = k(vkixj

+vkjxi

), ()i = xi

xk

nkni, (5)

2K = 2 21 |2|2

, 2 =(

x1,

x2

), (6)

T 1 = T 2, 1T 1

~n= 2

T 2

~n, (7)

x3 = h1,h2 : ~vk = 0, B1k Tk

x3+B0kT

k = bk (8)

: -

L, T , , A, . ~vk = (vk1 , v

k2 , v

k3) , p

k ,

T k , k = kT /L2, Ck = kT /L2, k = kAL, k - , ,

, Q0 = g0T 2/L, C = 0T 2/L3 , M = TAT 2/L2 .

72 .., ..

, , :

h2

v21dx3 +

h1

v11dx3 = 0 (9)

. . -

. , , b . , [1[4, [7, , .

~v0k = 0, 0 = 0, T 0k = Akx3, ~W0k = (cos

ck kAkx31 kAkx3 , 0, 0),

0k = (1 ck) cosx1 + (x3 kAkx23/2) sin, 1A1 = 2A2,

q0k = 0kkAkx23/2Q0 +Re2/20k cos2 c2k, 01(c1 1) = 02(c2 1), (10)

c1 = 1 0212A1A2(h1 + h2)011A1 ln(1 + 2A2h2) 022A2 ln(1 1A1h1)

(9)

, .

,

, [3, 4.

, ~vk = ~v0k + ~uk, = 0 + , T k =T 0k + k, ~W k = ~W 0k + ~wk, k = 0k + k, qk = q0k + P k.

, , uk1 =k

x3, uk3 =

k

x1, wk1 =

k

x3,

wk3 = k

x1, P k k

(k(x1, x3, t), k(x1, x3, t),

k(x1, x3, t), (x1, t)) = et+ix1 (k(x3),

k(x3), ik(x3), i).

Lk = kL2k k

[2kQ0 Re2Ak

(2 sink

i cos ck 11 kAkx3D

k

)] 2k2Re2 cos2

Ak(ck 1)2(1 kAkx3)3

k, (11)

Lk = k

[ 2 sink + i cos ck 1

1 kAkx3Dk +

+Ak(x3Lk +Dk)

]+ 2ki cos

ck 1(1 kAkx3)2Ak

k, (12)

73

k Akk = CkLk, L = (D2 2), D = /x3, (13)x3 = 0 :

1 = 2, D1 = D2, k = , (14)01D

1 02D2 = 2 sin(01 02) + i01(c1 1) cos(11 22), 1 2 = i cos(c1 c2), (15)1 + A1 =

2 + A2, 1D1 = 2D

2, (16)

1D21 2D22 + 2(1 2)1 = 2M(k + Ak), (17)

32(1 2)D1 + (01 02)D1 (1D31 2D32) + (18)+2(Q0(01 02) + C2) +Re2(01 02)[2 sin1 iD2 cos(c1 1) 22 cos2 01

02(c1 1)22] = 0, (19)

x3 = h1,h2 : k = 0, k = 0, Dk = 0, (20)B1kD

k +B0kk = 0 (21)

.

-

.

.

[1, 2 .

[9 . 1, ,

M() . , - ,

.

. 1. 10 FC-70. : -

, Re2 = 10. 1 = /2, 2 = /3, 3 = /4, 4 = /6, 5 = 0.: . g = 0, Re2 = 5 105, = 0. - . 1 , 2

, 3 .

. 1, , -

.

-

1 = 0.011, 2 = 0.01, Re2 = 5 105, = 0.

74 .., ..

, -

,

, , .

-

(05-01-00587, 07-0100099-a).

[1 .., .. -

-

// . 29.06.2007 683- 2007, 60 .

[2 Zenkovskaya S.M., Novosiadliy V.A. Inuene of high-frequeny vibration on the onset

of onvetion in two-layer system // C.R.Meanique 336 (2008). P. 269274.

[3 .. -

// . 20.08.2008 . 715-2008. 60 .

[4 .., .. -

-

// . 20.08.2008 . 716-2008. 42 .

[5 ..

. 01.02.05 , , // -

- . :

. . -, 1994. 415 .

[6 Lyubimov D.V. Thermovibrational ows in a uid with a free surfae // Mirogravity

Quart. 4. 1994. P. 117122.

[7 .., .. -

// . .

. 2002. . 66. . 4. . 573583.

[8 .., .. -

// . 2008. . 48. 9.

. 1710-1720.

[9 Zhou B., Liu Q., Tang Z. Raleigh-Marangoni-Benard instability in two-layer uid system

// Ata Mehania Sinia. 2004. V. 20. No. 4. P. 366373.

Novosiadliy V.A., Zenkovskaya S. M. Two-layer uid onvetion in high-frequeny

vibration eld . Current researh is onentrated on investigation of inuene of high-frequeny

progressive osillations on the onset on thermoapillary onvetion in two-layer immisible

uids system. Outer boundaries are assumed to be solid with heat exhange onditions in

general form.

Fluids are assumed to be weakly unisothermi, whih allows the use of generalized

Boussinesque approximation (this onept was introdued by Luybimov D.V.) as

75

mathematial model of onvetion. Averaging method was applied. Transition to lassial

Boussinesque approximation was made in derived averaged equations. Quasiequilibrium

solution, satisfying the full enlosure ondition for both slow and pulsational veloity

omponents, of aquired equations was obtained. The solution stability was extensively

studied for the ase of homogenous uids. Vertial vibration was asymptotially shown to

reate eetive surfae tension and to smooth the interfae. Horizontal vibration was shown

to have destabilizing inuene. Inhomogenous uids with undeformable in average interfae

were studied as well.

. .

, --

-

.

-

.

.

,

. -

, -

. -

, ,

, -

.

1.

( ) [1 3]

+ p = 0, = ik/Xk (1) + + q = 0, (2)

- , - , p -

, q - -

, - -, Xk (k=1,2,3) - , ik - , : = (skis ik) = skis ik

V, V. (1), (2)

, V

N dO +V

pdV = 0 (3)

V

(N + T N)dO +V

(q + p)dV = 0 (4)

(3), (4)N - V, = Xkik - - , dV - , dO - .

2. ,

+,

77

, O . -

y( = 1, 2). - R(y), R,R

.

R = R/y, R R = , R N0 = 0

N0 - , - .

-

= R+ ZN0, H1(y) Z H2(y), H1 +H2 = H, H1 > 0, H2 > 0. (5)

Z - , , H - . - ,

dV = A(Z)ddZ,A(Z) = det(G ZB), = (G ZB)1 Grad+N0/Z,G = E N0 N0 = GR R , G = R R,B = GradN0 = BR R, Grad R/y

(6)

d = Kd, (7)

K+ = A(H2)1 + (GradH2) (GH2B)2 (GradH2)

K = A(H1)1 + (GradH1) (G+H1B)2 (GradH1)

NdO = A(Z)(G ZB)1 dSdZ (8) (6)-(8) G , B - -

, E - , G B - - [4], Grad - ,N - O, - - , N0 = 0, dS - .

, O, - p q, , -

f l.

N = f, N = l (9)3. (3),(4), V

, Z = H1(y1, y2), Z = H2(y1, y2)

78 ..

O, - , . (6)-(9),

T dS +

fd = 0 (10)

(M T R)dS +

(l +R f )d = 0 (11)

f =

H2H1

A(Z)pdZ +Kf +K+f+ (12)

l =

H2H1

A(Z)qdZ+N0H2

H1

A(Z)pZdZ+K(lH1N0f)+K+(l++H2N0f+)

(13)

T

M

T =

H2H1

A(Z)(G ZB)1 dZ (14)

M =

H2H1

A(Z)(G ZB)1 ( Z N0)dZ (15)

(12), (13) ,

, ,

d , - ,

f l .

, (15) -

, N0, : -

.

, (10), (11) (Div - )

DivT + f = 0, DivT R T /y (16)

DivM + T + l = 0, (17)

79

(16), (17) [5, 6] , (..

) . [5] ,

, [6] . - , .. -

, y. - r, r, r

.

D = J(Grad r)T T , L = J(Grad r)T M

[4]

div(J(Grad r)T ) = J(trB)N0, J =

G11G22 G212q11q22 q212

, q = r r, , = 1, 2

(16), (17)

divD + Jf = 0, divL+[(grad R)T D]

x+ Jl = 0 (18)

div grad - .4. (q = 0), -

, .. ,

. (15)

M = N0, N0 = N0 = 0 (19) (13)

l =N0 , N0 = 0 (20) (14)

T = + V N0, N0 = N0 = 0, V N0 = 0 (21) (19)-(21) V - , - ,

, . (19)-(21) (16)

Div +N0Div V V B + f = 0 (22) (17) :

(Div ) G V + = 0 (23)

+ (B )T = T +B (24)

80 ..

= + (B )T , (24), (22),(23) V ,

Div BDiv (+T )RT (B/y)+N0Div (GDiv )+f+Div(N0) = 0(25)

-

[7], , (25) . , - (16), (17)

:

Div [ B + (Div ) GN0 + N0] + f = 0, = 12( + T ) (26)

[8] -

, .

5.

, . -

(1), (2) , [9]

V

tr( T + T )dV = V

(p + q )dV + V

(f + l )dO(27)

= +E , =

- -

[10], f l - . , (27) -

(1), (2).

, -

, -

, -

.

(y, Z) = 0(y) + Z0(y

)N0, (y, Z) = 0(y) (28) 0(y

), 0(y) - , -

0 = 0, 0 = 0 . (28) (27) (6), (12)-(15),

tr[T T (Grad 0 +E 0) +MT Grad 0

]d =

(f 0 + l 0)d

81

(16), (17). ,

-

(28) ,

.

[11, 12] -, [4] - (26). , (28)

, -

- . -

(28) -

.

[1 . . // . 1964. . 28. .3. . 401-408.

[2 Toupin R. A. // Arh. Ration. Meh. and Anal. 1964. V. 17. 2. p. 85-112.

[3 Zubov L. M. Nonlinear Theory of Disloation and Dislinations in Elasti Bodies. B.:

Springer, 1997. 205 p.

[4 . . . --

.: - , 1982. 143 .

[5 . . . // (Advanes in Mehanis). 1988. .

11. 4. . 107-148.

[6 . ., . . . .: , 2008. 287 .

[7 . . . .-.: -

. T. 1, 1947. 512 .

[8 . . // . 1950. . 14. . 5. . 558-560.

[9 . . .: , 1983. 400 .

[10 . ., . . // . . . 1994. 3. . 181-190.

[11 .. . .: , 1962. 431 .

[12 . . . .: , 1976. 512 .

Zubov L. M. Equations of equilibrium of the miropolar shells for the resultant stress

and ouple . Equations of equilibrium shells for the resultant stress and ouple is dedued as

impliation of equilibrium equations three-dimensional ontinuum Cosserat.

,

. .

,

-

,

( )

.

, . -

,

.

,

, -

.

, , -

, . , -

, , -

, , , .

-

, -

. a - P = {0, P eit}, . {r a, z = 0} . q(r) . h, r0, h0, h0 r0. ( ) - .

() -

Xr, Xz, - . , [1, 2,

q(r) = q1(r) + q2(r),

q1(r) = q(r, 0) .. .. [3, 4

a0

k(r, )q(, )d = J0(r), 0 a,

83

. 1. a = 5, 10, 15, 20 , r0 = 20 , h0 = 10 , 1 q1(r); 2 q2(r); 3 q(r)

. 2. a = 15 , r0 = 20 , 1 h0 = 2 ; 2 h0 = 10 ; 3 h0 = 18

84 ..

. 3. a = 5, 10, 15, 20 , r0 = 20 , h0 = 10 , 1 q1(r); 2 q2(r); 3 q(r)

. 4. a = 15 , r0 = 20 , 1 h0 = 2 ; 2 h0 = 10 ; 3 h0 = 18

85

;

q2(r) = ik=1m

(Q2(k)

(k)q(r, 0)J0(kr)kH

(1)0 (kr0)

) , -

(Xz = fz(z) (r r0));

q2(r) = ik=1m

Q2(k)(k)

q(r, 0)J0(kr)

r00

fz(r)rH(1)0 (kr)dr

k

, -

(Xz = fz(r)(z+h0)).

: = 1.4 103 /3, c1 = 0.2 103 /2, c2 = 0.12 103/

2, h = 20 , = 2.0 . -

. ,

, , -

, ,

(.1) (.2). -

: .

, , -

,

,

(.34).

(08-08-00144, 06-08-

00671), __ (06-01-96600, 06-01-96639), (-

2298.2008.1).

[1 .., .., .., .. -

-

/ .

. --, 2005. .2. . 201

205.

[2 ..

/ -

. V -

. , 2008.

.2. . 115116.

86 ..

[3 .. -

. .: , 1984. 254 .

[4 .., .., .. -

. .: , 1999. 246 .

Kapustin M.S. Analysis of inuene of internal loadings in the layer on the ontat

stresses, arising under the stamp. Axis symmetrial ontat problem about joint utuation

of a weightless stamp on surfaes layer, with the absene of frition in the eld of ontat

and system of rigid inlusions (vertially or horizontally foused) with the set ontat,

was explored. Integral equations and their approximate solutions were reeived, numerial

alulations and analysis of this results were organized. Estimation of inuene of internal

loadings in a layer on the ontat stresses, arising under a stamp is given.

..

...

, -

.

.

-

.

-

. -

,

, -

, , -

. , ( 50%),

. -

[1, -

,

. , -

[2-5, -

, , ,

, -

() , ,

.

, ,

,

L , - [6.

Q

[1:

Q

t= Y cp

x,

Q

x= Y

c

p

t(1)

Y , . , Y=Y(x)

=(x) ,

x = x,

88 ..

. - , -

,

.

, , -

[1,7, L >> , [7. [8 -

,

:

A

t+

x(A0u) = 0,

u

t+p

x= 0, p (A, x) = p0 + k (AA0)

A (t, x) , A0 (x) - , u , p (t, x) - ,p0 ( ),k(x) = p/A |p=p0. (R=R(x)):

P (t, x) =

Y (0)

Y (x)P 0eit

e x0 (x)dx + e L0 (x)dxLx (x)dx,

Q (t, x) =Y (0)Y (x)P 0eit

e x0 (x)dx e L0 (x)dxLx (x)dx (2)

Y (x) = S (x) (c (x))1, - , , (x) = (x) + i/c (x), (x), c (x) - , .

,

, , [7.

-

,

. -

[8,9. ,

(ow limitation), ,

ptm, . - ptm.

, [8. , -

,

[9.

-

89

, -

, -

.

, .

- -

, ,

- :

div (~v) = 0,~v

t+ (~v)~v = 1

fp+ ~v,

s2~ut2

= ps + div, 1 t

+ = 2G

(e+ 2

e

t

) (3) ~v, p, f , , - , ~u , s ps , G, 1,2 e , - .

- ,

:

r = 0 : vr = 0, |vx|

90 ..

u0r/R1, [12 . , (2), -

, .

. 1.

() j- ().

. 2. P (V ) E = 107; 5 106; 106; 5 105; 105( 1-5); h = 0.2d () h = 0.05d ().

Rj, hj , Ej, [7.

P (V ) R1 = R0 + (RL R0)x/L .2. , , ,

Re(Yt), Im(Yt), - [4. ,

,

, , -

, , . -

,

91

, .

[1 . . .: , 1980. 320 .

[2 ..

// . .. 2003. 5. . 129139.

[3 ..

// .. 2003. .15, 6. C. 6571.

[4 .. - -

// . . 2004.

.7, 1. C. 5061.

[5 .. -

// . .: - ,

2006. . 11. . 4463.

[6 Taylor M.G. Wave travel in a non-uniform transmission line, in relation to pulses in

arteries //Phys. Med. Biol. 1965. v.10. p. 539550.

[7 . .: , 1983. 400 .

[8 Shoerenberg M. Pulse wave propagation in elasti tubes having longitudinal hanges in

area and stiness // Biophys.J. 1968. v.8. p. 9911008.

[9 Bertram CD, Chen W. Aqueous ow limitation in a tapered-stiness ollapsible tubes

// J. Fluids Strut. 2000. vol.14. p. 11951214.

[10 Kamm RD, Patel NR, Elad D. On the eet of ow-indued utter on ow rate during

a fored vital apaity maneuver // FASEB J. 1993. vol.7. p.11.

[11 Shim EB, Kamm RD. Numerial simulation of steady ow in a ompliant tube or hannel

with tapered wall thikness // J. Fluids Strut. 2002. vol.16. p. 10091027.

[12 ..

// . C.. 2006. 3. C. 125139.

Kizilova N.N. Wave propagation in nonuniform tubes with variable ross-setions . A

brief review of theoretial and experimental data on the wave propagation and reetion in

tapered wall thikness and wall-stiness distensible tubes in appliation to blood irulation

and respiration systems is given. The orresponding problems for the plane and axisymmetri

waves are formulated and disussed. Original results on the axisymmetri wave propagation in

wall thikness and wall-stiness visoelasti tubes with variable ross-setion are presented.

.., ..

. .., ,

,

-

.

, .

-

. ,

, -

.

-

-

. -

,

.

, .

-

-

, , . -

, -

,

.

.

[1, 2.

-

,

[3.

h - 6 8 20 .

. b = 2 v, - .

(60 90), v (3 - 12 /), h (0.5 - 3 ). , -

93

,

103 . -

, ,

.

-

P (t), n(t) Ns(t), - ,

.

, , -

P (t), , - , . -

P (t) n(t) , P (t) -

.

P (t) v - h, h v, , [4.

, , ,

v = 10 / P (t) 220 280 h 1.5 3 . P (t) . 1.5 .

(-

), ,

,

. , -

, . 1.

.

. 1 ,

v = 10 /, h = 1.5 = 90

, . ,

, 55 . h 3 (. 1) -

,

35 ( -) H 2.0 . , , .

. , ,

, s. - ()

94 .., ..

) )

. 1. ( v = 10 /, b = 2 , = 90) 200, 400, 600 :) h = 1.5 ; ) h = 3.0 .

, , () , .

, , -

,

, .

-

s h. -

, [4, s h, .

. 2. -

,

v = 10 /, h = 1.5 = 90 300 ,

. , ,

70 H 0.5 . .

3

, .2,

. , , -

22 H 1.5 L 0.5 . ,

. ,

, -

65. S 4.0

95

)

)

. 2.

-

( v = 10 /, b = 2 , = 90) t :) h = 3.0 , t = 200, 400, 600 ;) h = 1.5 , t = 300 .

.

-

,

.

, , , -

, , . 2, -

s < S. h = 3.0 , S 4.0 . m = s/h 1.33. - ,

, , [4, m, .

, m 1, 33 - ,

,

, ,

. ,

, . -

-

m 1. , , -

. -

,

.

, ,

96 .., ..

-

. , -

, -

.

[1 .., .. - -

- -

- // i i: ii.

. . . . 30. i: I , 2001. . 7176.

[2 .. -

// 17852 07.09.2006 -

. -. -2006.

[3 .., .. -

// . . . .29 -

2007. . 169175.

[4 .., .. -

// XVII . . "

". , , 17-23 2007 . -: . .

-, 2007. . 146148.

Kostandov Yu.A., Lokshina L.Ya. Development of dynami harater of frature of

mining roks. Numerial researhes of frature proess at alareous and siliate materials in

dependene on the iruit and parameters of utting are exeuted. The omparative analysis

of utting eort and distribution of the destroyed elements of environment at utting as single

utters and group of utters is arried out. The basi regularity of inuene of a positional

relationship of utters and sequenes of their aeting on power onsumption of frature of

materials at their utting are established. It is shown, that eieny of eet of utters group

depends on magnitude of a ratio of values of a utting longitudinal spaing and a utting

depth. It is development of dynami harater of frature of mining roks at their utting.

. .

-, . --

- -

,

, -

.

; . ,

,

, .

. -

[6 ,

[7 60- .

-

- .

,

- ,

-

[3, [4.

. -

6mm

.

:

u2 = 0,uix2

=

x2=

x2= 0 . (1)

x2 = const [l, l] [H,H ], x3 = H - :

13|x3=H = 33|x3=H = 0; |x3=H = 0;

x3

x3=H

= q;

13|x1=l = 11|x1=l = 0; D1x1=l

= 0; q1|x1=l = 0.(2)

q x1, = 0, Fi = 0, w = 0, I = 0, 0 :

ll

D3(x1, H) dx1 = 0. (3)

98 ..

[6, 33 - . , , :

u1(x1, s) = u10(x1, s) x3u30(x1, s), u3(x1, s) = u30(x1, s), =

x3H0(s), = T1(x1, s) +

x3HT2(x1, s).

(4)

-

[4. -

,

- . -

, [1,

:

1: (-,

u10 T1)

u10 22 s2 u10 1 T 1 = 0, 1su10 k11 T 1 + s T1 = s1g 0 q1,(

u10 1 T1)

x=1= e 0 , T 1

x=1

= 0.(5)

2: (,

u30 T2)

2 u(4)30 222 s2 u30 + 322 s2 u30 + 1 T 2 = 0,

1su30 k11 T 2 +(s+

3

2

)T2 = 3q2

,(

u30 + 1T2

)x=1

= 0, T 2x=1

= 0,(u30 22 s2 u30

)x=1

= 0

(6)

q1, 2 = (q+ q)/2.

(3) :

11

(e(u10 u30

) 0 + 1g( T1 + T2)

)dx1 = 0. (7)

, -

1, , 2, . -

(5), (6).

- (5),

q1 x, q2 - ; T2 = u30 = 0. (7) :

11

(eu10 0 + 1g T1

)dx1 = 0. (8)

99

1

[5.

q1, , [4.

q1 = q(s)Q, Q = const,

0 s, - 0.

u10 =2i=1

AiPi sh(ix) , T1 =2i=1

BiPi ch(ix) + 1g 0 q1s

,

Ai = 1i , Bi = 2i 22 s2, Pi :

P1 =

((21g e) 0

1

sq1

)(2)sh(2)

1,

P2 = ((21g e) 0

1

sq1

)(1)sh(1)

1,

(i) = (2i 22 s2)i, = 122 s2 ((2)sh(2)ch(1) (1)sh(1)ch(2)), i

: A4+B2+C = 0, A = k11, B = s(k1122 s+1+ 212), C = 22 s3 1 ,2 =

B B2 4AC

2C.

(8)

:

0 =1 1q

s

g+ P

(21g2 1) + (21g e)P

=q1W (s), (9)

P = ((eA1 + 1gB1)(2) (eA2 + 1gB2)(1))sh(1) sh(2). W (s) -

s.1. s, 0, :

W (s) =1s

(W1 +

W2s

)+O(s2), (10)

W1 =g

21g2 1 , W2 =

2gb1( ge)(21g

2 1)2a2 , a1 = 2, b1 =21

2

2k112, a2 =

1k11

,

b2 = 21

2

2k3/211

22

.

,

1W , 1g, .

, W2.

100 ..

2. s 0, , :

W 0(s) =1s

(W 01 +W

02 s)+O(s) , (11)

W 01 =g(1 + 21

2) e(21g

2 1)((1 + 212)) + 21g e,

a01 =

1 + 21

2

k11, b01 =

1 + 21

2

k11

2221

2k11(1 + 21

2)2.

, 0() = lims0 sW (s) =1W

01 .

[1 .

,

, (5) -

, .. 2 = 0:

u10 1 T 1 = 0, 1su10 k11 T 1 + s T1 = s1g 0 q1,(

u10 1 T1)

x=1= e 0 , T 1

x=1

= 0.(12)

.

:

0 =1 1q

s

g + e

(1 + 212)(1 + e2) + 21(g + e)2=

1q

W (s).

, -

:

W (s) =1sW 1 , W

1 =

g + e

(1 + 212)(1 + e2) + 21(g + e)2(13)

W (s) W (s) s, W 0(s) W (s) s 0. s

lims

[W W ] s = (e 21g)( ge)

(1 21g2)(1 + e2 + 21(2 g2 2ge))= m1

s 0lims0

[W 0 W ] s = m2 ,

m2 =1g(1 +

21

2)( ge)((1 + 21

2)(21g2 1) + e(21g e))(1 + e2 + 21(2 g2 2ge))

.

101

m1 0, 41, m2 2, 1 104 . 1 , m2 = 0, m1 =

e( ge)1 + e2

.

, s 0 W 0(s) W (s) , , -

. -

W (s) W (s) s , s1,

-

.

-

.

[2 -

,

t .

. .

. . , . . . . .

[1 . . -

// . . - . 1999. 3. . 28-31.

[2 . . . // -

. /. . 2001.

T. 1. N1(7). . 82-88.

[3 . ., . .

//

. 2002. 1. . 43. . 196201.

[4 . . //

, .

III -, --, 15-19 2004, . --: -

, 2004. . 92-95

[5 . . .: . 1970. 256 .

[6 . . .: . 1986. 160 .

[7 Mindlin R. D. On the equations of motion of piezoeletri rystals // Problems of

ontinuum mehanis/ Ed. J. Radok. Philadephia: SIAM. 1961. P. 282-290.

Lavrinenko V. V. On a Non-stationary Problem of Coupled Thermoeletroelastiity for

Thin-walled Member . problem of the motion of the thermoeletroelasti strip plate is

onsidered. As a result of a heat load on the fae potential dierene is indued. It is

measured through an additional ondition of swithing-in of a member. A simplied model

is built in a Laplae image spae. The properties of solutions are studied. It is shown that

the potential dierene is determined on the initial stage by a value depending only on a

parameter that haraterizes thermoeletri onnetivity of the problem.

..

, ..

, ..

. ..

, --

-

- ,

.

- , -

ANSYS,

APDL ANSYS. ,

.

,

, ,

. -

,

-

.

.

O1r. , r = R1 0 1 0 + 1, r = R2 - 02 0+2, 0 = /2+(n1) 0 = /2+(n1) - ( = /2 - r ). . 1 = /4, 2 = arcsin(R1/R2 sin1), = 2/N , = /2, n = 1..N - , N - .

, r = R2 , r = R1 R0 = R1 h O2, h 0. P (R0h, /2), . , - .

. 1 (

).

- -

ANSYS APDL.

103

. 1.

"-" : -

, ;

, ; -

; -

. -

- , E; - v, Ev - s, Es.

, -

- . -

- -

. , -

, -

PLANE82 . -

CONTA175,

TARGE169 [1.

-

.

hk -, ntd - , ntv - . nbd - - nbv - . , -

. -

. 2.

. 3 -

q() = rr(R1, ) - (

104 .., .., ..

. 2. -

= 0 r ). - P = 1.7 , R1 = 0.025 , R2 = 0.031 , R0 = 0.02491 . -

= 0.3 E = 1011 , s = 0.3 Es = 2.1 1011 . - , -

v 0.4, Ev = E/n, n = 1, 2, 5, 10 (n = 1 ). . 4 - .

, -

q(), || - , .

. 3. -

, -

, -

105

. 4. -

,

[2, [3.

-

( 06-08-01257, 08-08-00873).

[1 .. ANSYS: . .: , 2005. 640 .

[2 , , . . 2. .

.. .. . .: , 1968. 464 .

[3 .., .., ..

// -

: X -

, --, 5-9 2006 . . 2. --: - , 2007.

. 190196.

Laypin A.A., Chebakov M.I., Kolosova E.M. On the omputation theory of the

binary ylindrial bearing . The ontat problem of elastiity theory in two-dimensional

formulation on the interation elasti shaft with pieewise-heterogeneous ylindrial layer

by insertions, simulated the work of binary self-lubriated slider bearing, is onsidered. The

method of nite element analysis, realized using the program system ANSYS, for whih the

appropriate program are developed by the language APDL ANSYS, is employed for solution.

Computation of ontat stresses, magnitude of ontat zone under various material parameters

of insertions is onduted.

-

. .

, --

. -

-

-

.

. -

, .

. -

(. 1).

. 1.

, . ,

(. . 1). ,

.

?

. -

, . -

( ).

. -

: .

/1-4/. -

, -

(). . 2 -

.

- 107

. 2.

-

(. 3).

. 3. -

, ,

, - -

.

-

.

. -

. -

, - -

, (. 1).

-

( 9 .)

-

( ) . 4. , -

-

108 ..

-

-

-

%

1 17,0 22,0 5,0 22,73

2 18,3 21,6 3,3 15,28

3 20,6 19,5 1,1 5,64

4 20,1 19,6 0,5 2,55

5 20,6 20,8 0,2 0,96

6 20,1 20,7 0,6 2,90

1,78 8,34

3.

3

. 4.

, .

-

.

- . -

. , . 5

. -

1,5 .

, .

1. -

,

- .

2. 3 -

, 50 .

- 109

. 5.

3. -

, -

.

[1 .-., .-. . : .

.- .: , 1989.-190.

[2 .. -

. //" ". XI

, , --, 2008. .2., . 123-127.

[3 ..

. //" -

". VII , , -

-, 2001. .1., . 157-159.

[4 .. -

. // "-2003". -

, , --, 2003., . 126-127.

Maystrenko A. V. The nite-element analysis of the formation onditions of the stress

onentration in the longeron. Objet of researh is the longeron of a vehile. The omparative

analysis of internal eorts in a zone of harateristi uts in longerons with the normal and

hanged form of ross-setion by means of a nite-element modeling method is arried out.

Degree of prole hange inuene of a longeron surfae on stress redistribution in it parts is

established. Plaes of onentration of the stress, o

urrenes of raks being the reasons and

destrutions of longerons are dened.

..

..

, ..

. . . ,

-

, .. 1940 ,

.

-

, .

, -

, ,

.

, -

-

. , -

,

[1. [2

, -

-

. [2, 3 -

,

, -

-

.

,

[1.

, .. [4, 5 -

-

. -

, , [5,

[6, . . . . . . -

:

1

. [7, -

,

2

,

1

(.)

2

.. (.)

.. 111

, -

. , ,

, -

, -

. ,

.

[8

,

.

.

, ,

, .

0 x < l, |y| 1 F :

22F + C2T = 0, (1)

F = f0(y),F

x= g0(y), x = 0, 1 y 1,

F = fl(y),F

x= gl(y), x = l, 1 y 1

(2)

x = 0, l , , y = 1:

F = 0,F

y= 0, y = 1, 0 x l. (3)

2 ; T ; C E E/(1) ; , .

(1)

F = F + FT ,

FT

2FT + CT = 0

, F , (1) (2),

(3). FT [9 . , -

, l , fl = gl = 0

112 .., ..

( x = 0 ). , .. [4, 5,

F =

k=0

{Yk(y)ake

kx + Yk(y)akekx}, (4)

; k -, sin 2 = 2 k sin 2 = 2 ; Yk(y) , :

Yk =cosky

cos k y sin