Parametric oscillations of a viscous-elastic orthotropic shell of variable thickness Текст научной статьи по специальности «Строительство и архитектура»

Magazine of Civil Engineering. 2023. 120(4). Article No. 12010

Magazine of Civil Engineering issn

2712-8172

journal homepage: http://engstroy.spbstu.ru/

Research article UDC 539.3

DOI: 10.34910/MCE.120.10

Parametric oscillations of a viscous-elastic orthotropic shell of

variable thickness

D.A. Khodzhaev1 B R.A. Abdikarimov2 , M. Amabili , B.A. Normuminov1

1 Tashkent Institute of Irrigation and Agricultural Mechanization Engineers National Research University, Tashkent, Uzbekistan

2 Tashkent Institute of Architecture and Civil Engineering, Tashkent, Uzbekistan

3 McGill University, Montreal, Canada

M [email protected]

Keywords: thin walled shell, viscoelasticity, composite materials, variable thickness, nonlinear vibrations, dynamic stability, Galerkin method, numerical method

Abstract. A solution to the problem of parametric oscillations of a viscous-elastic orthotropic shallow shell of variable thickness is presented. Dynamic loading acts along one side of the shell in the form of a periodic load. Unlike linear problems, the nonlinear problem under consideration could not be solved by applying analytical methods; therefore, approximate methods were used. The mathematical model of the problem is built within the Kirchhoff-Love theory. In this case, tangential inertial forces and geometric non-linearity are taken into account. Deflection and displacements approximation is performed using the Galerkin method in higher order approximations, which allows reducing the problem solution to a system of nonlinear integro-differential equations (IDE) with variable coefficients. The weakly singular Koltunov-Rzhanitsyn kernel with three rheological parameters is used as the relaxation kernel; it describes the viscous-elastic properties of the shallow shell. A numerical method based on the use of quadrature formulas is used to obtain a resolving system of equations for the problem. To obtain numerical results, a computer software was compiled in the Delphi environment for a computational algorithm of the problem solution. The effects of viscous-elastic, orthotropic, nonlinear properties of the shell material, thickness variability, and other physical, mechanical, and geometrical parameters on the dynamic strength of a shallow shell are studied.

Citation: Khodzhaev, D.A., Abdikarimov, R.A., Amabili, M., Normuminov, B.A. Parametric vibrations of a viscoelastic orthotropic shell of variable thickness. Magazine of Civil Engineering. 2023. 120(4). Article no. 12010. DOI: 10.34910/MCE.120.10

1. Introduction

The problem of parametric oscillations of elastic and viscous-elastic thin-wall structures (plates and shells of variable thickness) is one of the most relevant problems in the mechanics of a deformable rigid body. The solution to such problems is of great importance for the modern aerospace industry, rocket technology, and mechanical engineering. Structural elements (plates and shells of variable thickness) can be found in many engineering and building structures, in aviation and motor transport, and in various units.

The first studies devoted to the problem of parametric oscillations of plates and shells of constant thickness, within the framework of the theory of thin plates, include the research work by V.V. Bolotin [1]. To solve problems, he used methods based on the variational approach.

There are a great number of articles in the literature devoted mainly to the dynamic stability and parametric oscillations of elastic thin-wall structures (plates, panels, and shells) under the impact of periodic load.

© Khodzhaev, D.A., Abdikarimov, R.A., Amabili, M., Normuminov, B.A., 2023. Published by Peter the Great St. Petersburg Polytechnic University.

An analysis of the study of oscillation problems of shells made of various materials, conducted in the period from 2003 to 2013, can be found in [2], [3]. It also contains a review of publications related to parametric oscillations.

Analytical and numerical solutions for different types of structures (plates and shells) were considered in [4].

In [5], the dynamic stability of truncated-conical shells under dynamic axial load was studied. To solve the problem, described by a differential equation of the Mathieu-Hill type, the Galerkin method was used.

Reference [6] is devoted to the dynamic stability of a linearly elastic thin rectangular plate subjected to a bi-axial time-varying load. The differential equation of plate motion was solved using the finite difference method (FDM). To identify domains of dynamic stability, the Mathieu-Hill equation was derived.

The dynamic stability of a viscous-elastic rectangular plate subjected to constant and variable loads in the plane of the plate was considered in [7]. The equation of motion is described by an integro-differential equation with respect to an unknown time function. The effect of the viscous-elastic characteristics of material on the dynamic instability zone was shown.

Reference [8] considers the dynamic instability of laminated non-homogeneous orthotropic truncated-conical shells under periodic axial loading. The problem was reduced to solving the Mathieu equation. Bolotin's method was used to evaluate the behavior of the shell for various parameters.

The dynamic instability of layered composite panels of variable stiffness under non-uniform periodic excitation was studied in [9]. The Ritz method was used to obtain the resolving system of equations for the problem. The domains of dynamic instability were constructed by the Bolotin method.

In [10], the behavior of a footbridge under rhythmic loading was studied. The footbridge was considered a shell of variable thickness. The problem was solved by the FEM. The influence of different mass distributions along the footbridge on its dynamic behavior was analyzed.

In [11], the dynamic stability of a cylindrical shell with linear variable thickness was considered under axial forces and pulsed external pressure. The Bubnov-Galerkin method was used to solve the problem. A resolving system of equations for the problem was derived in the form of an infinite system of homogeneous algebraic equations.

In [12], the dynamic instability of toroidal shells was studied. The Galerkin method was used to obtain a semi-analytical solution to the problem. The results obtained were compared with the ones available in the literature. The effect of various geometric and mechanical parameters on the dynamic instability of shells was studied.

A study of the nonlinear dynamic stability of a cylindrical shell of variable thickness is given in [13]. The equation of motion was derived based on the classical theory of shells in a geometric non-linear formulation. The solution to the equation was obtained by the fourth-order Runge-Kutta method and the Galerkin method. The effect of the characteristics of material and geometrical parameters on the dynamic behavior of a shell was investigated.

In [14], the impact behavior of an elastic spherical shell under step pressure was considered. Initial geometric imperfections were introduced.

The study in [15] concerns the analysis of the nonlinear dynamic behavior and stability of heterogeneous axisymmetric shells of variable thickness. The equation of motion was constructed based on the Kirchhoff-Love hypothesis. The Ritz method was used.

In [16], the dynamic behavior of a three-layer (sandwich) conical shell under the action of a periodic load was studied. At that, various boundary conditions were considered. The problem was reduced to solving an equation of the Mathieu-Hill type, the solution of which was obtained by the Bolotin method. The results obtained were compared with the results obtained by other authors.

The behavior of a sandwich plate under periodic load was considered in [17]. Using the constructed mathematical model of the problem, the effect of various geometric and mechanical parameters of a plate on its dynamic behavior was studied.

The study in [18] is devoted to the dynamic instability of a cylindrical shell made of a thin-walled composite material. The ABAQUS program was used. The influence of the parameters of periodic loading and initial geometric imperfections on the dynamic behavior of the shell was investigated.

In [19], the dynamic stability of sandwich panels under periodic load was considered. The solution to the problem was obtained by reducing the obtained equation of motion to an equation of the Mathieu type

and applying the Bolotin method. The effect of different geometric and mechanical parameters of panels on their dynamic behavior was studied.

Reference [20] is devoted to the study of the dynamic stability of cylindrical composite shells under the action of a pulsed loading. The FEM was used with the ABAQUS program. The impact of different parameters on the dynamic behavior of the shell was shown.

In [21], an annular plate under external harmonic excitation was studied. Resolving equations were derived based on the non-linear von Karman theory. New mechanical effects were observed.

A brief analysis of the available scientific publications showed that there are almost no studies of nonlinear oscillations and dynamic stability of thin-wall structures (viscous-elastic plates and shells of variable thickness) [22]-[24]. In the article below, nonlinear parametric oscillations of viscous-elastic shallow shells of variable thickness are numerically studied.

The object of the research is various viscous-elastic thin-wall constructions of variable thickness.

The purpose of the study is to develop effective methods, algorithms and a computer program to evaluate the dynamic behavior of thin-wall constructions, taking into account the viscoelasticity of the material properties and variable thickness.

The following problems were solved to achieve this goal:

- to obtain resolving systems of nonlinear integro-differential equations with singular kernels of viscous-elastic thin-wall constructions of variable thickness under the impact of periodical loads;

- to develop an effective approach to numerical solution, computational algorithm and software products for evaluating the strength of viscous-elastic thin-wall constructions of variable thickness under periodical influences.

2. Methods

A rectangular viscous-elastic shallow shell of variable thickness h(x,y) is considered with an

account fort the geometric nonlinearity based on the Kirchhoff-Love hypotheses. Let the shell be

dynamically loaded along side a by a periodic load P(t) = P0 + P1 cos(0t), P0,P1 = const, 0 - is the

frequency of external periodic load (Fig.1). A coordinate lines x and y of the curvilinear orthogonal

coordinate frame is directed along the lines of principal curvatures, and the z -axis - along the internal normal of the middle surface.

Figure 1. Shallow shell of variable thickness.

The system of equations of motion in the framework of the chosen theory has the following form [25]

dNx dN.

■ + -

xy

d 2u

dx dy

+ Px-Ph~Y = 0 ■

dt2

dNy_ + dNy

d 2v

dx dy

+ Py-PhTT = 0

y dt2

d 2Mx d2Mv ^ d 2H

y

+ 2-

dx dy dxdy

+

dy

df ? ^

dy

d

+ kxNx + kyNy + — x x y y dx

2 w

dw

dw

A

x xy ^

dx dy

d 2w

,.T dw ,r5w ^nd w , u w

N-y&+Ny m)+P(t h?+q-ph p-=

where kx = 1/Rj and ky = 1/Rj are the principal curvatures (R and Rj are the principal radii of

curvature) of the shell along the x and y axes, respectively; px, py and q - external static loads applied to the shell element in directions x , y and z .

The system of equations (1) is supplemented by the corresponding boundary conditions [25], which will be used in the solution to the problems:

1. All edges are simply supported:

at x = 0, a : u = 0, v = 0, w = 0, Mx = 0 ; at y = 0, b : u = 0, v = 0, w = 0, My = 0 .

2. All edges are fixed:

dw dw

at x = 0, a : u = 0, v = 0, w = 0, — = 0 ; at y = 0, b : u = 0, v = 0, w = 0, — = 0 .

dx dy

3. Two opposite edges are simply supported, the other two edges are fixed:

dw

at x = 0, a : u = 0, v = 0, w = 0, — = 0 ; at y = 0, b : u = 0, v = 0, w = 0, My = 0 .

dx y

The initial conditions at t=0 are as follows:

u(x, y,0) = u0 (x, y ), u(x, y,0) = щ (x, y ), v(x, y,0) = v0 (x, y ), v(x, y,0) = v>0 (x, y ),

w(x, y ,0) = w0 (x, y ), w (x, y ,0) = w 0 (x, y ) .

Here, u0 (x, y), v0 (x, y), w0 (x, y), щ (x, y), i>0 (x,y) and 1^0 (x,y) are given functions.

The components of the vector of forces {N }= (Nx, Ny, Nxy ) and moments {M }= (Mx, My, Mxy ) for symmetric structure shells in matrix form can be written as:

{N }=Nx ; Ny ; Nxy } =[C]-H {M}=Mx ; My Mxy }=[D\{x}, (2)

here

{S}=(Sx, , Sxy ^ {X}=(Xx, Xy, Xxy JT

du

sx =--kxw + —

x dx x 2

dw

42

dv

— I , sv =--kyw + —

_dx J v dv v 2

2

Xx =

д 2 w

dx2,/L y dy Stiffness matrices [C ] and [D] have the following form:

X y

д 2 w

v дУ J

2 ' Xxy =

du dv dw dw

sxv = — + — +--, (3)

dx dy dx dy

д 2w

dxdy

f Cil Ci2 Ci6 ^ f Dii Di2 Di6 Л

C = Cl2 C22 C26 , D = Di2 D22 D26

V Ci6 C26 C66 J V Di6 D26 D66 J

(4)

where the coefficients of the stiffness matrix Cj, Dy (ij = 11,22,12,16,26,66), depending on the mechanical characteristics of the material and coefficient m are determined as follows:

h( x,y) h( x,y ) h( x,y)

2

Cj = 2 Bj (l - rj

h( x, y ) 2^

Dj =

J By (l - Г*■ )z2dz, (i, j = 1,2,6), m = 2pdz. (5)

h( x, y )

Y

h( x, y )

* *

Here Bij are the stiffness coefficients [26], r , r - are the integral operators with relaxation

kernels r(t ) and rj (t ), respectively:

t t T*ç = jr(t - z)v(z)dz, j = Jr (t - т)ç(т)dт, i, j = 1,2 .

0

0

In operator form, the system of equations of motion (1) is written as:

d 2u

d 2v

Luu + L12V + L13W = -L14W - px + ph—t- , L21U + L22V + L23W = -L24W - py + ph

dr y dr

d 2 w

L31U + L32V + L33W = -L34w - q - P(t)—2- + ph

d 2 w

dx'

dt

2

(6)

Here u,v and w - are the components of displacement vector {U} in the directions of the Ox, Oy , and Oz axes, respectively.

52 ^2 ^2 ^2 ^2 ^2

o o o o o

L11 = + 2C16 + C66—J-. L12 = L21 = + (C12 + C66)^ZT + C26~2,

dx1 cixdy dy2 dx2 dxdy dy2

f d d^

L13 =-L31 =- (k1C11 + k2Cu)^- + (k1C16 + k2C26)T-l dx dy

d

2

d

2

L22 = C66 +2C26 'dxdy

+C

d2

22 TT dy

f d d Y

L23 = L32 = (k1C16 + k2C26^ + (k1C12 + k2C22)^~ ,

V dx dy

d

2

d

2

L22 = C66 +2C26 'dxdy

+C

d

2

22

dy

d 4 d 4

L33 = Du —— + 2(D 12 + 2 D66 )

dx4

dx2cy 2

+ 4D

16

dx3dy

+ 4 D

d4

26

dxdy3

+ D

22

dyA

- (C11kx + 2C12kxky + C22kl )

d

L14(u, v, w, C ij ) = dx

dw

v2

C111 I + C19

2 11l dx J 2 12

1 ^ f dw Y v^ j

„ dw dw

+ C16--

dx dy

+

+ -

dy

1 fdw

2 C161 ax

+ 2 C26

fdw Y

V^ J

+ C

66

dw dw dx dy

+

(7)

+

du , 1 f dw

--kjw + — I —

dx

21 dx

f dC11 dC Y

dx

+ -

-16

dy

+

dv , 1

--kyw + —

dy y 2

fdw Y2

v^ J

dC

12

dx

+ -

dC16 Y

dy

+

+

f du + dv + dw dw ^ ^ dC16 + dC66 ^

v

dy dx dx dy JV dx dy J

4

d

2

2

d

2

d

L24 (U v W Cj ) = -dk

1 ^ fdw f 1 _ T dw ^ „ -, dw dw

"Clfil I + C26 2 16v dx J 2 26

v^y J

+ C

66

dx 5y

+

+ -

d_ dy

1

dw

\2

— C19I - | + C22

2 12 lax J 2 22

1 ~ Tdw^

v^y j

dw dw

+ C26 - -

dx dy

+

+

du , 1 T dw

--kvw + — I —

dx

2{dx

T dC12 dC ^

dy

+ -

-16

dx

+

dv

--kyw + —

dy y 2

T dw ^

dy.

dC

22

dy

+ -

dC26 ^

dx

+

+

t du + dv + dw dw ^^ dC26 + dC66 ^ dy dx dx dy A dy dx

^ w, Cij, Dij ) = 2

dxJ

+ -

dx dy J dxdy

3

+ 2

d3w

dDi

'12

dx

+ 3

dD-

'26

dy

+ 2

dD,

66

dx

+

31(, f

+ 2

d 3w

dx 2dy

dD

12

dy

+ 3

dD

16

dx

+ 2

dD,

66

dy

3.J

+ 2

d 3w

dy3

dD22 , dD26

dy

+ -

dx

+

+ -

d 2 w

dx2

+ 2

d2 Du + d2 D12 + 2 d2 D16 ^

dxL dyL dxdy

+ -

d 2 w

d 2 w

dxdy

d2 D16 + d2 D26 + 2 dx2 dy2

dy

d 2 D66 dxdy

d2 D12 + d2 D22 + 2 d2 D26 ^ dx2 "

1 T dw

+ —I 21 dx

dyz dxdy

(kxC11 + kyC12 ) +

+

+ -

d 2w dx2

+ -

^dw^

C

11

du , 1

--kxw + —

dx x 2

vdy j

2

2 ^ dw

(kxC12 + kyC22 ) + d- (kxC16 + kyC26 ) +

dx dy

dw dx

+ C

12

dv , 1

--k,,w + —

dy y 2

vdy j

+ C

16

du dv + dw dw ^ dy dx dx dy y

> +

+ 2

d 2 w dxdy

C

16

du

--k„w + —

dx

2

dw dx

+C

26

dv , 1

--kyw + —

dy y 2

vdy j

+C

66

r du + dv + dw dw ^ dy dx dx dy y

+

+ -

d 2 w d^

C

12

du 1 T dw --kvw + —

dx

2

dx

+C

22

dv

--kyw + —

dy y 2

vdy j

+C

26

f du + dv + dw dw ^ dy dx dx dy y

!> +

+ dw [L11(u) + L12 (v) + L13 (w) + L14 (w)] + [L21(u) + L22 (v) + L23 (w) + L24 (w)] dx dy

If the shell under consideration has orthotropic properties, then the coefficients are C16 = C26 = 0 and D16 = D26 = 0 . In relationships (2), the stiffness matrices have the following form:

2

2

2

1

2

2

2

f C11 c12 0 > f D11 D12 0 1

c — C12 c22 0 , D — D12 D22 0

10 0 C66 J 1 0 0 D66 J

(8)

Here, coefficients Cj and Dj (ij = 11, 12, 22, 66) are expressed in terms of elastic constants Ej,

E2, ^12, M1 , M2 as follows:

1l = hB^ - r'n)= Mi^, C22 — hB22(l - r22)= ÎltlZâ],

1 -M1M2 1 -M1M2

, — B12(1 -^)h = M2El(l'/t2h, C66 = B66(1 -r6*6)h = hG12(1 -r6*6)

1 -mm

Du — Bn\1 -

D12 — B12U -

(1 - n h — EÎ^M,

1 11 12(1 -M1M2)

(1 - r*2 h=M2Ei^3

1 12/ 12(1 -M1M2)

D22 = B22'1 -

(1 - ^ h = E2( - r22 ^

1 227 12(1 -M1M2)

^66 = B66 (1 - r66 )h3 h3.

12

(9)

Here Ei,E2 - are the moduli of elasticity in the direction of the x and y axes; G^ is the shear modulus; /, / - are the Poisson's ratios.

If the shell has isotropic properties ( Ej = E2/ = / ), then the elements of the stiffness matrix take a simpler form with two elastic constants E (modulus of elasticity) and / (Poisson's ratio):

Cn = C„ = B (1 -

-22

(1 - r* h = , c12 =4 - r* >

1 -m

ßEh

c66 ='

^ b(1 - r )h — E (1 - r h, D11

2(1 + M)

*V.3

D„ — D22 — b( - r ) — =

3

D12 =

12 12(1 -u2) 2

By introducing into equation (6) the following dimensionless quantities

1 - M

3 E (1 - r * )h

12 12(1 -m2) 3

4 - r • )h- = ME^h3, D66 = i-MÄ - r • h = E (1 - r .

V 12(1 - m ) 2 12 24(1 + m)

(10)

_u_v_w_x_y- T h a „ b - a u — — ; v — — ; w — — ; x — — ; y — — ; t — at ; h — — ; A — - ; o — — ; kx —

.2

h

0

h

k y —

ho ^2

0

; q =

h

0

f u V

VE1E2 I h

; Px =

b

Px

h

0

b

h

0

h

VEE

Py =

2

py ; 0 = 0 ; o A ; 0— P_

4E1E2' 0 Pcr ' 1 Pcr

and taking into account relationship (9) for orthotropic shells and keeping the previous notation, we obtain a dimensionless system of nonlinear integro-differential equations for problems of parametric oscillations of a viscous-elastic orthotropic shallow shell of variable thickness. Here, operators Lit take the following form:

Lii(u) = 4-r* )d22 -U/U2 )g (1 -r* )d22 + f 4-r" )r +

dx

dy2 dx

dx

a

2

b

q

+* % &w)«(i-r* .

dy dy

Li2 (v) = Ah W2a(i - r*2)+ (1 - ww )g(l - r* )]

^ + A^ w2A(l-r;2 № +

dxdy dx dy

A

dh

dy

(1 -ww )g (i-r* R

dx

L13(w) = h

kxA

AS

(1 -r* )+Aky W2 a(1 -r*)

dw dh

dx dx

kxA' AS

(1 -r* )+AkWA (1

S

L14(w) = £1 -11*1 )dW dw A M -Ii2 )+(1 -W1W2 )g (1 -r* )

dx dx2 S

dw d w

dy dxdy

- +

+ (1 -ww£(1 -r*)dx dx

*\dw d w dh +

A

L Y dw ]

1 -r11 — I + 2AS idx J

+

AwA11

* 1 dw 12 -dy

A2

J

+ -

dh /, \ Ag L dw dw

dy

(1 -ww )£ (1

-r* r^—

dx dy'

L21(u)=

w (1 -r*1 )+(1 -W1W2 )g (1 -r*)

d 2u

dxdy

- +

+(1 -ww )A % (1 -r* )du+w dh (1 -r-, )du.

AcT dy AA dy dx

L22 (M ) = f 1 -f2*2 fe

2

+ (1 -ww )£ (1 -r* n +

A dx

A \g dh/ dv 1 dh t \dv

+(1 "ww £ ix(1 +A dy(1 -r22

L23(w) = h

k± w (1 -r*1)+ £ (1 -r*2)

AS A

SA

dw dh — + —

dy dy

1 (1 -^22)+ "x

kx w

AAS A

(l- r* )w

dw d 2 w

L24(w) = ^ )dy dy2

+

w

A

(1 - r*1 )+(1 -www, (1 -r* )

a2s

dw d 2 w

dx dxdy

+

+(1 -ww (1 -r* % ^ (1 -ww (1 -r* )a"d"'

+ -

A2S

dh

dy

dy dx

-L (1 -r* (dw

2SA 22 \dy

+

A2S

w (1 -r2*1 Jfdw

dx dy

+

2AASA

dx

L31(u ) = -12h

ASAkx(1 -r' )+A3S^(1 -r*1)

du dx

- +

2

2

dh_

dx

+^ äsa( - r* № +^ (1 - ww lsg (l - r* +

\du dy

+

dh (1 ls (1 -r* ^+

dx L32 (v ) =

dy dy A

X2öß2 Akx (1 - r*2 )+ ^ (1 -F2*2 )

1 -r

21;

\du

dx

A

dv dy

+

f ^(1 -r* )dv+f (1 -w* v* (1 -r-

dx

,dhh \i2~L r*\ dv dh ÄS + - M1M2)Ä Sl1 - r f— + -——

dx dx dy A

1 -r

22

)dv dx

dv dy

■ +

L33 (w) = h3 U -r* fe + 4(1 -W1W2 )g( -r* )+W2A(1 - r*2 )+W (1 - r-1)

I dx A

dx

Ä2^ +

+ -

ÄÄ A

(1 -Fi )

d 4 w

dy4

+ 3

o, dhY2 ,2 d2h 2h — | + h2—-

dx 2

A(1 -FA rW +Ä2W2 A(1 -r* )

d 2 w

dx

dy2

dx 2dy 2

+

+6h2 dh u -r* )d3w+w a(1 -r*)+ 2(1 -ww )g (1 -r* l2^ K

dx I dx dxdy

+6h2 dh Jl (1 -

dy | A

(1 -r*2 L

CTW

dy2

- +

w (1 -r*)+ 2(1 -WW2 )g (1 -r*)

L^w- >+

dx dy

+ 3

2h

f dh Y

vdy j

+h

d 2h dy 2

2

A (1 -r* l di(1 -r*1 l

22 34 d w + d w

dy2 A

dx'

2

+

+12

., dh dh j2 d h 2h--+ h2

dx dy dxdy

(1 -ww )g (1 -r* )ä

,2 d w

dxdy

- 12h

Ahl (1 -r* )+L kxky

W2A(1 - r*2 )+W (1 - r*1)

4,2

Ä4k

+

A

(1 -r*2 )

>w -

dh

dx

kx A(1 -r* )+Ä2kyW2A(1 - r*2 )

w + -

dh

dy

Ä4ky

_y

A

(1 -r2*2 )+

A

1 -r2*1)

w,

L34 (u, v, w)=1

kx a(1 -r*)+iWy (1 -r2*1,

+

2

2

1

+ — 2

AAkx

wzA(1 -f1*2 ^)+AAky- (1 - r*2)

A

'dw ^

ldy J

-12

dx

hJ A1 -

(1 -r* (

„ _du dw d w

AS—y +--T

dx2 dx dx1

kx A (1 -r*1)+ A wAky (1 -r* )

dw dx

- +

+

w2 a(1 - r*2)+ (1-M1V2 )g (1 -r*) + (1 -ww )g (1 -r*)

2„, A

.2S d v 2 dw d w A o--+A

dxdy dy dxdy

+

,3 e du n2 dw d w AS—it + A

d/

dx dy2

(11)

^d 2 w,

-12—r h

dx 2

4

. h „* . , „du 1 (dw A 1 -r11 AS—+ -| — 1U dx 2 lex

+

A(1 -r1*2 )

2S dv A2 (dwA

A o--1--—

dy 2 {dyJ

2

-[kx a(i -r* )+A2 kyw a(

2. .A

1 -r

12

■ -12^J hJ-1 (1 - r2*2 )( A4Sd-2 + A4 ^

dy | |AV 22/l dy dy dy

^ (1 -r2*1 (1 -Ii)

dw dy

■ +

w (1 -r2*1 )+(1 w )g (1 -r*)

( .3„ d2u .2 dw d2w A

AS-+ A--

dxdy dx dxdy

+ (1 -ww )g (1 -r*)

2 e d v .2 dw d w A o—A ----

dx2 dy dx2

2. .A

+

d 2 w, -12—r h

dy 2

(1 -r*)

w (1 -r

A

A3S— + A2 2

dx 2 l dx

+

+A (1 -r*)

,4 c dv A4 (dw^ AS— +

dy 2

vdy j

A

(l-r*1 (l-F2*2 )

w

0. d w *\( .3- du 2 s- dv .2 dw dw

24—— h(1 - ww )g(1 - r J| A3S—+ A S—+ A ———

dxdy l dy dx dx dy

+

dh

+ —J dx

4

A(1 -rn| -| —

+w2a(,

,1 -r

12 J

A

2(dwA2

+

dh dh|(l -ww)A2g(l-F*)dw dw dh

+

dx dy

+ —J

dx dy dy

L (1 -ri)

A

A4

vdy j

vdy j

J> +

+

Hl

A

(l-r2*i ^

2

dw dx

n2

here Pcr = -t

3(1 -ww)

( h

■sjelE2 1 is the static critical load; a> =-sjn2^E1E2h2P*crj(pb4) i

is

> —

2

X

X

2

> —

2

2

2

the frequency of the fundamental tone of oscillations; P* =

n

4ÊE2 (b / h )2 3(1 — И1И2 )

; a= E

E2

The system of equations (6), (11) with the corresponding boundary and initial conditions describes the motion of a viscous-elastic orthotropic shallow shell of variable thickness under a periodic load

P(t) = P0 + P1cos(0t) .

(12)

In calculations, the singular Koltunov-Rzhanitsin kernels [27] are used as relaxation kernels: r(t) = Ae~ptta-\(0 <a<l), P*(t) = A^^f*-1, (o <ajj <l)

Let the shell thickness change following the law h(x) = lho(l + a*x), i.e., it leads to a linear increase in the shell thickness (Fig.2).

* 7

Here, a is the parameter characterizing the thickness variability; ho is the shell thickness corresponding to a = 0 .

a) b)

Figure 2. Change in the shell thickness depending on the value

of parameter a* :

a) a* = 0.2 ; b) a* = 0.5

The solution to the obtained IDE system that satisfies the boundary conditions of the problem is sought with respect to the displacements u and v, and the deflection w in the form

N M N M

^ t )=Z I Unm (t }Pnm y I v(x ^t )=Z Z vnm (t )<Pnm y I

n=i m=1 n=1 m=1

N M

^ t )=Z I wnm (t )Wnm (x, y) , n=i m=i

(13)

where unm = unm (t), vnm = vnm (t A wnm = wnm

(t) - unknown functions of time; 0nm (x, y), (Pnm(x,y), Wnm(x,y), n = 1,2,...,N; m = 1,2,...,M - coordinate functions that satisfy the given boundary conditions of the problem.

Substituting (13) into the system of equations (6), (11) and performing the Bubnov-Galerkin procedure, we obtain the following system of basic resolving nonlinear IDEs:

N M

Z Z aklnmunm n=1 m=1

-Jl Z {| -r*1 hlnm + (1 -r*d

ln=1 m=1

2klnm

i1 - Г12 ])d3klnm + ( - Г )d4klnm. Vnm + ( - Г11 ^iklnm + ( - Г12 )d6klnm. Wnm }+

N M Г/

+ z z [(1 -гцd

n,i=1 m, j=1

Iklnmij

+ 1 - Г1 2 )d8klnmij + t - Г )d9klnmij Ï1

iklnmij iWnmWij - W0nmW0j

, )j=0,

+

nm

+

I Mbklnmvnm-niil I {| -r2*1 \klnm + ("^)

klnm nm

n=1 m=1 ln=1 m=1

2klnm

i1 - r22\klnm + (1 - r )e4klnm. vnm + ( - F21 \iklnm + (1 - P22 )e6klnm. wnm }+

(14)

+

1 1 f r22 )Slklnmij +(1 r21 ^8klnmj +r \klnmij Ï1

n,i=1 m, j=1

9klnmjïwnmwj - w0nmw0j

, )}=0,

N M NM 2

^^ ^^ Cklnm wnm + % I 1 Pklnm (1 - 2Vklnm C0S ®tw nm n=1 m=1 n=1 m=1

-n31 II I{[(1 - r1*1 )frnnm + (1 - r2*1 )f2klnm\nm + |- ^2)f3klnm + t- ^2)f U=1 m=1

4klnm

v +

nm

+

❖ ❖ ❖ ❖ ❖ I

r11f5 klnm + r12 f6 klnm + r22f7 klnm + r21 f8klnm + r f9klnmr0nm /

n31 1 wnm {[(1 - r11 )%1klnmij + (1 - F21 )^2klnmij +C1 - F )^3klnmij ]iij +

n,i=1 m, j=1

+

K1 r222 \4klnmij +(1 r12 ^5klnmij + (1 F ^6klnmij \ij }

+1 -r

1 1^Iklnmij + (1 - r1 2 ^Zklnmij + (1 - F22 )%9klnmij + (1 - F2 1 )%10klnmij twij - w0ij )}+

+ Z Z i1 -r11 \rnnmjrs + (1 -r12 )g2klnmjrS + (1 -I21 \mnmjrs + ^-r22 )g4klnmjrS )(wnmwj - w0nmw0j )h n,i=1 m, j=1

+ E E wnm f1 - r11 )g5klnmjrs + (1 - F12 )g6klnmijrs + (1 - F22 )glklnmjrs + n,i,r=1 m, j,s=1

+(1 - r21 )g8klnmijrs + (1 - r * )g9klnmijrs j(wij wrs - w0ijw0rs )| = 12% (1 - U1U2 KVkl ,

unm (0) = u0nm , unm (o) = u0nm , vnm (o) = v0nm , vnm (o) = v0nm , wnm (0) = w0nm , wnm (0)= w0nm , k = l2,-,l = 1,2,..., M ,

where the constant coefficients included in this system are related to coordinate functions and their derivatives and have the following form:

11

aklnm = U WrmAldxdy ;

00

11

00

d1klnm , xx + hx$nm,x

}ßkldxdy ;

d2klnm = j j(1 - ß1ß2 (h€m,yy + h'y$'nm,y }ßkldxdy ;

00

+

nm

+

1 1

d3k/nm

— jjH2A4h<nm ,xy + hxpnm, 00

d4k/nm — J J (l - H1H2 )s^h<nm

, xy + hypnm, x

00

d5klnm — -11 Akx (h ¥rim,x + ^Knm К^Ф ; 00 Лд

d6klnm — - J J AkH^ (h y/'nm,x + ^K™ К^Ф ;

00 5

il A Г 1 Л

dlk/nmij — JJ^ôVh ¥nm,x¥?j,xx + ^hXm,xW'ij,xJ^/^ ;

d8k/nmij — J íUдAЛ^hWnm,yWii■,xy + 1 h'xV'nm,yW'ij ,y \kAdy 00 д V 2 J

d9klnmij — J J (l - HlHl )g -Г (h ^nm, yV"ij, xy + h Km, xWij ,yy + hy¥nm, x¥ij,

oo

1 1

bklnm — J J h<nm<kldxdy ;

00

e1klnm — J ÍT7 (h€m,xy + ^m,x ;

00 АЛ

e2k/nm — J J(l H1Hl )g (h€m,xy + hi^m,y КАФ ; 00 Л

11

e

J J 7 (h<nm,yy + h'yP'nm,y )Pkldxdy >

00 A

e4klnm — J J(l Hl2H2 )g (h<nm,xx + ^P^x)Pkldxdy > 00 Л

J J (h W'nm,y + h'yWnm )Pkldxdy e6k/nm — -J J (h ^nm,y + h'y¥nm )Pk/dxdy > 00 Л 5A 00oa

ii 1 г 1 Л

— J Jdl hW'nm,yW'îj,yy +ÖhW'nm,yW'ij,y \<kldxdy 00dA V 2 J

j — JJ^"ÍhK'nm,xW'j,xy + 1 h'y¥rim,xW'ij,xIpkl^ ;

00 ЛдАч 2 J

J J(l H2lH2 )g (h¥'nm,x¥?j, xy + h¥'nm,y¥¡,xx + h'x¥'nm,x¥¡j,y )<k/dxdy

elklnmij

e

8klnmij

e9k/nmij — J J .2 e Vh¥nm,x¥ij,xy + h¥nm,y¥ij,xx + 00 Лд

11

Ck/nm — JJh¥nm¥k/dxdy ; 00

11 11 Л-frK

f1klnm = -12í l^^xH'nmWkl^y ; f2klnm = -12ÍÍ-Л-4'nm)kldxdy ;

00 00 Л

11 11 Л4-к

f3 klnm = ~12\ \ Л -12Лк xh^nm,yW kld^dy f4klnm = Wnm,y¥kldxdy

00 00 л

f5klnm = Í íл(h3 Wnm , xxxx + 3[2h(hX )2

+ h hxx tf^nm, XX

+6h1h')W4im XXX + 12kXh Wnm Wkldxdy ;

00

f6klnm = Í Jl2 ЛЛ2 (h3vZ,xxyy + 3 2h(hX )2 + h 2 hXx 1)'!п,уу +6h 2 h'xWnm,xyy + 12kxkyhWnm 00

flklnm = 1(h3wZ,yyyy + 6h2h'yWnm,yyy + 3 2h(h,^ + h2^ ^m,yy +12кУh Wnm \к^У > 00 л

11 ,, з2

00 Л

f8klnm = í í^^" ^vOmxxyy + 6h 1^у)!гп,хху + 3 2h(h, ^ + ^ ^üm,,, +12kxkyhWnm ^Ы^У >

11 í

f9klnm = íí(l -Ill2 )gЛ2 (4hVnm,xxyy +12h^L,xyy +12h2h'yW^m,xxy + 00

+ n^hhXhy + h 2h"Xy V'nmxxyV kldxdy ;

)1klnmij = 121!Л-Ф W'nm,xфj,XX + hX)^m,хфф\x + hW^m,ххфф,x Vkldxdy > 00

11 лз-

)2klnmij = 12í Í I1"!-(h W'n m, Уфф , xy + hyW пп,уфу , x + h) nm,yyфij, x

00 л

11

)3klnmij = 12íí(l -Ill2 )gЛ3—(h Wnm, фф ,yy + h'yW'nm,y^ij y + hV'nm,yfij, xy + 00

+ h'xW'nm, уфф

,y + 2h Wnm, xy фу, y ))kldxdy ;

ll Л4-/' \

)4tt"j= «И"^», У^, УУ + V», y^j, y + hV'm,yyVlj, У W** ;

+ h'w' +J

00

)5klnmij = , xy + hxWnm, x@ij,y + h)nm, xx@ij, y \)kl(dxdy

'5klnmij 2 у Vnm, xVj, xy ^ "хУ nm, хУу,у ^"r'nm, ххУу,у

00 ;

11

)6klnmij = 12íí(l -Ill2 h^^V'nm,xVij,xy + xV'ij,x + h )!п,yV'îj,xx +

00

+ h'xV'nm,yP'ij,x + 2h Wnm,xy^^^,x ))kldxdy ;

11

)l klnmij -12ÍÍ kx4h W!m,xWij,x + hXWnm,xWij + hW^m,xxWij )^kldxdy ; 00

$

8klnmij -12J J ÄU2 ky A(h W'nm, xW'ij, x + Kv'nrn, xWij + h W'hrn, xxWij }^kldxdy i 00

9klnmij = -12J^—T^ {hW'nm,yWij,y + h'yW'nm,yWij + hW"nm,yyWij }wkldxdy i

00 11 ^2

A

$

10 klnmij

11 AU k i \

= -12J J-(h W'nm,yWlj,y + h'yW'nm,yWij + h Wnm, yy Wij }^kldxdy i

00 A

1 1

11

g 1klnmij = 6JJkxAhW'nm,xWij,x¥kldxdy i g2klnmij = 6iiÄ U2kxAhW'nm,yW'ij,y¥kldxdy i

00

nmij

00

11Ä u1ky tt'""y

g3klnmij = 6JJ-^— hWnm,xW'ij,xWkldxdy i g4k/nmi/ = hWnm,yW'ij,y¥kldxdy i

00 A 00 A

11Ä4 k

11 t 1 1 Y

12JJAI h Wnm,xW'ij,xw"rs,xx + 2hXm,xW'ij,xW'rs,x + ^hW™,xxW'ij,*W«,x j^kldxdy i

g 5kh

nmijrs

00 v 11 7

11 2T 1 1 Y

g 6klnmijrs = 12 J J U2 AA21 hWnm,xWij,yWrs,xy + _ hxWnm,xWij,yWrs,y + - hWnm,xxWij,yWrs,y Wkldxdy i 00 V 2 2 j

11 ä4 T 1 1 Y

glklnrnijrs = 12JJx hX'nrn,yW'ij,yW"rs,yy h'yW'nm,yW'ij ,yW'rs,y + ~hW"nm,yyW'ij,yW'rs,y \Wlddxdy i 00 A V

nmijrs

22

12J J u1l (2h W'nmyW'ij, x¥"rs , xy + hyWnm,yWij, xWrs, x + h Wnm,yyWij, x V«,x Vkldxdy i

00

1 1 /

g^klnmijrs = 12J K1 - U1U2 )gÄ (h W™,xW'ij,yW"rs,xy + h W™,xWij,xWn

,IV ij,xY rs,yy

+

00

+ hyWnm, xW ij, xW rs, y + hWnm,yWij, xWrs,xy + hWnm,yWij ,yWrs, xx +

1 1

+

h'xW'nm,yW'ij,xWro,y + 2h Whm,xy Wj,xWrs,y )Wkldxdy i qkl = q JJwkldxdy i

00

2 i2 *

2 _ r r r r r A 2 J2 * c — 2^ J Phlnm c

Pklnm = J 5klnm + f6klnm + f7klnm + f8klnm + J 9klnm - j pklnmd0 ; <%lnm = 2 C1 .

pklnm

System (14) was integrated using a numerical method based on the use of quadrature formulas [28]. Assuming harmonic oscillations, system (14) in integral form is obtained by integrating it twice over time t:

N M

Z Z aklnmunm n=1 m=1 n=1 m=1

NM t t f N M (w (A

= Z Zaklnm(u0nm + u0nmt)+V1 JHZ Z {[(1 -r11 Klnm + (1 Ji2klnm

n=1 m=1 0 0 l«=1 m=1

i1 -r*2 \3klnm + (1 -r* )d4klnm \nm + |-r*1 )d5klnm + ^-r1*2 )d6klnm ]Vnm }+

u+

nm

ZZ Z I-r1*1 Wlnmy +(1 -r1*2 Wlnmy -r' Wlnmj \wnmwij - w0nmw0ij )[dT<ds ,

n,i=1 m, j =1

+

+

NM NM t t f N M rn * \ ZA

I I bklnmvnm = I Ibklnm(v0nm + WK^jjJ I I {[(l^21 faklnm + (l-F >

n=1 m=1

NM

II

n=1 m=1

t t f N M

nfi i

0 0 U=1 m=1

"2klnm

u +

nm

+

(l - r22\klnm + (l - r \4klnm. vnm + (l - r21 \klnm + (l - F22 )e6klnm. wnm } 1 1 ((l - r22 ^7 klnmij + (l - ^21 )?8klnmij + (l - F )e9klnmij ]l

(15)

n,i=1 m, j=1

e9klnmij Wnmwij - w0nmw0j

j )ldrds,

NM NM t t f NM 2

I I cklnmwnm = I I cklnm (w0nm + ^W jj{ I I PMnm (l - 2wklnm cos ©t K

n=1 m=1 n=1 m=1 00 Ln=1 m=1

ii i {[(l-riL)flklnm + (l-r2*i)f2klnm\nm + [(l-ri2)f3klnm + (l^)f n=1 m=1

' 4klnm

v +

nm

ri1f5klnm + ^12 f6klnm + r22f7klnm + r21 f8klnm + F f9klnm

w

0nm

}-

- 1 1 wnm { [(l - ri 1 ^iklnmij + (l - r2 1 )^2klnmij +(l - F )^3klnmij \ij + (l - r22 ^4 klnmij + (l - 2 )%5klnmij + (l - r iklnmij \ij

n,i=1 m, j=1 +

+

+

N M

(l-ri 1 iklnmij + (l-ri 2iklnmij + (l-F22iklnmij + (l-F2 1 \l0Mnmij]wij -w0ij)}+

+

ii i f1 - ri L)g1klnmijrs + (l - ri 2)g2klnmijrs + (l - f2 L)g3klnmijrs + (l - f22)g4klnmijrs ^nm^ij - w0nmw0ij ) +

n,i=1 m, j=1

+ H 1 wnmfl l)g5klnmijrs + (l 2\6klnmijrs + (l F22)g7klnmijrs + n,i,r=1 m, j,s=1

(l - r2*i )gmnmijrs + (l - r *)g9klnmijrs ]^ijwrs - w0ijw0rs )- 12^3 (l - ßlß2 )A Vkl |dTs ■

By the formula for replacing the double integral with a single integral the system (15) is given in the following form:

N M

1 1 aklnmunm n=1 m=1 n=1 m=

NM t f N M (W * \ ( *\

= I Iaklnm(u0nm + u0nmt)+nij(t-t) I I {[(l-ri ,^m + (lfa^

n=1 m=1 0 ln=1 m=1

u +

nm

(l-r,2 hlnm + (l

N M [7

+ I I (l-r,ijd.

n,i=1 m, j=1

I

w 1+

nm

N M

I I

n=1 m=1

I I bklnmvnm = I Ibklnm (v0nm + v0nmt) +

(l - r, i L)d5klnm + (l - ri 2 )d6klnm 7klnmij + (l - ri2iklnmij + (l - ^9klnmij lwnmwij - w0nmw0ij ■

<bklnm(v0nm + WK^j(t-t){ II I {[(l^l\klnm + (l-r*)e2kln

1 0 |n=1 m=1

}

NM

II

n=1 m=

(l-r*2 }3klnm + (l

=1 m=1

(i-r2i )e 5klnm i

w 1+

nm

u +

nm

(16)

+

+

+

Z Z ( r22 )elklnmij +(1 r21 ^Mlnmy +1 r )S9klnmij Ii1

n,i=1 m, j =1

^UnmijrnmWy -w0nmw0j

NM NM t f NM 2

Z Z cklnmwnm = Z Z cklnm (w0nm + w0nmt) - J (t - T)j Z ZPklnm i1 - 2Uklnm cos ©tK

n=1 m=1 n=1 m=1 0 Ln=1 m=1

Z Z { i1 r111)f1klnm + (1 r21 )f2klnm unm + j1 F12 )f3klnm + 1 F22 )f4klnm n=1 m=1

v +

nm

r11 f5klnm + r12 f6klnm + r22 flklnm + ^21f8klnm + r f9klnm

m }

- Z Z 'wnm { [i1 - r1 1 )$1klnmij + i1 - r2 1 )$2klnmij +11 - r )$3klnmij \ij K1 - r22 2$4klnmij + i1 - r1 2 )$5klnmij + i1 - r )$6klnmij \ij +

+

n,i=1 m, j=1 +

+ LI1 -

N M

1 - r1 11$iklnmij + i1 - r1 2 )$8klnmij + i1 - r22 )$9klnmij + i1 - r2 1 )$10klnmij ](wij - w0ij )}+

+ Z Z i1 -r1 11)g1klnmijrs + (1 -r1 2 ])g2klnmijrs + (1 -r2 1gklnmijrs + (1 -r22 )g4klnmijrs )(wnmWi/■ - w0nmw0ij )+ n,i=1 m, j=1

+ Z Z wnm f1 r1 1 \5klnmyrs +(1 r1 2 2)g6klnmijrs + (1 r22 )glklnmyrs + n,i,r=1 m, j,s=1

+(1 -r2*1 )g8klnmijrs +(1 -r ^klnmijrs ]^ijwrs - w0ijw0rs )- 12% (1 - U1U2 )Ä4qk/ ^|dT ,

lnm (0) = u0nm , unm (0) = u0nm , vnm (0) = v0nm , vnm (0) = v0nm , "0nm , wnm (0)= w0nm , k = 1,2,•••,l = 1,2,--,M ■

wnm (0) = wn„m, w„

Assuming that t = t{, ti = iAt,i = 1,2,... (where At is the integration step) and replacing the integrals with quadrature trapezoidal formulas to calculate the unknowns winm = winm(ti),

uinm = uinm (ti) and vinm = vinm (ti) , a system of recurrent formulas is obtained.

Based on the developed algorithm, a program was compiled in the Delphi algorithmic language.

+

3. Results and Discussion

The results of calculations for various physical and geometric parameters are shown in graphs in Fig. 3-7. Numerical results are compared to the ones available in the literature.

The effect of orthotropic properties of the material on the behavior of a shell was studied (Fig. 3). As seen from the figure, an increase in parameter a that determines the degree of anisotropy (curve 1 -A =1; curve 2 - A =1.5; curve 3 - A =2.0) leads to an increase in the oscillation amplitude and a phase shift to the left.

0 5 10 15 0 25 0 35 t

Figure 3. Dependence of the deflection vs time for

k = 1; 5 = 25; kx = 10; ky = 10; q = 0; px = 0; py = 0; a* = 0.5; 0 = 1.1;

A = A* = 0.05, i, j = 1,2; A =1 (1); 1.5 (2); 2.0 (2)

Figure 4 shows the results obtained from different theories. Here, curve 1 corresponds to the case when the shell material is elastic, curve 2 - to the case when the viscosity of the material is taken into account in the direction of shear (a = 0.05, AiJ- = 0, i, j = 1,2), and curve 3 - to the case when the viscosity

is taken into account in all directions (a = A* = 0.05, i, j = 1,2).

Figure 4. Dependence of the deflection vs time for

j = 1; 5 = 25; kx = 10; ky = 10; q = 0; px = 0; py = 0; a* = 0.5; 0 = 1.1; A = 1

The results obtained confirm that viscous-elastic properties of the material should be considered not only in the shear direction but also in other directions.

The influence of the shell thickness on its behavior is studied. Figure 5 shows the results obtained

for various values of the thickness change parameter a*. It can be seen that with an increase in this parameter, the oscillation amplitude increases. In particular, the results obtained for a shallow shell of

constant thickness (a* = 0) coincide with the results obtained in [29].

Fig.5. Dependence of the deflection vs time for

1 = 1; S = 25; kx = 10; ky = 10; q = 0; px = 0; Py = 0; 0 = 1.1; A = A = 0.05, i, j = 1,2; A = 1 a* =0 (1); 0.5 (2); 0.8 (3)

Figure 6 shows the results obtained for various values of the curvature parameter kx. An increase in this parameter leads to an increase in the amplitude of oscillations.

Figure 6. Dependence of the deflection vs time for

1 = 1; s = 25; ky = 10; q = 0; px = 0; py = 0; a* = 0.5; 0 = 1.1; A = Aij = 0.05, i, j = 1,2; A = 1 kx =10 (1); 15 (2); 20 (3)

Figure 7 shows the results obtained for various values of the frequency of the external periodic load 0 . An increase in this parameter leads to an increase in the amplitude of oscillations.

0.01

0 5 10 15 20 25 30 35 (

Figure 7. Dependence of the deflection vs time for

1 = 1; S = 25; kx = 10; ky = 10; q = 0; px = 0; py = 0; a* = 0.5;

A = Ay = 0.05, i, j = 1,2; A = 1 0 =1.1 (1); 1.3 (2); 1.5 (3)

4. Conclusion

A mathematical model, method, and computer program were developed to estimate parametric oscillations of a viscous-elastic orthotropic shallow shell of variable thickness, taking into account geometric nonlinearity under periodic loads.

The dynamic stability of a viscous-elastic orthotropic shallow shell of variable thickness was described by a nonlinear system of IDEs.

The application of Galerkin method with the discretization of spatial variables at each time point reduces the problem of dynamic stability of a viscous-elastic orthotropic shallow shell of variable thickness to solving a non-decaying system of ordinary nonlinear IDEs with weakly singular kernels with variable coefficients.

The impact on the amplitude-time characteristics and the SSS of a change in the physical-mechanical and geometric parameters of the shell material was estimated.

The method proposed in this article can be used for various types of thin-wall structures (plates, panels, and shells of variable thickness).

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Information about authors:

Dadakhan Khodzhaev, PhD in Physics and Mathematics ORCID: https://orcid.org/0000-0001-5526-8723 E-mail: [email protected]

Rustamkhan Abdikarimov, Doctor of Physics and Mathematics ORCID: https://orcid.org/0000-0001-8114-1187 E-mail: [email protected]

Marco Amabili, PhD

ORCID: https://orcid.org/0000-0001-9340-4474 E-mail: marco. amabili@mcgill. ca

Bakhodir Normuminov,

ORCID: https://orcid.org/0000-0002-5018-3533 E-mail: normuminovb [email protected]

Received 11.10.2022. Approved after reviewing 20.03.2023. Accepted 23.03.2023.