Mathematical model of oscillations of a three-layered channel wall possessing a compressible core and interacting with a pulsating viscous liquid layer Текст научной статьи по специальности «Физика»

UDC 517.958

DOI: 10.18698/0236-3933-2019-6-4-18

MATHEMATICAL MODEL OF OSCILLATIONS OF A THREE-LAYERED

CHANNEL WALL POSSESSING A COMPRESSIBLE CORE

AND INTERACTING WITH A PULSATING VISCOUS LIQUID LAYER

1 Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation

2 Balashov Institute of Saratov State University, Balashov, Saratov Region, Russian Federation

The paper deals with the formulation of a mathe- Modeling, oscillations, hydroelastic-matical model to study a dynamics interaction of a ity, three-layered plate, compressi-three-layered channel wall with a pulsating viscous ble core, viscous liquid fluid layer in a channel. The narrow channel formed by two parallel walls was considered. The lower channel wall was a three-layered plate with a compressible core, and the upper one was absolutely rigid. The face sheets of the three-layered plate satisfied Kirchhoff s hypotheses. The plate core was considered rigid taking into account its compression in the transverse direction. Plate deformations were assumed to be small. The continuity conditions of displacements are satisfied at the layers' boundaries of the three-layered plate. The oscillations of the three-layered channel wall occurred under the action of a given law of pressure pulsation at the channel edges. The dynamics of the viscous incompressible fluid layer within the framework of a creeping motion was considered. The formulated mathematical model consisted of the dynamics equations of the three-layered plate with compressible core, Navier — Stokes equations, and the continuity equation. The boundary conditions of the model were the conditions at the plate edges, the no-slip conditions at the channel walls and the conditions for pressure at the channel edges. The steady-state harmonic oscillations were investigated and longitudinal displacements and deflections of the plate face sheets were determined.

E.D. Grushenkova1 L.I. Mogilevich1 V.S. Popov1 A.V. Khristoforova

,2

[email protected]

[email protected]

[email protected]

[email protected]

Abstract

Keywords

Frequency-dependent distribution functions of amplitudes of plate layers displacements were introduced. These functions allow us to investigate the dynamic response of the channel wall and the fluid pressure change in the channel. The elaborated model can be used for the evolution of nondestructive testing of elastic three-layered elements contacting with a viscous fluid layer and being part of the lubrication, damping or cooling systems of Received 16.04.2019 modern instruments and units © Author(s), 2019

This work was supported by the Russian Foundation for Basic Research,

project no. 18-01-00127-a

Introduction. Modeling the interaction of elastic elements of structures with a fluid is of great importance for the development of instrument making. For example, in one of the first papers [1], in which free vibrations of a circular plate in contact with water were considered, its author H. Lamb noted that this study is extremely important for transmitting signals underwater. Using Rayleigh strain-energy method, he obtained expressions for the plate's oscillation frequencies. Further development of this issue was carried out in [2] based on of the formulation and solution of the hydroelasticity problem in joint consideration of the equations of motion of an elastic element and fluid. In these works, autors relied on the contact of one plate surface with an unlimited volume of an ideal incompressible fluid, and it was shown that oscillation is damped due to the conversion of energy to wave formation in a liquid, and the effect of increasing inertia, estimated by the added mass, is observed, resulting in a decrease in vibration frequencies. Further studies on this issue are aimed at studying the influence of related factors. For example, the study of the attached masses of a fluid, which take into account its inertial properties, with oscillations of plates of different shapes and with various methods of their fixing was performed in [3]. In [4], free vibrations of a circular plate on the free surface of an ideal non-compressible fluid, the volume of which is limited by a rigid cylindrical wall and bottom, were investigated. A similar problem with the immersion of a plate under the free surface of an ideal fluid was considered in [5]. The problem of chaotic oscillations of a plate that is in contact on both sides with the flow of an ideal fluid is considered in [6]. Mathematical models for the study of the dynamics of elastic elements of pressure sensors and vibration devices based on the formulation of the problem of hydroelasticity of a plate interacting with an ideal fluid were considered in [7, 8]. In [9], a model was proposed for determining the width of the contact zone between solid surfaces and a thin-

walled cooling channel, having a flat-oval section, when it is hydroelastic deformed under the influence of internal static pressure.

At the same time, studies of vibrations of elastic elements of structures interacting with a viscous fluid are extremely important, since viscosity determines the damping properties in an oscillatory system. For example, in [10, 11], the interaction of a viscous fluid with the structural elements of gyroscopic devices was considered and the effect of this interaction on the vibration resistance and accuracy of devices was shown. In [12], the study [1] was generalized for the case of taking into account the viscosity of the fluid. Hydroelastic vibrations of the elastic elements of the vibratory machine interacting with a layer of a viscous fluid are considered in [13]. The problem of oscillations of a cantilever-fixed beam immersed in an unlimited volume of a viscous fluid was solved in [14]. A similar problem for a piezoelectric beam in a viscous fluid flow was studied in [15]. The transverse oscillations of two coaxial disks interacting with a layer of a viscous incompressible fluid between them are studied in [16].

In connection with the development of aerospace technology, the three-layered structural elements in the form of beams and plates are becoming more widely used in modern aggregate instrument making. These elements have the necessary rigidity and low weight, and protect against aggressive effects (temperature, radiation, etc.). Approaches to the study of their statics and dynamics are well developed and presented, for example, in the review part of the monograph [17]. However, studies on the simulation of their oscillations in the interaction with the liquid are not enough. The oscillations of composite plates interacting with an ideal liquid were investigated in [18]. In [19-22], vibrations of three-layered beams and plates prossessing an incompressible core and interacting with a viscous fluid were studied. In this paper, a mathematical model is proposed for studying the longitudinal and bending vibrations of a three-layered channel wall having a compressible core and interacting with a pulsating layer of viscous fluid, taking into account normal and tangential stresses on its side.

Formulation of the problem. Consider a narrow channel formed by two parallel walls, the bottom wall of which is a three-layered plate with compressible filler (Figure). The plate is simply supported on the ends, the thickness of its outer bearing layers are hi and h2, the thickness of the core is 2c. We (or let us) associate the Cartesian coordinate system xyz with the middle plane of the core in the undisturbed state. The upper wall of the channel is rigid. We assume the size of the channel in terms of 2l x b and assume b » 2l, that is, we consider the plane problem. The distance between the channel walls in the undisturbed state is

ho and 2l » ho. The channel is filled with a viscous incompressible fluid, the pressure in which pulses due to the harmonic law of pressure pulsation given at the ends p* = po + pmsin (ra t). Here po is constantly pressure level, pm is amplitude of pressure pulsation, ra is frequency, t is time. The elastic movement of the plate is much less than ho. We investigate the steady-state oscillations since the viscosity of the fluid cause's considerable frictional forces in the channel, which leads to a rapid decay of transients.

у//////////////////////.

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сц

A narrow channel, the bottom wall of which is a three-layered plate: 1,2 are upper and lower bearing layers; 3 is core

The plate consists of upper and lower bearing layers, which perceive the main loads, and core, ensuring their joint work as a single package. We assume [17] that the bearing layers are isotropic, incompressible in the transverse direction, and satisfy the Kirchhoff conjectures. The core is considered hard given its compression, the exact relations of the theory of elasticity hold for it, and the dependence of the displacements of its points on the transverse coordinate z appears linear. The deformations are assumed to be small, and at the boundaries of the layers of the plate, the conditions for the continuity of their displacements are satisfied. For the above assumptions, the stress-deformable state of the plate is fully described by the longitudinal displacements and deflections of the middle planes of its bearing layers, and the equations of its dynamics are as follows [17]:

d2u1 d2u2 dw1 F + a U1 - «1 U2 - «4 —— - as —— + a2 ——1-

dx

dw2 . diw1 + аз--2 ü6-— + Й7

dx

F2 - ai щ + ai щ - as

dx3 д2щ

dx2

-a<)

dw2 d3w1 „ - a2 —--a6 —— + 2a7

dx

dx3

dx 2 ô 3w2 dx3 d2Ü2 dx 2 d3w2 dx3

dx

~ qzx;

dw1

(1)

a3

= 0;

dx

du1 du2 . diu1 d3u2 d2w1 d2w2 F -ai7~— + aw—— + 2a6 —— + a6—— + an—— -ai2 —— +

dx

dx

dx

dx3

dx

dx2

d4w1 „ d4w2 L dqzx

+ ai5—~— 2ai^—— + as wi -as W2 = qzz +-m——;

dx4 dx4 2 dx

du1 du2 diu1 d3u2 d2w1 d2w2 F4 ~ a1s——ь a19 —--a7 —-— 2 a7 —-— a12 —— + a14 •

dx

dx

dx3

dx3

dx

dx

d4w1 d4w2 ■ a16 —7- + a13--— as W1 + as W2 = 0.

dx4 dx4

Here ui and u2 are the elastic longitudinal displacements of the upper and lower bearing layers of the plate, wi and w2 are the deflections of the upper and lower bearing layers of the plate, qzz, qzx are the normal and shear stresses acting on the upper supporting layer of the plate from the liquid side, these stresses are written as

qzx = -pv

duz dux , ,

—- + —- I at z = c + h1 + w1; dx dz

duz

(2)

qzz =~p + 2pv- at z = c + h1 +w1,

dz

where p, v are the density and kinematic viscosity of a fluid, Ux, Uz are the projections of the velocity vector of the fluid on the coordinate axes, p is the fluid

pressure, p^ is the material density of the k-th plate layer, Gk, Kk are the shear and

4

volume strain moduli of the k-th plate layer, K£ = Kk + — Gk,

Kk = Kk - -Gk. 3

In addition, the following notation has been introduced:

_ 2 G3 ( a1 —-; a2 = 2 G3

1 h11 K3- _ f _ h2л

1 + —--; a3 = 2 G3 1 + —

2 c 2 2 c

KL 2

a4 = h1 +

2 K 3+c

as =■

K 3+ c

_K3+ ch1 K3+ ch2 a6 --; a7 -

6

6

as =

K+ ^ 2 K+c

3 ■ a9 = K| h2 + 3

2 c

G3

-—; a10 = — 3 2

hL 2 c

KL 2

an =

K 3- h1

G3 c ( h1 Л

1 + —

2 c

G3 c

c

3

ai2 =

K3 (hi + h2 ) Gi<

f

i +

hL 2 с

Л

f

I +

2 с

G3 с

K+ h23 K+ chf K3 h2 G3 с

ai3 —--1--; fli4 —---

13 12 6 14 2 2

h3 K3+ ch2

i+hi

2 с

2

G3 с

hi K3" ^2 hi G3

ai5 - —~—1--;—; ai6 ----; ai7 - —

i2 6 i2 2

v

f

i + ü 2 с

KL

2

a G3 ai8 =

(

i + hL

2 с

^ K3 G3 ^

——; ai9 - — 2 2

i + h.

2 с

Л

KL 2

d2 f „ ÖW2 F = —г I rni ui + msu2 + 2 m5--m7-

dt v dx dx

d2 ( dwi „ ÔW2

F = —2 I ms ui + m2U2 + m5—--2 ——

dt v ox ox

F =

F4 =

ô ôt2

ôt2

0Щ

ÔU2

-2 m5 —- - m5 ■

ôx ôx

mi wi + ms w2 - m3 ■

m7

5ui ôx

ô2 wi

ÔX2

ôu2 ô2wi 2m7-+ ms wi + m2 w2 + m6 ——

5x

-m6-

m4

a 2w2 ^

ôx2

Ô2W; ^

ôx2

mi =pi hi h P3 с; m2 = P2 h2 + -P3 с; m3 =

3 3 i2

m Pih ,p3^ m Pзchl m4 =-—Л--— ; ms =

pi hi3 p3 с hi2

i2

6

P3 chl h2 P3 ch2 P3 с m6 =-; m7 =-; ms =

12 6 The boundary conditions of equations (1) are

duk d2Wk

3

Wk =

dx dx2

= 0 at x = ± l, k = i, 2.

(3)

In narrow channels and slots for modeling the dynamics of a viscous fluid, creeping motion [23] can be considered and the equations of motion of the fluid can be written as

1 dp _ f d2ux d2ux ^ ; 1 dp _ f d2uz d2uz ^ 8ux duz _ 0 (.)

_ - V ^ ^ ^ ) _ - V ^ ^ ^ ; " ^ " - 0. (4)

p dx

dx

dz2

p dz

У dx2 dz2 у

ôx

ôz

Equations (4) are supplemented by the boundary conditions of fluid adhesion on an absolutely solid wall and the upper bearing layer of the plate

ux = 0, uz = 0 at z = c + h1 + ho,

du\ dw\ (5)

ux =-, uz =-at z = c + hi + wi,

St St

and boundary conditions of coincidence of pressure in a fluid with a given pressure in the end sections of the channel

p = p0 + p* at x = ± l. (6)

Determination of the dynamic response of the three-layer channel wall.

We introduce the following dimensionless variables:

x z - c - h\ l

T = ® t, —, C =-;-, ux = wim Q — U^,

l ho ho

„ * pVWimffl l2

uz = Wim ®UC, p = p + F 1m — P, (7)

ho ho

and given the equations (5) and boundary conditions (6), (7), we obtain

ap_(hoY g2 u^ | d2u^; l) + ac2 ;

DP (ho^2

dQ V l

ho}? d2UC d2UC

I J d^2 + a;2

dU4 6UQ _ ; + ;

(s)

TT n TT n + f i TT ho uim dUi

= UQ= o at C =i; u^=~---—;

l wim dx

= W at ;= W^ W (9)

OT ho OX

P = o at $ = ± i.

Here, it was assumed, where ui = uim Ui (£, x), wi = wim Wi (£, x), where

uim, Wim are the amplitudes of the longitudinal displacement and deflection of the upper bearing layer.

In the considered formulation uim/wim ~ i, and parameters ho/1 ^ i, wmi/ho ^ i, i.e., with the given parameters in (8) and (9), can be omitted, as a result we have

ÔP _ d2U4 ; ЙР _ ; dU^ dU£ _ ;

a£, aç2 aç дdC

aw,

(10)

U^= o, Uq= o at C = i; U^ = o, Uc = —i at C = o; (ii)

P = o at £ = + i. (i2)

Taking into account the comments made earlier, the stresses qzz and qzx in the variables (8) are written as

qzx =- (pv Wim wl/h;°) dU4 o;

qzz = -p* - (pv Wim ral2/h%) P. Solving problem (ii)-(i3), we find that

P = i2 JJ^W—1 + 6 i) }jf — di di;

\o -io

(13)

ÔU ?

aç

Ç=0 0 ^r

h, a fau^

-10

ax

(14)

hi aw, = — 6-

C=0

i ax

l an ^

The solution of equations (i) is represented as

<x>

uk = Z Tk" (a t) cos ((2n + i)^/2);

n = 0

œ

(15)

wk = X cos ((2n +2), k = 1, 2.

n = 0

Substituting (i5) into (i4) and decomposing the pressure p* in a series of cos ((2m + 2), we obtain

qzz = Z

pvl2

n = 0 h0

4(-1)

n+1

p

(2n +1) л pvl2

h3

h0 - 12

2 2 > dRn

(2n +1) п dt У

2n + 1 x

cos - л — ;

2 l

v Pyl<

qzx - - ^ -y 6

n = 0 h0

(2n +1) я

dRn 2n +1 x

—— sin-ж—

dt 2 l

(16)

1 , aqzx 1 h0 h1 » pvl2 d^n — h1^— =---^ „ 6 —— cos

8x 2 l l n " 0 h0

dt

2n +1 x

- л —

2 l

h1 dqz

qzz ^ 1, i.e., the term

From expressions (16) it follows that--,qzz

2 dx I

h d

— ^zx in (1) can be neglected in compression with qzz. Then, substituting (15),

2 dx

(16) in (1), and equating the terms in the resulting system with the same trigonometric functions, we proceed to the system of ordinary differential equations in time, which includes two homogeneous algebraic equations. Using

them, we will find a connection T2n, through T1n, R":

Til = (Tn (b24b41 - b44b21 ) + Щ (Ы>43 " ^44^23 ))/(b22&44 " ^24^42 ) ; Щ. = (Tf (b42b21 - b22b41 ) + R1 (b42b23 ~ &22&43 ))/(Ь^2&44 " ^42 ) .

(17)

Further, given that for harmonic steady-state oscillations d2Rn /dt2 = - ra2Rf, we finally get

dRn

b*iTin + bi3R" + 2K1 —1 = 0; 1 1 dt

* * dRn 4(-1)n+1 *

b3iTn + b33Rf + 2 Kn —L = — p . 31 1 33 1 n dt (2n +1)rc F

The following notation is introduced here:

A = b22b44 - b24b42; b*1 = b11 + b12 (b24b41 - b44b21 )/A + bu (b42b21 - b22b41 )/A ;

b*3 = b12 (b24b43 - b44b23 )/A + ¿13 + 614 ^b23 " b22b43 )/A ;

b31 = b31 + b32 (b24b41 - b44b21 )/A + b34 (b42b21 - b22b41 )/A;

b33 = b32 (b24b43 - b44b23 )/A + b33 + b34 (b42b23 " b22b43 )/A ;

-|2

(18)

2Kn = 12

pv/2 h0

(2n + 1)л

2К1 = 6

pv/

"h2" ho

bn = a1 + 04

( 2n + 1 >

b13 =

V

2n +1 2/

2/

я

-я

- т1ш2; b12 = -a1 + a5

(2n + 1) л ( 2n + 1 У

2/

■я

ш8ш2;

—a2 _ 2a6

2n + 1 >

2/

я

+ 2ш5Ш2

2n + 1 >

2/

я

2n + 1

b14 =-Ж

14 2l

-Ü3 + 07

( 2n + 1 ^

ж

2l

т7

J

b23 =

b21 = -01 + 0-5

b22 = 01 + 09 2n + 1

( 2n + 1 ^

2l

■ж

т8ш2;

J

( 2n + 1 ^

ж

2l

2

У

b24 =

b31 =

b32 =

-Ж

2l

2n + 1 2n + 1 2n +1

( 2n + 1 >

03 - 06

2l

-ж

m2œ

+ т5ш2

я

ж

02 + 207

Г 2n + 1 ^

2l

■ж

017 _ 2 06

( 2n + 1 ^

2l

■ж

b33 = 08 - 011

2l

Г 2n + 1 У

я

Г 2n + 1 ^

010 _ 06

21

-ж

- 2т7ш2

+ 2т5 ш2

+ т5 ш2

2l

-я

+ 015

2n + 1

4

2l

-ж

т1 + т3

2n + 1 Y

2l

я

2

b34 = -08 + 012

2n + 1 ^

2l

ж

- 016

2n + 1 ^

2l

я

т8 -т6

2n + 1 Y

2l

я

ш

ш2;

2n + 1

b41 =-ж

41 2l

2n + 1

b42 =-:— n

-018 + 07

( 2n + 1 ^

~1T

b43 = -08 + 012

b44 — 08 — 014

2l

2n + 1 ^

Ж

019 + 2 07

2n + 1 ^

2l

я

т7 ш

- 2т7ш2

2l

Г 2n + 1 ^

016

2n + 1 >

2l

ж

2n + 1 Y

2l

ж

+ 013

2n + 1 ^

2l

я

т8 -тэ

т2 + т4

2l

ж

2n + 1 Y

2l

я

ш2;

ш2.

From (18) we have

dRn dt

2Kn 2Kn

Ы

11

+ Rn

b33 _ b31 b*1

4(-1)

n+1

1

(2n + 1) ж b

p ;

31

Tn _ _ b*3 Rn _ 2Kn dR'n

T1 ~ 7* R1 7* A+

b11 b11 at

(19)

Given the linearity (19) and representing Rf = Rf0 + Rf, Tf = Tn0 + T1n, where the superscript 0 corresponds to a static solution, we have

4(-1)n +1 1 (2n + 1)tc d1

Ri10 = p0

ю = 0

Tn0 _ pnü b13 T1 "" R1 -~ bn

4(-1)n Г b*3 1 ^

m= о P0 (2n + 1) b*1 d1

(20)

ю = 0

and for harmonic oscillations we find that

Rn =

4(-1)

n+1

Tn =

1 (2n + 1) ^d2 + d2Q2 4(-1)n Vb*32 + (2K1rn)2

e^ рт eifflt ;

(21)

,¿9 p* piwt

ti — i e e pm e

(2n + i) ^d2bi*i2 + d°ffl2bii2

where di = b33 - b3i b*3/bn; d2 = 2Kn - b3i2.KM/bn; tgy = - d2ra/di;

tg 0 = 2k1 ©/b*3 .

Taking into account (2o) and (2i) in (i5), we obtain expressions for the deflection and longitudinal displacement of the first bearing layer of the plate during its oscillations:

W1 = p0 Z

4(_1)n+i 1

n = 0 (2n + 1) ж di

ю = 0

2n + 1

cos-ш +

2l

Щ = p0 S

+ ртПW1(q, x)sin(ot + 9w1); 4(-1)n ( b*3 1 1

= 0 (2n + 1) ж I bn d1

2n + 1 sin-жx +

(22)

ю = 0

2l

+ ртП„1(ш, x)sin(ot + ф„1).

Here

nw1 (ш, x) = ^Ep + Fp ; ПМ1 (ш, x) = <JAp + Bp ;

n

Ep = S

4(-1)

n +1

Fp = T

4(-1)

n+1

&d

=о (2n + 1) л df + dfra2

Ap

4(-1)n

n = о (2n + 1) я

d1

b13

5p = S

4(-1)n

(2n + 1) л

d2 + dfю2 bn d1

1 2n + 1 — cos-ш; djш2 21

d2 2n + 1 2 „ cos-го; dl®2 2l

2Kn& d2 ш sin 2n + 1

bn d2 + d^2 2l

b*3 d2 ю sin 2n + 1

b*1 df + d2Q2 2l

•тсх;

roc;

d2 +

9wi = -arctg(Fp/Ep), 9„i = arctg(Bp/Ap).

The functions introduced into consideration nw1(w, x) and nu1(w, x) are the frequency-dependent distribution functions of the amplitudes of the deflection and the longitudinal movement of the first bearing layer along the channel. Note that using (17), we can write similar functions for deflection and longitudinal movement of the second bearing layer of plate.

Conclusion. A mathematical model is proposed for modeling bending and longitudinal vibrations of the channel wall as a three-layered plate possessing compressible core and interacting with a pulsating layer of a viscous incompressible fluid. The expressions for the elastic displacements of the layers of the plate, which completely determine its intensely deformed state, are determined. Frequency-dependent deflection amplitude distribution functions are constructed nw1 (q, x) and longitudinal movement nu1 (q, x) the first bearing layer, allowing to analyze the dynamic response of the channel wall. In particular, these functions with a fixed value of the longitudinal coordinate represent the amplitude-frequency characteristics of the corresponding cross-section of the channel wall. It can be noted that the proposed functions allow us to investigate hydroelastic vibrations of the channel wall and determine the distribution of fluid pressure in the channel. For example, they can be used to determine its resonant vibration frequencies and the corresponding amplitudes of longitudinal displacements, deflections of the bearing layers of the plate and pressure in the liquid. In addition, the developed model can be used for the further development of methods for non-destructive monitoring of the state of three-layered walls of channels filled with a pulsating viscous fluid. In particular, if, with the well-known harmonic law of pressure pulsation at the ends of the channel, in some fixed section, the amplitude-frequency characteristics of the

n

m

deflection and longitudinal displacement of its second bearing layer are experimentally determined, then using relation (i7), we can recalculate the characteristics the first bearing layer of the channel. Comparing the result obtained with the previously known reference result, one can judge the state of the channel wall. Thus, the developed mathematical model can be used both for studying the dynamic response of three-layered channel walls, in various devices and assemblies and for developing non-destructive testing technologies for three-layered structural elements in contact with fluid layers in lubrication and damping systems or cooling according to the parameters of forced oscillations of their bearing layers.

Translated by K. Zykova

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Grushenkova E.D. — Post-Graduate Student, Department of Applied Mathematics and System Analysis, Yuri Gagarin State Technical University of Saratov (Politechnicheskaya ul. 77, Saratov, 410054 Russian Federation).

Mogilevich L.I. — Dr. Sc. (Eng.), Professor, Department of Applied Mathematics and System Analysis, Yuri Gagarin State Technical University of Saratov (Politechnicheskaya ul. 77, Saratov, 410054 Russian Federation).

Popov V.S. — Dr. Sc. (Eng.), Professor, Department of Applied Mathematics and System Analysis, Yuri Gagarin State Technical University of Saratov (Politechnicheskaya ul. 77, Saratov, 410054 Russian Federation).

Khristoforova A.V. — Cand. Sc. (Phys.-Math.), Assoc. Professor, Department of Mathematics, Balashov Institute of Saratov State University (Karla Marksa ul. 29, Balashov, Saratov Region, 412300 Russian Federation).

Please cite this article as:

Grushenkova E.D., Mogilevich L.I., Popov V.S., et al. Mathematical model of oscillations of a three-layered channel wall possessing a compressible core and interacting with a pulsating viscous liquid layer. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2019, no. 6, pp. 4-18. DOI: 10.18698/0236-3933-2019-6-4-18