12394


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 22, No 3, Special Issue, 2024, pp. 473 - 484  

https://doi.org/10.22190/FUME231219010K 

© 2024 by University of Niš, Serba | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

LONGITUDINAL-RADIAL VIBRATIONS OF  

A VISCOELASTIC CYLINDRICAL THREE-LAYER STRUCTURE 

Khayrulla Khudoynazarov 

Samarkand State University, Uzbekistan 

ORCID iD: Khayrulla Khudoynazarov   https://orcid.org/0000-0001-8994-9738  

Abstract. The paper considers a cylindrical three-layer structure of arbitrary thickness 

made of viscoelastic material. It consists of two external bearing layers and a middle 

layer, the materials of which are generally different. The problem of nonstationary 

longitudinal-radial vibrations of such a structure is formulated. Based on the exact 

solutions in transformations of the three-dimensional problem of the linear theory of 

viscoelasticity for a circular cylindrical three-layer body, a mathematical model of its 

nonstationary longitudinal-radial vibrations is developed. Equations are derived that 

allow, based on the results of solving the vibration equations, to determine the stress-

strain state of a cylindrical structure and its layers in arbitrary sections. The results 

obtained allow for special cases of transition into cylindrical viscoelastic and elastic two-

layer structures, as well as into homogeneous single-layer cylindrical structures and 

round rods. 

Key words: Three-layer structure, Vibration, Stress, Torsional displacement, Load-

bearing layers, Non-stationary 

1. INTRODUCTION 

Three-layer structural elements are widely used in aviation and shipbuilding, construction 

of buildings and structures, the space industry and other industries [1,2]. Therefore, the 

problem of developing effective methods for calculating the stress-strain state of three-layer 

structural elements, as well as generalizing classical theories using refined models reflecting 

the dynamic behavior of modern materials, is urgent [3,4]. In this regard, cylindrical structures 

and round rods are one of the main elements of various engineering structures [5] and studies 

of their dynamic behavior have important applied values [6,7]. Such elements are often under 

the influence of dynamic loads during operation, which lead to their vibrations [8,9]. 

Recently, special attention has been given to study vibrations of layered structures made 

of homogeneous and functionally graded materials (FGM) [10]. These include works 

where vibrations of three-layer plates are considered, considering imperfect [11], slipping 

 
Received: December 19, 2023 / Accepted March 03, 2024 

Corresponding author: Khayrulla Khudoynazarov 
Samarkand State University, 100104, 15 University Boulevard, Samarkand, Uzbekistan 

E-mail: kh.khudoyn@gmail.com  

https://orcid.org/0000-0001-8994-9738
https://e.mail.ru/compose/?mailto=mailto%3akh.khudoyn@gmail.com


474 KH. KHUDOYNAZAROV 

[12] contacts between layers, cylindrical [13] and conical [14] FGM shells. In studies 

devoted to the dynamic behavior of elements of engineering structures, it is important to 

develop mathematical foundations for such structures [15,16] for design of new 

generations of improved lightweight structural materials [17]. 

One of the main problems in the study of the static and dynamic behavior of shells and 

rods is the choice of vibration equations, which should be implemented based on the 

specific physical and mechanical properties of their materials [18]. Various methods of 

derivation of vibration equations are used. One of these methods is the method of using 

general solutions in transformations of three-dimensional problems of elasticity theory 

[19,20]. The essence of the method is to study the constructed solutions for various types 

of external influences [21] and to clarify the conditions under which the displacements or 

their “main parts” satisfy simple vibration equations, and to find an algorithm that allows 

calculating approximate values of displacement and stress fields in any cross section for an 

arbitrary a moment in time. Similar studies, but considering more complex physical and 

mechanical properties, in particular viscoelastic ones, are considered in [22,23]. 

The analysis of the behavior of elements of engineering structures, taking into account 

the layering and heterogeneity of structures based on computational models, is relevant for 

applied problems, as evidenced by publications [24,25]. In addition, a fairly large number 

of studies of shell dynamics are carried out, which take into account the influence of 

hyperelastic [26], anisotropic, temperature and other physical and mechanical properties of 

the material. 

Thus, it can be argued that at present there is a very limited number of works devoted 

to the practically important task of studying non-stationary longitudinal-radial vibrations 

of cylindrical three-layer structures of arbitrary thickness. Therefore, the problem of 

creating models for the dynamic calculation of such systems under the influence of 

dynamic loads, taking into account various physical and mechanical properties of their 

material, is urgent. In this article, a circular cylindrical three-layer viscoelastic structure of 

arbitrary thickness with sticking condition between the layers is considered. The task is to 

study its nonstationary longitudinal-radial vibrations based on the above-mentioned 

method of exact solutions in transformations. It is envisaged to build a mathematical model 

of it, including the derivation of general and refined vibration equations and the creation 

of an algorithm that allows determining the stress-strain state of an arbitrary section of the 

structure in coordinate and time using the field of desired functions.  

2. MATHEMATICAL MODEL OF THE PROBLEM 

2.1. Formulation of the Problem 

In the cylindrical coordinate system (r, θ, z), a three-layer circular cylindrical structure 

made of viscoelastic material is considered. It is assumed, that the structure consists of two 

layers, hereinafter called load-bearing layers, which are separated by a certain distance 

using the third layer. The intermediate layer holds the load-bearing layers at a distance. 

The axis Oz of the coordinate system is directed along the axis of symmetry of the structure 

perpendicular to the cross section and we schematic picture of layers is shown in Fig.1. 

Through a and b we denote the inner and outer radii of the cylindrical structure, and through 

r1 and r2 the inner and outer radii of the middle layer. When deriving the vibration 

equations, we assume that both the cylindrical structure as a whole and its layers separately 



 Longitudinal-Radial Vibrations of a Viscoelastic Cylindrical Three-Layer Structure 475 

strictly obey the mathematical theory of viscoelasticity and are described in an accurate 

formulation by its three-dimensional equations in a linear formulation. 

 

Fig. 1 Cross section of a three-layer structure 

With longitudinal radial vibrations of the cylindrical structure, only the components of 

displacements wm, um, and stresses 
( )m

rr ,
( )m

 ,
( )m

zz ,
( )m

zr  (m=0,1,2), will be different from 

zero [20]. Accordingly, the equations of motion of points of a viscoelastic structure are 

taken in the form of wave equations with respect to the potentials of longitudinal φm and 

transverse χm waves in the layers of the structure: 

 
( )

( ) ( )

2

0 2

2

0 12

, 2 ,

, 0,1,2 , ,

m m

m

m
m m m m

m
m m

R R R R
t

R m a r r
t

 
 = = +

 



 = =  

 

 




 


 

 (1) 

where 
2 2

0 2 2

1

r rz r

  
 = + +

 
, while 

m
R

,
m

R
 are the integral operators defined by the formula 

( ) ( , ),

0

( ) ( , ) ( ) ( ) ( )
mm

t

m mR t K t d
 

= − − 
 

   
        , 

λm, μm are the (Lame)coefficients  of layer materials, ( , ) ( )
m

K t −    is the kernel of the 

integral operators. It is assumed that the viscoelastic operators ( , ) ( )
m

R     are reversible, 

and their kernels are arbitrary. Here and everywhere else, the index m takes values 0, 1, 2. 

Therefore, in the following, this will not be emphasized every time, implying that this is 

always the case. 

It is assumed that the cylindrical structure is at rest prior to loading, and at the moment 

t=0, external surface’s stresses 
( ) ( , )i

rF z t , ( ) ( , )i

rzF z t (i=1,2) are applied, i.e. it is assumed 

that the boundary conditions have the form: 

 

(1) (1) (1) (1)

(2) (2) (2) (2)

( , , ) ( , ), ( , , ) ( , ) ,

( , , ) ( , ), ( , , ) ( , ) .

rr r rz rz

rr r rz rz

a z t F z t a z t F z t at r a

b z t F z t b z t F z t at r b

 = = =


= = =

 

 
 (2) 



476 KH. KHUDOYNAZAROV 

In addition, the sticking conditions must be met on the contact surfaces between the 

layers of the structure, which require equal displacements and stresses, i.e. contact 

conditions have the following form at r=ri, i=1,2: 

 
0 0

(0) ( ) (0) ( )

( , , ) ( , , ), ( , , ) ( , , ),

( , , ) ( , , ), ( , , ) ( , , ), ( 1,2).

i i i i i i

i i

rr i rr i rz i rz i

w r z t w r z t u r z t u r z t

r z t r z t r z t r z t i

= =


= = =   
 (3) 

The initial conditions of the problem are considered zero, i.e. at t=0 

 0, 0, 0, 0m m
m m

t t

 
= = = =

 

 
  . (4) 

2.2.  Derivation of Vibration Equations 

To solve the formulated problem of torsional vibrations of a three-layer cylindrical 

viscoelastic shell, the functions ( ) ( , )i

rF z t , ( ) ( , )i

rzF z t (i=1,2) of external influences under 

boundary conditions (3) are considered in the class of functions represented as [21]: 

 ( ) ( ) ( ) ( )

0 ( ) 0 ( )

sin cos
( , ) , ( , )

cos sin

i i pt i i pt

r r rz rz

l l

kz kz
F dk f k p e dp F dk f k p e dp

kz kz

  
= = 

−  
    . (5) 

Here (l) is an open contour in the plane p adjacent to the right section (–iω0, iω0) of the 

imaginary axis. Furtgermore, the functions 
( ) ( , )i

rF z t , ( ) ( , )i

rzF z t  are assumed to be such 

that the functions ( ) ( , )i

rf z t , ( ) ( , )i

rzf z t  are negligibly small outside the domain {0<k<k0, 

Im|p|<ω0} 

Let’s assume the potential functions φm and χm also in the form of Eq. (5). Substituting 

them into the equations of motion (Eq. (1)), we obtain ordinary Bessel differential 

equations for the transformed potential functions m , m (m=0,1,2):  

 
2 2

2 2

2 2

1 1
0, 0m m m

d d d d

r dr r drdr dr
   

   
+ − = + − =   

   

 (6) 

where

0

2 2 1 2 2 2 1 2

, ,

, ,

( ) ( ) , ( 2 ) 1 ( ) 1 ( )

( ) 2 ( )
( )

2

m m m m

m m

m

pt

m m m m m

m m

m m m m m m

m m

p R k

K p K t e dt R K p R K p

K t K t
p R k K t


−

− −

   
   

= +

= = + − = −

+
= + =

+

   

 



  

 
  

 

. (7) 

The general solution of Eq. (6) is 

 (1) (2) (1) (2)

0 0 0 0( ) ( ) ( ), ( ) ( ) ( ),m m m m m m m m m mr A I r A K r r C I r C K r= + = +       (8) 

where ( ) ( , )i

mA k p , ( ) ( , )i

mC k p , (m=0,1,2) (i=1,2) are arbitrary, regarding variable r, integration 

constants. The number of these constants is generally 12. It can be reduced based on the 

following considerations. The general solutions given by Eq. (8) for all three layers have 

the same structure. They should be finite in the limits r → 0 and r → ∞. At the same time, 

the boundaries of the inner layer are smooth a and r1, that is a ≤ r ≤ r1. It is bounded from 



 Longitudinal-Radial Vibrations of a Viscoelastic Cylindrical Three-Layer Structure 477 

below (from the inside) by the surface r = a, which may tend to zero in the limit, but exceed 

in no way the values of r1, i.e. it cannot strive for infinity. Therefore, in general solutions 

for the potential functions of the inner layer 1( )r , 1( )r , we can limit ourselves to taking 

into account their layer in the form of: 

 1 1 0 1 1 1 0 1( ) ( ), ( ) ( ).r A I r r D I r= =     (9) 

Similarly, the boundaries of the second, outer layer are cylindrical surfaces with radii 

r = r2 and r = b that is r2 ≤ r ≤b. This layer is bounded externally by the surface of radius 

r = b, which may tend to infinity, i.e. b → ∞. On the other hand, the inner surface of this 

layer cannot be tightened to a straight line, because this would lead to a homogeneous rod 

of circular cross-section of radius r = b. Therefore, in general solutions for the potential 

functions of the outer layer 2 ( )r , 2 ( )r , we can limit ourselves to taking into account its 

limitations only at r → ∞. Based on this, we will take the general solution, Eq. (8), for the 

outer layer as: 

 2 2 0 2 2 2 0 2 2( )( ) ( ), ( ) ( ), r r br A K r r D K r  = =    . (10) 

For the middle layer, we will take general solutions. Eq. (8), considering that these 

solutions, in the absence of the two outer layers, should transform into known solutions for 

a homogeneous cylindrical layer, limited at r → 0 and r → ∞ at the same time: 

 0 1 0 0 2 0 0 0 1 0 0 2 0 0 1 2( ) ( ) ( ), ( ) ( ) ( ), r r rr B I r B K r r C I r C K r  = + = +       (11) 

Thus, the number of integration constants to be determined from the contact conditions 

is reduced to eight. 

Let us assume the stress components, both ( ) ( , , )m

rr r z t  and ( ) ( , , )m

rz r z t  as 

 
( ) ( )

( ) ( ) ( ) ( )

0 0

sin cos
,

cos sin

m m pt m m pt

rr rr rz rz

l l

kz kz
dk e dp dk e dp

kz kz

  
= = 

−  
       . (12) 

Substituting Eqs. (12) and (5) in the boundary conditions, Eq. (2), and expressing them 

in terms of solutions given by Eqs. (9) and (10), we obtain: 

1

1

2 2 2 1 (1)1 1
1 0 1 1 1 1 1 0 1 1 1, 1

2 2 1 (1)

1 1 1 1 1 1 1 1 1

2 2
( ) ( ) ( ) 2 ( ) ( ) ( , ) ,

2 ( ) ( ) ( ) ( , ) ,

r

rz

k
k I a I a A k I a I a D R f k p

a a

k I a A k I a D R f k p

−

−

   
 + − − − =     

   

 − + =  





 
     

    

 (13) 

2

2

2 2 2 1 (2)2 2
2 0 2 1 2 2 2 0 2 1 2 2

2 2 1 (2)

2 1 2 2 2 2 1 2 2

2 2
( ) ( ) ( ) 2 ( ) ( ) ( , ) ,

2 ( ) ( ) ( ) ( , ) ,

r

rz

k
k K b K b A k K b K b D R f k p

b b

k K b A k K b D R f k p

−

−

   
 + + − + =     

   

 − + = −  





 
     

    

 (14) 

Let us express the contact conditions also through general solutions. In this case, the 

number of unknown constants will be only 8. Therefore, due to the fact that we use four 

boundary conditions, Eq. (2), it is sufficient to take four of the eight contact conditions 

given by Eq. (3). In addition, the boundary conditions, Eq. (2), are dynamic, therefore, for 

contact conditions we will use only the kinematic part of the conditions given by Eq. (3). 

Let us also assume the displacements in form of Eq. (5) and apply together with Eq. (12) 



478 KH. KHUDOYNAZAROV 

to the contact conditions, Eq. (3), in the displacements. Expressing the obtained contact 

conditions through general solutions, Eqs. (9-11), we find the constants Ai, Di (i=1,2). 

Substituting the found expressions of the integration constants into Eqs. (13) and (14) gives 

the following system of four equations: 

 

2
1 2 2 ( )

1 0 2 0 2 2

2
1 2 2 ( )

3 0 4 0 1 3 3

2
( , ) ( ) ( , ) ( ) ( ) ( , ) ,

( )

42
( , ) ( ) ( , ) ( ) ( ) ( , ) ,

( )

i

i

ii
i i i i i i i i i r

i

ii
i i i i i i i i rz

i

r
G u r G w r R k f k p

r a b

r
G u r G w r R k f k p

r a b

−

−

 + = + 

 + = + 





    

    

 (15) 

where 
2 2 2 2 2 2

1 2

2 2 2 2 2 2

3 4 1 1

1 1
( , ) , ( , ) (2 )

4 4

( , ) ( 3 ), ( , ) (2 )

i i i i i i i i i i

i i i i i i i i

G k k G k

G k G k k

    
= + + =  −    

   
 = + = − −

       

       

 

 

2

2 2

2

, 1,
( )

, 2.

a if i
a b

b if i

 =
= 

=

 

Let us highlight the main parts of the longitudinal radial displacement of the middle 

layer of the shell. For this, the displacements of the points of the median layer ũ0(r,k,p) and

0w (r,k,p) transformed by Eq. (5) are expressed in terms of general solutions, Eq. (11). We 

will have 

 
   

   

2

0 1 0 0 2 0 0 0 1 0 0 2 0 0

0 0 1 1 0 2 1 0 0 1 1 0 2 1 0

( , , ) ( ) ( ) ( ) ( ) ,

( , , ) ( ) ( ) ( ) ( ) .

u r k p k B I r B K r C I r C K r

w r k p B I r B K r k C I r C K r

= + − +

= − − −

    

     
 (16) 

We introduce the radius of some intermediate surface of the middle layer of the 

structure according to the following formula [27]: 

 1 1

22

r r

r

 
= − 

 
  .  

Writing Eqs. (16) through the standard decompositions of the Bessel functions Ii and Ki 

(i=0,1) into series according to the degrees of the radial coordinate r and, limiting them to 

the first terms at r=ξ, we obtain the indicated main parts, which we denote by 0,0 0,1,u u  and 

0,0 0,1,w w : 

 

2 2

0,0 10 0 10 0,1 2 0 2

2 2

0,0 0 10 0 10 0,1 2 2

, ( ) /

, ( ) /

u kB C u kB C

w B k C w B kC

= − = −

= − = −

  

  
 (17) 

where: 

 0 0
10 1 2 10 1 2

1 1
ln (1) , ln (1)

2 2 2 2
B B B C C C

   
= − − − = − − −   

   

   
  , 

and ψ(n), n=1,2, … is the logarithmic derivative of the Gamma function. 

Writing Eqs.(16) for the transformed displacements ũ0(r,k,p) and 0w (r,k,p) through the 

standard decompositions of the Bessel functions Ii and Ki (i=0,1) into series according to 



 Longitudinal-Radial Vibrations of a Viscoelastic Cylindrical Three-Layer Structure 479 

the degrees of the radial coordinate r and substituting in them the values of the constants 

B10, C10, B2 and C2, found from the system of Eqs. (17), we obtain 

 





2 (0) (0) 2

0 0 0,0 1 0,0 0 0,0
0

2
2 2 (0) (0)

0 0,1 0 2 0,1 0,1 3, 2

( , , ) ( )

( 2)
( ) ( ) ,

( !)

n

n
n

n
n

n n

u r k p u q Q kw u

r
u q Q kw u r

n



=

= + − −

 − + − 

  

   

 (18) 

 





2 2 (0) (0)

0 0,1 0 0,0 0 1 0,0 0,0
0

2 1
2 2 (0) (0) 2

0 0,1 2 1 0 0,1 0,1 1,

( , , ) ( )

( 2)
( ) ( )

!( 1)!

n

n
n

n
n

n n

w r k p w w q Q w ku
r

r
w q Q w ku r

n n



=

+
+

+

= − +  + − −

 − + −  +




 

   

 (19) 

where: 

 
0 0

(0) 1 (0) 1

1 0 2 0 3,
1

1 1
1 ; 1; ln ;

2

n

n
k

r
q R R q R R

k

− −

=

= −  =  − = + −  


  

 
( )

2 2 1
2 1(0) 2 (0) (0) (0) 2 20 0
0 0 0 1 2 0 02 2

10 0

; 0; 1; .
n n n

n k k

n
k

Q Q Q Q
−

− −

=

−
= = = = = +

−


 
   

 
  

Let us introduce the following representations for functions u0,i(r,z,t) and w0,i(r,z,t): 

 
0, 0, 0, 0,

0 ( ) 0 ( )

cos sin
,

sin cos

pt pt

i i i i

l l

kz kz
u dk u e dp w dk w e dp

kz kz

  
= = 

− 
     (20) 

and operators ( ), ( )n n

m m     , (m=0,1,2; n=0,1,2,…)  

 2 2

0 ( )

sin
( ), ( ) ( , ) ( , , ) ,

cos

n n n n pt

m m m m

l

kz
dk r k p e dp

kz

 
 =   − 

         (21) 

Similarly to Eqs. (18) and (19), we express Eq. (15) in terms of the main parts of the 

transformed longitudinal 0,0 0,1,u u  and radial 0,0 0,1,w w  displacements. Limiting ourselves 

to the first approximations of (n=0) in infinite series in the obtained equations and applying 

operators given in Eqs. (20) and (21) to them, we obtain: 

 

1

0,0 0,1(0)

1 2 0,0 1 3,0 2 2

2 2
(0) 1 ( )

2 2 0 0,12 2 2 2

( ) ln

2
(1 ) ln ( , )

( )

i
i i i i i

ii i
i i r

i

u ur
G G w G r G q

z z

r r
G q w R F z t

r a b z

−

  
− + + − − 

  

    
− + + = −   

     


 


  


 (22) 

 

2

2
0,0 (0)

3 0,0 4 3 3,0 4 2 0,12

2 2
0,1(0) 1 ( )

4 2 0 2 3 3 2

( ) ln

42
(1 ) ln ( , ) ,

( )

i
i i i i i

ii i
i i rz

i

w r
G u G G r G q u

z z

wr r
G q R F z t

zr a b z

−

  
− − + + 

  

    
+ + + = −   

      


 


  


 (23) 



480 KH. KHUDOYNAZAROV 

Based on Eqs. (7) for the values αm and βm (m=0,1,2), it is easy to show that the 

operators n

m  and n

m , m=0,1,2; n=0,1,2,… introduced by Eqs. (21) during the reverse 

transition along coordinate z and time t have the following form: 

 

2 2 2 2
1 1

2 2 2 2
( ) , ( )

0,1,2; 1,2,3,...

m

n n

n n

m m m m mR R
t z t z

m n

− −
         

= − = −      
            

= =



   
     

 (24) 

The system of Eqs. (22) and (23), in accordance with Eqs. (21) for operators n

m , 

m=0,1,2; n=0,1,2,… are integro-differential equations. These equations contain the main 

parts u0,0, u0,1 and w0,0, w0,1, respectively, longitudinal – u0 and radial – w0 displacements 

of points of some “intermediate” surface of the middle layer of a three-layer cylindrical 

structure. The specified “intermediate” surface has a radius, the values of which are 

enclosed in the range r1 ≤ ξ ≤ r2 In accordance with the numerical values of radius ξ, this 

“intermediate” surface can turn into a median surface at ξ = (r1+r2)/2 and contact surfaces 

between layers of the structure at ξ = r1 and ξ = r2. Therefore, the equations are a system 

of Eqs. (22) and (23) depending on the values of the radius ξ. The equations of vibration 

of a three-layer cylindrical structure relative to the main parts of the longitudinal and radial 

displacements of the points of the median or contact surfaces of the median layer can be. 

These equations, in the absence of external layers, are general equations of longitudinal-

radial vibrations [19] of a circular cylindrical viscoelastic structure, relative to the main 

parts of the longitudinal and radial displacements of points on the intermediate surface of 

the structure. 

In addition, Eqs. (22) and (23) in their right parts correctly considering the forces acting 

on the outer and inner surfaces of the three-layer structure, reflect (approximately) the 

relationship and mutual influence of the layers-through the middle layer. It is not difficult 

to see in these equations the dependence on the viscoelastic operators of 
m

R (m=0,1,2) 

layers. 

If specific kernels Km and 
m

K are given for viscoelastic layer material operators, then it 

is not difficult to transform these equations for specific shell layer materials. Note that the 

operators given by Eqs. (22) and (23) are derived for the general case of arbitrary operator 

kernels 
m

R (m=0,1,2). 

Consider the following special case: suppose that the Poisson coefficients of the materials 

of all three layers of the structure are constant in time. Then the kernels of the viscoelastic 

operators Rm and
m

R will be the same, and taking this fact into account, the operators n

m  and 

,n

m take the form: 

 

2 2 2 2
1 1

2 2 2 2 2 2

1 1
( ) , ( )

0,1,2; 1,2,3,...

n n

n n

m m m m

m m

M M
a t z b t z

m n

− −
         

= − = −      
            

= =

   
     (25) 

Here the operator Mm is defined by the formula: 

 
0

( ) ( ) ( ) ( ) ;
t

m mM t K t d= − −       (26) 



 Longitudinal-Radial Vibrations of a Viscoelastic Cylindrical Three-Layer Structure 481 

am and bm (m=0,1,2)  are propagation velocities of longitudinal and transverse elastic waves 

in layer materials: 

 ( 2 ) ; .m m m m m m mа b= + =      (27) 

In this case, the operators (0)

1q and (0)

2q  are computed in then following way: 

 ( ) ( ) ( )(0) (0)

1 0 0 0 2 0 0 0 0, 2q q= + =− + +        (28) 

Taking into account Eqs. (25-28), the equations of longitudinal-radial vibrations of a 

three-layer cylindrical viscoelastic structure are significantly simplified. 

2.3.  Determination of Displacements and Stresses 

To determine the stress-strain state of the structure, it is necessary to determine the 

displacements and stresses. This procedure must be performed for all three layers. To do 

this, it will be necessary to express all displacements and stresses through the main parts 

u0,0, u0,1 and w0,0, w0,1 respectively, longitudinal – u0 and radial – w0 displacements of points 

of some “intermediate” surface of the middle layer of a three-layer cylindrical structure. 

First, it is necessary to determine the displacements u0, w0 and the stresses (0)

rr , (0)

zz ,
(0)

 , (0)

zr  points of the middle layer. To find u0, w0 it is enough to reverse the expressions 

in Eqs. (18) and (19) for 0 ( , , )u r k p and 0 ( , , )w r k p  by p and k. Applying transformations 

given by Eqs. (20) and (21) to Eqs. (18) and (19) and limiting ourselves to the zeroth order 

approximation, we obtain: 

 0 0,0 0,1 0 0,1 0,0( , , ) ; ( , , ) .
2

r
u r z t u u w r z t w w

r
= − = − +


  (29) 

Similarly, formulas for stresses are derived, for example 

 

0

2
0,0(0) (0) (0) (0)

1 0,0 1 3, 0 2 2
0

2
0,1(0) (0)

1, 0,1 2 3, 2 1, 0,12 2

(1 ) ( )

2
( ) 2 ( ) ( )

rr n
n

n n n

u
R q w q r q

z z

u
r w q r q r w

zz r



=

  
 = − − + − + −       

     
 − − − −        

   

    

 (30) 

The displacements and stresses of the other layers are found in a similar way, for example: 

 

2 22 2 2

2
2 2 2 2 2 0 22 2 2

2 2

0 22
2 2

2

( , , ) ( , , )
8

( , , )
( )

4

r r
u r z t u r z t

z z z

w r z tr r

r z

    −  
− = − + − +           

−
+ −



   

 

 (31) 

Solutions of Eqs. (22) and (23) for u0,0, u0,1 and w0,0, w0,1, can be used for calculating 

the displacements u0, w0 of points of an arbitrary cross-section of the median layer for any 

moment in time using Eq. (29) with the desired accuracy along the radial coordinate r, for 

an arbitrary moment in time. Similarly, the stresses of the remaining displacements and 

stresses at the points of the layers are determined in the case when m=1 and m=2. 



482 KH. KHUDOYNAZAROV 

3. RESULTS AND DISCUSSIONS 

The obtained equations of longitudinal-radial vibrations of a three-layer circular cylindrical 

viscoelastic shell, Eqs. (22) and (23), are general. In the following, we consider several limiting 

cases and particular types of vibration equations. 

3.1.  Vibrations of a Two-Layer Viscoelastic Structure by a Layer,  

with Sticking between the Layers 

At a = r1 the three-layer cylindrical structure transforms into a two-layer structure. The 

intermediate surface of the structure of radius  passes into the intermediate surface of the 

inner layer of the structure. In this case, it should be assumed that the external action 

function (1) (1)( , ), ( , )r rzF z t F z t  acts on the surface r = r1 and the operators λ1 = 0, δ1 = 0. 

Then, from the system of Eqs. (22) and (23), we obtain a system of equations of a two-

layer cylindrical viscoelastic structure. 

Similarly, it is possible to obtain a system of equations of longitudinal-radial vibrations 

of a two-layer cylindrical viscoelastic structure, where the main unknowns will be the main 

parts of the longitudinal-radial displacements of the intermediate surface of the outer layer. 

To do this, it is enough to assume that there is no outer layer and put b = r2  assume that 

the functions of external action (2) (2)( , ), ( , )r rzF z t F z t  act on the surface r = r2, and the 

operators λ2 = 0, δ2 = 0. 

If there are no inner and outer layers of the cylindrical structure, i.e. it is homogeneous 

(single-layer), then a = r1, b = r2 and λi = 0, δi = (i=1,2) should be put into general Eqs. (22) 

and (23). In this case, we will have a system of equations for longitudinal-radial vibrations 

of a cylindrical homogeneous layer, which exactly coincides with the system of equations 

derived by Khudoynazarov et al. [19]. 

3.2. A Three-Layer Cylindrical Viscoelastic Structure  

with a Thin Middle Layer and Rigid Contacts between the Layers 

If r2 = r1(1+ε), where ε > 0 is a small parameter, then the middle layer of the structure is thin 

(for example, a thin layer of glue, usually applied between layers). In this case, the values of 

ln(ri/ε) can be assumed to be zero. Then Eq. (19) for η3,n(ri) is simplified and takes the form:  

 ( )3,
1

1 1
, 0,1,2,..., 1,2.

2

n

n i
k

r n i
k=

= − = =  (32) 

Consequently, Eqs. (22) and (23) are also equations of longitudinal-radial vibrations of 

a three-layer cylindrical structure with a thin middle layer, but with a different value for 

η3,n(ri), determined by Eq. (32). 

3.3.  Three-Layer Cylindrical Elastic Structure with Sticking between Layers 

If the materials of the layers are elastic, then the expressions for viscoelastic operators 

will have the equality Kμm(t)=0, and therefore we will have
ь mR =  . Then we get a system 

of equations that coincides in structure with the system of Eqs. (22) and (23), where the 

integral operators 
i

R
,i=1,2, are replaced by the Lame coefficients μ1 and μ2, respectively. 



 Longitudinal-Radial Vibrations of a Viscoelastic Cylindrical Three-Layer Structure 483 

In this case, the integral-differential operators ( )n

m   and ( )n

m  , defined by Eqs. (25), pass 

into the following differential operators: 

 
2 2 2 2

2 2 2 2 2 2

1 1
( ) , ( ) , 1,2,3,...

n n

n n

m m

m m

n
b t z a t z

         
= − = − =      

            

   
      

The results obtained coincide with the results by Filippov and Filippov [20]. In 

particular cases, from the equations obtained in this way, it is easy to obtain equations of a 

two-layer elastic structure and a three-layer elastic structure with a thin middle layer, like 

the limiting cases discussed above. 

4. CONCLUSIONS 

A new mathematical model of longitudinal-radial vibrations of a circular cylindrical 

three-layer viscoelastic structure with sticking condition between the layers is proposed. 

The model includes new vibration equations and an algorithm for calculating the stress-

strain state of an arbitrary point of the structure. The vibration equations of the considered 

structures, including the influence of the moments of inertia and transverse shear deformation, 

are derived for arbitrary external dynamic loads acting onto the structure surfaces. In the 

absence of external layers, the results obtained completely coincide with the results of 

Filippov and Filippov [20]. 

A new method has been developed for the dynamic calculation of circular cylindrical 

three-layer elastic and viscoelastic shells for the action of various external dynamic loads. 

The method consists in the derivation of vibration equations, both refined ones of the 

Timoshenko type and classical ones of the Kirchhoff-Love type, and in the development 

of an algorithm for calculating the stress-strain state system. 

As special cases of the obtained results, new equations of longitudinal-radial unsteady 

vibrations of a circular cylindrical three-layer elastic structure are proposed. With restrictions in 

infinite series, their approximations of different order are followed by Kirchhoff-Love type 

vibration equations [21], Hermann-Mirsky and other refined equations of higher orders. 

Equations have been developed that enable to determine the stress-strain state at any 

point of an arbitrary section of a circular cylindrical three-layer viscoelastic structure based 

on the results of solving longitudinal-radial vibrations. 

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