Longitudinal vibrations of a cylindrical shell filled with a viscous compressible liquid

This article investigates the longitudinal vibrations of a semiinfinite circular cylindrical elastic shell filled with a viscous compressible fluid. It is believed that the vibrations are excited by a suddenly switched on longitudinal displacement at the end. To solve the problem, the refined equations of longitudinal vibrations of a circular cylindrical elastic shell interacting with an internal viscous compressible fluid, previously proposed by the authors, were taken as the main resolving equations. In this case, the lateral surfaces of the shell are considered free from external loads; in addition, considering purely longitudinal vibrations, it can be assumed that the radial displacements of the points of the shell are equal to zero.


Introduction
The dynamic theory of shells is one of the sections of the three-dimensional theory of elasticity, in which such problems of calculating shells are considered, under which the boundary conditions on the lateral surfaces are specified in stresses in the form of external loads. In this case, the boundary conditions on the lateral surfaces are set in stresses in the form of external loads [1]. Usually, problems about transient processes in shells and in plates and rods are usually carried out based on approximate equations of oscillation [2][3][4][5]. Therefore, the basic relations of the theory of shells are developed by reducing a threedimensional problem in spatial coordinates to a two-dimensional one, using various methods and approaches using hypotheses and premises [6]. Depending on the factors taken into account, the methods of the derivation of the approximate equations of vibration, based on the dynamic theory of elasticity, are divided into several directions [7].
One of these areas is the method of using general solutions in transformations of threedimensional problems of the dynamic theory of elasticity, which was developed in the works of IG Filippov et al. [8,9]. In this case, in [8], approximate equations of transverse and longitudinal-radial vibrations of a cylindrical shell were constructed, and in [9], the questions of formulating the boundary conditions to them were discussed. This method has been successfully applied in the works of Kh. Khudoinazarov et al. To the problems of unsteady vibrations of three-layer cylindrical shells [10] and plates [11,12], in which general equations of vibrations of layered plates and shells were obtained taking into account the viscoelastic properties of the material. *Corresponding author: kh.khudoyn@gmail.com The refined equations of vibrations of shells and plates were applied in the works of RI Khalmuradov to solve the problem of transverse vibrations of a reinforced plate [13] and frequency analysis of longitudinal-radial vibrations of a cylindrical shell [14]. Based on the solution of the obtained frequency equations, the frequencies of natural vibrations of the shell are determined. A comparative frequency analysis of longitudinal vibrations of a circular cylindrical elastic shell is carried out based on the classical Kirchhoff-Love theory, refined theories of Hermann-Mirsky, and Filippov-Khudoinazarov. Based on the results obtained, conclusions were made regarding the applicability of the studied oscillation equations, depending on the waveform and shell length.
It should be emphasized that the essence of this method for deriving the oscillation equations is reduced to studying the constructed solutions [15,16] for various types of external influences and to fin-ding out the conditions under which the displacements or their "main parts" satisfy the simple oscillation equations, and to finding the algorithm, allowing for the field of these "main parts" to calculate the approximate values of the fields of displacements and stresses in any section for an arbitrary moment in time.
In recent decades, the study of non-stationary interaction of structural elements [17] and, in particular, cylindrical shells [18][19][20] with a viscous fluid are becoming increasingly important. In these works, various theories of vibrations of plates and shells interacting with a fluid are presented, and an analysis of their applicability in solving various problems of wave dynamics, including transient deformation processes. This article is devoted to the study of longitudinal vibrations of a cylindrical shell interacting with an internal viscous compressible fluid.

Methods
It is believed that the cylindrical shell under consideration, as a cylindrical layer (threedimensional solid), strictly obeys the mathematical linear theory of viscoelasticity and, in its exact formulation, which is described by its three-dimensional equations. Moreover, it is assigned to a cylindrical coordinate system , where is the axis -directed along the axis of symmetry of the cylinder. It is assumed that the vibrations of the layer, like fluids, are small. In this case, the smallness of the oscillations implies the smallness of the displacements of the points of the layer and the fluid.

Basic equations and formulas for stress-strain state
To solve the problem, we apply the refined oscillation equations derived in [6] for nonstationary longitudinal-radial oscillations of a circular cylindrical elastic shell, which in the absence of external influences take the form   For the correct formulation of the boundary conditions of applied problems, when truncating the number of terms in the series of equations (1), one should adhere to the same accuracy as in the oscillation equations.

Statement of the problem
We study the process of longitudinal deformation of a semi-infinite elastic circular cylindrical shell filled with a viscous compressible fluid excited by a suddenly included longitudinal displacement at the end . 0  z To solve the problem, for the main resolving equations, we take the specified equations of the longitudinal radial vibrations of a circular cylindrical shell with a viscous compressible fluid. To obtain such an equation, we use equations (1). Сonsidering purely longitudinal vibrations, we can assume that the radial displacement (and its main parts) is equal to zero. This condition can be achieved, for example, by dressing a rigid ring that is not deformed in the radial direction on end face subjected to an external load. Then, the equations of longitudinal vibrations of the cylindrical shell obtained from (1), taking into account the above remarks and in dimensionless variables, will have the form Where the fluid reaction is equal In the expression for the fluid reaction (4), the second term is responsible for the reaction of the fluid due to viscosity. In this case, the kinematic viscosity coefficient for most fluids is of the order -4 10 and below. In addition, the Lame coefficient present in the denominator of the coefficient of this term is of the order 10 10 and higher. Therefore, we can assume that the contribution of the second term in (3.2) is very small, and it can be neglected. Then the boundary conditions of the problem recorded for the main part of the longitudinal displacement will have the form where   t g is impulse of a given displacement. The initial conditions are assumed to be zero.

Solving of the problem
We assume that the displacements of the cylindrical shell have sufficiently small quantities. Then the main parts of the displacement z U and their derivatives of the third and higher orders will be small quantities of a higher order. Therefore, in the first equation (1), we neglect the derivatives of third-order functions, and solving the system with respect to 1 , z U we will have Where: To solve equation (6), we apply the Laplace transform. Then the general solution (6), due to (7) (8) In case of the absence of fluid takes place Based on (7), we find  Сonducted to verify the reliability of the obtained results, a comparative analysis is showed that the refined equations of axisymmetric vibrations of the hydroelastic system under consideration, in the particular case of the absence of an interacting viscous fluid, coincide in structure but differ in coefficients with the corresponding results of [ 8,21].
The proposed approximate equations of longitudinal vibrations of elastic circular cylindrical shells interacting with a viscous fluid are proposed, suitable for solving applied engineering practice problems.