Free Oscillations of a Toroidal Viscoelastic Shell with a Flowing Liquid

On the basis of the method of orthogonal sweep and the Mueller method, the solution of the problem of intrinsic oscillation of a Toroidal shell with a flowing liquid is discussed. The problem of determining the frequencies and forms of intrinsic bending vibrations in the plane of curvature of curvilinear sections of thin-walled Toroidal shells of large diameter with a flowing liquid, with different conditions for fixing the end sections is solved. The behavior of complex Eigen frequencies as a function of the curvature of the shell axis is studied.


Introduction
Elements of structures in the form of a Toroidal shell are widely used in various areas of modern technology. In particular: in the construction of pipelines, power engineering, in the rocket and space industry [1][2][3][4][5], etc. To study the strength and load-bearing capacity of shell structures, it is first of all necessary to determine their stress-strain state, which leads to the need to develop effective methods for solving boundary value problems in shell theory [6][7][8]. For the first time, the problem of bending vibrations of a straight pipe with a flowing liquid was posed and solved in [9]. Using the equations of small oscillations of the beam, the authors [9] made errors when taking into account the inertial forces of the fluid flow and obtained the wrong result. The error was already corrected in [10], and the solution obtained by him on the beam theory with the help of the Bubnov-Galerkin method of the circular frequency of a hinge fixed at the ends of a pipeline with a steady flow of liquid still has practical application: where 0 , p p -the density of the pipe and liquid material, respectively, 0 , A A -cross-sectional area of pipe and liquid walls, EI -bending stiffness of pipe, L -length of pipeline, U -speed of a flowing liquid. As can be seen from the last formula, the increase in speed U reduces frequency ω .
When the speed reaches a certain critical value kp U the oscillation frequency vanishes, and the pipeline loses stability. The value of the critical velocity can be obtained from the last equality: In [11], this problem was solved by an analytical method, the result obtained was confirmed in [12]. Further studies in this field [13,14] have evolved in the direction of taking into account additional factors affecting the oscillations of the pipelines, and also in the direction of refinement of the solution. The solution of the geometrically nonlinear problem taking into account the influence of internal hydrostatic pressure is considered [15]. In this paper, we solve the problem of the actual vibration of viscoelastic Toroidal shells with a flowing fluid, based on orthogonal sweep methods, the Mueller and Gauss method.

Equations of Motion of the Toroidal Shell
The equation of motion of bending oscillations of the toroidal shell (Figure1) is derived on the basis of the general relations of the geometrically nonlinear theory of shells of the average bending described in [16]. This theory considers such a bending of shells, in which the maximum deflection is of the same order of magnitude as the wall thickness, or even exceeds it, but small compared with other linear dimensions of the shell.
In accordance with this theory, the equations of equilibrium of the moment's forces for the element of the toroidal shell, which is in the deformed state, have the form: , , X X X -components of external force vectors and indices 1 and 2 refer to toroidal coordinate's β and θ respectively.
The first three equations (1) are the equations of equilibrium of forces, the last two are the equations of equilibrium of moments.
Differential equilibrium equations for the shell element (1) are nonlinear, since they contain products of effort and deformation. In addition, they are obtained for a shell in a deformed state. Therefore, these equations include radii of curvature * 1 R and * 2 R deformed middle surface of the shell. Their connection with the curvature of the initial state is expressed in accordance with [17] by the following relations: Change in curvature of the midline of the cross section of the shell 2 χ and torsion θ are expressed in terms of the angle of rotation ϑ the following relations: In accordance with the assumptions (1) -(3) of the semifree shell theory in the first three equilibrium equations (1), we neglect the transverse force 1 Q , and in the last two Hthe torque. Then (1) of parameters (2) and (3) we obtain a system of equations of motion of the shell in the effort: To solve the dynamic problems of the pipeline section in question, it is necessary to obtain the equation of motion of the toroidal shell in displacements.
Therefore, we transform equation (5) , -referred to the radius r dimensionless displacement components; y W -projection onto the axis y Moving point A of the middle surface of the shell to the position * A as a result of deformation of its contour (see Figure 1); ϑ -angle of rotation of the tangent to the midline of the section of the shell as a result of deformation of the cross section; Е ɶ -operator modulus of elasticity, which have the form [17,18]: respectively, the cosine and sine Fourier images of the relaxation core of the material. As an example of a viscoelastic material, we take three parametric relaxation On the influence function ( ) R t τ − the usual requirements of inerrability, continuity (except for = ), sign-definiteness and monotony: u -vector of displacements of the environment of the j-th layer.
Substituting relations (2), (3) and (6) where * i X -components of inertia forces: -tangential components in coordinates β and θ ; w X rhp p t ∂ = − + ∂ p -internal pressure, including hydrodynamic, which occurs when the fluid moves, p -density of the shell material.
The equation of motion of the toroidal shell (7) is a differential inhomogeneous partial differential equation with four unknown quantities ϑ , , , w v u . Adding to it the three relations of the semi-membrane theory of shells: we obtain a complete system of equations with four unknowns. For a stationary fluid flow, the solution of equation (7), (8) allows us to determine the frequencies and shapes of the Eigen modes of the toroidal shell.

Determination of Hydrodynamic Pressure Caused by a Fluid Flow
One where c -scale factor.
The velocity field of an ideal incompressible fluid in the process of shell oscillation is an irrational potential field with a potential b. the equation of motion (Euler) where ( ) where p and 0 p -hydrodynamic and hydrostatic pressure, respectively. From (10) -(13), a relationship is established between the hydrodynamic pressure p and the potential of the disturbed velocities ϕ : Considering the velocity vector of the fluid flow U in toroidal coordinates, we write the expressions for its components by , , For the component of the velocity vector U α , directed to the normals to the deformed shell surface, the smooth flow around this surface by the liquid flow must be satisfied [20]: where w -categorized as radius r the dimensionless component of displacement of the points of the middle surface of the shell ( Figure 1). Thus, the problem of determining the hydrodynamic pressure of a liquid on the pipe wall reduces to finding the potential φ , satisfying the Laplace equation (10) and conditions (14), (16) for r α = .
The Laplace equation (10) As a result of the separation of variables after substitution we obtain from (17) the known equation of the torus: where , const n const The general solution of the torus equation (20) Taking into account that in the problem posed, we consider the domain bounded by the surface of the torus by the  (22) and the solution of the Laplace equation (17) taking into account (18), (19), and (22) will have the form: After substituting the value of this product in (23), we obtain an expression for the velocity potential: We find the hydrodynamic pressure of the flowing liquid on the wall of the shell from (14), neglecting small second-order ones arising in the calculation of the partial function φ by : In formula (26) for hydrodynamic pressure, the expression in parentheses, by analogy with the cylindrical [21], should be regarded as reduced acceleration (with allowance for the velocity U ) element of the shell, and the value 0 n p Ф , density-dependent 0 p , be regarded as an adjoined mass of liquid.

The Equation of Motion of Toroidal Shells with a Stationary Fluid Flow
In toroidal coordinates , β θ the equation of motion of shells takes the form: 3  2  3  3  3  2  2  3   3  2  2  2  3  3  2  2   2  3  2  3  3   2  2  2  2  2   3  3  2   2  3  2  2 cos cos cos sin cos sin  taking into account (29) and discarding the small nonlinear terms in Eq. (28), the following terms remain (the last three terms on the right-hand side) ( ) Adding to the equation of motion (28) the relationship of the semi muscular theory of shells and using formula (30) for the hydrodynamic pressure ж p , we obtain a complete system of equations for the problem in the displacements ( ) It should be noted that the displacement components , , u w ν (30), (31) are also dimensionless. The boundary conditions can be as follows.
1. Both ends are hinged. The boundary conditions corresponding to this fixation can be formulated as follows: at To simplify the form of equation ( In addition, we transform equation (36) using the curvature parameter of the toroidal shell µ , which characterizes not only the geometry of the shell, but also its material, since it includes the Poisson's ratio: ( ) Here it is taken into account that in the penultimate term of equation (31) The

Numerical Natural Oscillations of a Toroidal Shell with a Fluid Flow
The study of the frequencies of the proper bending oscillations of the curved sections of pipelines (steel) with a steady flow of liquid is carried out numerically [22]. As the relaxation nucleus of a viscoelastic material, we take a threeparameter core which has a weak singularity, where , , A α β -parameters materials [17]. We take the following parameters: 0, 048; 0, 05; 0,1 A Using the complex representation for the elastic modulus, described earlier. The roots of the frequency equation are solved by the Mueller method, at each iteration of the Muller method is applied by the Gauss method with the separation of the main element. Thus, the solution of equation (31) does not require the disclosure of the determinant. As the initial approximation, we choose the phase velocities of the waves of the elastic system.
In the pipeline, water flows at a velocity u from 0 to 50 м c .
The acquisition of the results allowed us to estimate the influence of the flow velocity on the frequencies of the first four waveforms ( 1, 2,3 m = , 4 at 1, 2,3 n = ). Calculations were carried out for curvilinear pipes with relative thicknesses   If there is no deformation of the contour of the cross sections of the pipe -that is, the pipe oscillates like a beam of tubular section. For the dynamic calculation of the pipeline, the most important is the shell mode (with 2 m = and 3), corresponding to the deformed contour of the pipe crosssection. With the increase in the curvature of the pipeline section, that is, the ratio r R , corresponding to the deformed contour of the pipe cross-section. With the increase in the curvature of the pipeline section, that is, the ratio Thus, the greater the curvature of the tube, the more rigid it becomes, and the thicker the pipe wall, the more rigid it is. This is also seen from the graphs in Figure 6, which shows monotonically increasing frequency dependence 21 ω in the form of oscillation.