Dynamic instability of vibrations of thin-wall composite curvorine viscoelastic tubes under the influence of pulse pressure

In this paper the results of a study of the dynamic characteristics of thin-walled composite curved viscoelastic pipes under the influence of internal pulsating pressures is presented. The relationship between stress and strain is described by the Aviary equations. Based on the principle of calculus of variations, the equations of the dynamics of curvilinear shells are obtained. The obtained integral and integro-differential equations are solved using the finite element method, the Mueller and Gauss methods. The results of calculations of the natural frequencies of the shell and the curved rod are compared. As a result of the calculation, it was found that, shell vibrations occur at higher frequencies than rod vibrations (manifestation of a boundary effect). With this difference, the natural frequency, which has no analogue in the rod model decreases. This is due to the fact that, in this case the transverse vibrations of the section itself become significant, the rod. Some mechanical effects were discovered when taking into account the rheological properties of the pipe material.


Introduction
The effect of a stationary fluid flow on the eigenparameters of curved pipes is considered in [1][2][3][4][5][6][7][8]. In them, by analogy with straight pipes, the dependences of the natural frequencies on the flow velocity are studied, critical flow velocities are determined taking into account the boundary conditions, and forms of stability loss are found. For this, both rod and shell computational models are used. The initial stress state induced by the stationary internal flow, as a rule, is not taken into account.
In the works of V.A. Svetlitsky [9,10] gives equations of small oscillations of an articulated tube as a spatial rod with a moving fluid. The dependence of the natural frequencies of the oscillations on the parameters of the tube is established. The initial stress-strain state is taken into account. In this case, the natural frequencies of a circular articulated tube depend on the velocity of the fluid flow only through the Coriolis forces.
The problem of studying the spatial oscillations of thin-walled curvilinear pipes of small curvature is considered in the works of V.F. Ovchinnikov, V. A. Prokhorovich and L. V. Smirnov [11,12]. The stationary internal flow is taken into account. One-dimensional equations of dynamics are obtained that describe the rod and shell forms of motion, taking into account the effects of flattening and cross-section deployment. The influence of the Karman effect on the spectrum of lower eigenfrequencies is analyzed. In this case, the stress state is not considered.
The free vibrations of thin-walled curved pipes with a flowing fluid are studied on the basis of a geometrically nonlinear semi-momentless theory of cylindrical shells, in an article of V. G. Sokolov [13]. The Bubnov-Galerkin method is used. In the article of O.B. Khaletskoy [14] for the analysis of free vibrations of a curved pipe with pivotally fixed end sections uses the semi-momentless theory of thin shells. It is shown that, with increasing pipe curvature the lowest natural frequency increases. In the article of V.P. Ilyin and B.F. Sokolov [15] to study the free vibrations of curved pipes with a stationary fluid flow, a geometrically nonlinear version of the semi-momentless theory of shells is used. This approach is traditional. In [16], based on the geometrically nonlinear version of the semimomentless theory of shells and the theory of potential fluid flow, the problem of free vibrations of thin-walled curved pipes is solved. The rod and shell forms of motion are considered. In this case, the rod forms associated with the bending of the pipe axis are shown independently of the shell forms (the effect of flattening is not taken into account). It is established that, with an increase in the curvature of the pipe, the natural frequencies increase. For polyethylene pipes, a significant dependence of natural frequencies on the internal flow rate was revealed. With increasing speed, natural frequencies increase markedly. In this case, the initial stress-strain state (SSS) is not taken into account. In the work of Yu.A. Kulikov [17] to calculate the parameters of free oscillations of thin-walled curved pipelines with a stationary internal flow, a dynamic model of the finite element method is used. The effect of flattening of the cross-section during bending (the Karman effect) is taken into account using the coefficients of increasing flexibility calculated on the basis of the static model [18]. One of the most characteristics and "destructive" loads on the pipe-wire is unsteady internal flow. Thin-walled viscoelastic composite pipes are more deformative than metal pipes. Therefore, they have an increased sensitivity to the effects of internal flow. Also, the action of unsteady internal flow causes variable loads on the pipeline. The main causes of pipeline failures are depressurization of joints and pipe ruptures, as a rule, in places of bends and fixtures (up to 90%). Most damage is pronounced fatigue. Fatigue strength significantly depends on structural and technological factors, primarily, initial geometrical irregularities, especially curved sections and rheological properties of the material.
This article is devoted to a partial solution to the above problems.

Statement of the problem and basic relations
Let a curvilinear viscoelastic composite pipe with an internal pulsating pressure, shown in figure 1, be given. In figure 1, in the Cartesian coordinate system x, y, z, the design diagram of a (multilayer) composite viscoelastic pipe is shown, the axial line of which represents an arc of a circle of radius R of length L with a central angle φ 0 . The pipe has a cross section with a nominal average radius r and wall thickness h. We restrict ourselves to thin-walled, rather long pipes of small curvature: h/r ≤ 1/20, L/r ≥ 4 and r/R ≤ 1/5. The dynamic stability of oscillations of thin-walled composite curved viscoelastic pipes under the influence of internal pulsating pressures is considered. A pipe is considered as an element of a pipeline that conducts a fluid. We consider the internal flow to be one-dimensional, and the fluid to be single-phase, ideal, and incompressible. The parameters, characterizing the unsteady flow regime are decomposed into constant (unperturbed) and variable (pulsation) components: Here p m , V m , p n (t), V n (t) -components of pressure and velocity, respectively, t is time. Variation deviations p n (t), V n (t) from average values p m , V m supposed small.
To calculate the dynamic characteristics of multilayer viscoelastic composite pipes, we use the finite element method (FEM) in displacements. Consider an 8-node isoparametric curvilinear finite element, the so-called "degenerate" shell element [19,20]. The element is designed to calculate shells of medium and small thickness with variable wall thickness and variable reinforcement parameters. Dependencies based on equations of the theory of viscoelasticity and hypotheses of the theory of shells are used. The geometry of finite elements represents a curved box in three-dimensional space with a ruled surface in thickness ( Figure 2). The physical equations for the relation between stress and strain are of the form [21]: where  , -instant elastic modules; , t-observation time,  -time preceding observation. It is required to study the stability of oscillations of composite curvilinear viscoelastic pipes under the action of pulsating internal pressure depending on the component of internal pressure, geometric factors, and rheological properties of composite pipes. We believe that the ends of the pipes are rigidly pinched.

Solution Methodology
Here i-anchor point number (i=1,2,....8); hi-knot wall thickness i; With a symmetrical structure of the package of layers, the coordinate surface is combined with the middle surface of the shell. The shape of the finite element (FE) is uniquely determined by the coordinates of the nodal points. In approximating the viscoelastic displacements of FE, the hypothesis of non-deformability of the wall element in the transverse direction and the Timoshenko kinematic hypotheses are used. The displacement vector of an arbitrary FE point is expressed through the displacements of nodal points: , -linear motions of nodes; normal rotation angles i n relative to axes -FE shape functions with quadratic approximation [22]. According to the isoparametric concept, when approximating geometry (2) and elastic where   B -quadratic symmetric matrix of third-order differential operators. Cartesian coordinate system connected with the middle surface of the shell, the axis z directed normal to the surface (Fig. 2). Global derivatives (3) where    -is the matrix of the guide cosines of the coordinate axes with respect to x, y, The stiffness matrix and the mass matrix of FE are determined by the following expressions: -elemental volume, J -Jacobean,   D a quadratic matrix whose elements satisfy the following operators: Here ij D 0 -instant elastic modules of composite multilayer pipes, Dij R  -relaxation core [23,24]. Integrals (5) (6). An operator matrix is also constructed by summing over the layers. Then, to transform the variables, the Jacobi matrices and Jacobian are compiled. After that, by summing over quadrature points, matrix elements of     axial directions, respectively. Then we obtain the following integro-differential equation in matrix form [26]: Here  

Numerical implementation of the integration method
A detailed analysis of existing methods for integrating dynamic state equations is presented in a monograph by K. Baiter and E. Wilson [27]. The methods used for solving dynamic equations by the finite element method (FEM) are divided into methods of direct integration and expansion in their own forms. The program package uses direct numerical integration, the Gauss method, and the Mueller method, which allows one to efficiently combine the solution of dynamic equations with the organization of the iterative process for physical problems [28]. This program is designed to solve the problem of intrinsic, parametric and non-stationary vibrations of thin-walled composite curved viscoelastic pipes, containing an ideal fluid. Equilibrium equation (7), with direct numerical integration, is satisfied not at any moment of time, but over some given (sufficiently small) time interval t  . Direct integration methods are divided into explicit integration methods, when velocities, accelerations, and displacements are calculated from equilibrium equations at a moment in time (central difference method), and implicit integration methods that use equilibrium conditions at a time t t   (linear acceleration method, trunk method,  -Wilson and Newmark method). The algorithm and the program implements an unconditional stable Newmark integration scheme [28], in which the following finite-difference relations are applied: where  and  -parameters, that determine the accuracy and stability of the integration scheme. In the calculations  and  they will take values of 0. 20 The ends of the pipes are rigidly pinched.
The results of calculations by the FEM were compared with the decisions of I.D Grudeva [29], who considered the oscillations of thin curved rods. Table 1 gives a comparison of frequencies (   0 ). The numerator shows the calculation data on the FEM. Regular mesh used: М=25 и N=350-the number of divisions in the circumferential and axial directions, respectively. Dimensionless parameters j  (j=1,2,3,4,5) are calculated for each proper form by numerically integrating the equations of motion using the Runge-Kutta method and the initial parameter method. In parentheses is the calculation error. Analysis of the table shows that, when 0  (during the "extension" of the pipe) the natural frequencies corresponding to the rod forms, increase and approach the natural frequencies of the straight pipe.

Damping of natural oscillations of pipelines
Consider the natural vibrations of a multilayer thin-walled curved pipe made of organoplastics Keplar 49 / PR-286. Then equations (7) take the following form We take the integral terms in (8)   -respectively, the cosine and sine Fourier images of the core relaxation of the material. On the function of influence ) (   t R the usual requirements of integrability, continuity (except   t ), sign-certainty and monotony. In this case, equation (8) from integrodifferential equations turns into a system of differential equations with complex coefficients of the form where ) , ( ) , ( . Then solution of the equation (10) is sought in the form Substituting (11) into (10), we obtain the following system of homogeneous algebraic equations: The system of homogeneous algebraic equations (12) is solved by the developed algorithm (Mueller and Gauss method) [25].
The pipe walls are formed by cross helical winding of two symmetrical fiber systems with reinforcement angles ± 0  . The number of unidirectional layers is six. The elastic characteristics of the wall are shown in Table 2. As a relaxation core, the three-parameter  Table 3 presents the four lowest complex eigenfrequencies, depending on the reinforcement angles. It is established that, with increasing reinforcement angles ± 0  and internal pressure рm , the real and imaginary parts of the natural frequency grow. Moreover, the higher the pressure, the weaker is the effect of the reinforcement angles. If the rheological properties of the material are taken into account according to formulas (1) and (6), then the real and imaginary parts of the complex natural frequency are reduced to 15%    Parametric resonances are suppressed more with increasing number. Fig.4 shows the formation of the region of the main parametric resonance depending on the excitation parameter. Figure 4 illustrates the instability region taking into account viscosity and without viscosity.
, in this case, the regions of dynamic instability degenerate in a line. Note, for practical applications, the accuracy of the formulas of the first approximation of the small parameter method is sufficient, up to the value 5 . 0   [32]. From the analysis of the obtained numerical results it follows that with a decrease in the dimensionless parameter regardless of the parameters of the relaxation core value. Also, with an increase in thickness within the indicated limits, the regions of dynamic instability are removed from each other. With increases in the amplitude of the relaxation core (Аm) the speed of removal of the areas of dynamic instability from each other also increase.

Conclusions
1. A mathematical formulation and methods for solving the dynamic instability problem of a curvilinear viscoelastic composite pipe under the action of internal pulsating loads using the theory of viscoelasticity and the theory of thin shells have been developed. 2. A computational dynamic model of a curvilinear viscoelastic composite pipe using the finite element method is proposed. 3. The reliability of the results obtained is shown by comparing the obtained numerical values of the calculations with the known results of the core theory. 4. Taking into account the viscous properties of the material has a noticeable effect on the region of dynamic instability. Moreover, viscous properties play the role of stabilizing factors for parametric vibrations of mechanical systems. 5. Taking into account the viscous properties of the material leads to a reduction and mixing of the geometric dimensions of the regions of dynamic instability, corresponding to the main parametric resonances. 6. It was established that the rheological properties of the material lead to the complete exclusion of parametric oscillations.