Natural and forsed osculations of pipelines in contact with the Wincler medium

. In this paper, the pipeline is modeled as a curved rod in contact with the Winkler medium. Linear oscillations of a curved viscoelastic rod lying on the Winkler base are considered. The general formulation of the problem of free oscillations of a spatially curved viscoelastic rod with variable parameters is reduced to a boundary value problem for a system of ordinary integro-differential equations of the 12th order with variable coefficients relative to eigenstates; it can be solved by the method of successive approximations. The relations allowing to present the solution of the boundary value problem for the rod in an analytical form are formulated. It is established that the dimensionless complex frequencies of natural oscillations of a spatially curved rod, while maintaining the elongation of the rod constant, do not depend on it. The Poisson's ratio has little effect on the dimensionless real and imaginary parts of the natural frequencies.


Introduction
In recent times, aeromechanics has become one of the developing applied areas in the field of mechanics, within which great attention is paid to the study of the strength of elements under vibration, shock and other types of external influences [1][2]. Biomechanical structures, in general, have a very complex structure and shape [3][4]. Their mechanical properties depend on the individual characteristics of the organism, age, functional state, external factors and are largely determined by the stress-strain state, since the biomechanical system adapts to external influences. An organism, as an object of mechanics, is a complex system in which a hierarchical organization is viewed [5]. Considering the general methods of studying complex systems, it can be argued that their mathematical modeling requires the compilation of models of elements of the lowest level of the hierarchy, that is, in relation to this case, bones, muscles and internal organs. From the given examples of structuring it follows that the elements of the skeleton, i.e. bones, which are the main supporting elements of the structure of the organism, are an indispensable element of modeling [6]. In the vast majority of cases, bones can be represented as spatially curved rods of variable cross-section. In addition, the elements of the skeleton have pronounced viscoelastic properties [7][8], varying both along the axis and in cross-section.
The method of modal analysis (modal decomposition) is currently used to solve problems on the dynamic behavior of viscoelastic bodies [9][10]. Its advantage is the possibility of using both analytical and discrete models, in which there is a weak dependence on the nature of external influence. To implement the method, the decomposition of the motion of a viscoelastic body according to the modes of vibrationsthe functional basis -according to the forms of free vibrations of an elastic body is used. The convenience of this basis is that it is a complete orthogonal system of functions, which simplifies the decomposition technique [11][12].
Therefore, in this paper we consider the question of representing solutions to dynamic problems through combinations of elementary transcendental functions, which in the limit give strict solutions to a system of equations for forms of free oscillations. The complex eigenfrequencies are determined by the Muller-Gauss method. Also, when the curvilinear pipeline lying on the base of the Winkler oscillates, the present work is considered.

Problem statement and solution methods
To derive the differential equations of motion of a curved rod, we select the elementary part of the curved rod ( Figure 1). Consider the element of a curved rod of length ds and all the acting internal and external forces, which is shown in Figure 1. In Figure 1, the following designations are adopted: where Q1= N axial force Q2=Qncutting force or normal force, Q3=Qb-shearing force or components of the shearing force vector by binormals;  The selected element is in equilibrium only when the sum of all forces and the sum of moments are equal to zero, which gives two vector equations: r is the main moment of inertia forces. The system of equations (1) can be written in the following form Here- is the main moment arising in the sections of a curved rod; p dM r --the main factor of the forces of external loads. Taking into account the Frensay-Serre formula [13], it is possible to write through projections of tangent vectors t , normal n and binormals b in the form of a system of three differential equations: Now we write the second equation (3) in expanded form. To do this, we use the expression of moments through displacements: Where , , t n b J J J are moments of inertia of the rod relative to the axes. To calculate the moments from the forces, we use the well-known ratio from vector analysis If we use the Fresne trihedron [13], then we will have: Vector multiplication (6) can be written through the components of vectors in scalar form. Now using equations (5) and (6) we obtain three more differential equations in scalar form: Thus, we obtained six differential equations with 12 unknowns: , , , , , , , , , , , To obtain a closed system of equations, additional geometric and physical equations are needed. Normal stresses at any point of the rod, taking into account the viscoelastic properties of the rod material, are represented by Hooke's law [14][15][16]: Where E 0n is instantaneous values of the modulus of elasticity, Here- is deformation of the rod axis; b u is movement of the rod particle along the binormal; n u is movement of the rod particle along the normal. The deformations of the axis of the rod are determined by (9) and takes the following form: If we use Hooke's law (8), then for internal points we get known formulas of power factors: Here- , ( ) t  -an arbitrary function of time. The force factors (11) can be brought into a form convenient for calculation using the above ratios (8)-(10): Here , , axial, centrifugal and polar moment of inertia of the section.
If we use (1), (7) and (12), then we get the following system of integro-differential equations

Conclusion
Thus, the paper has developed a solution technique and an algorithm for studying the natural oscillations of curved deformable rods. With the growth of its own motion, the attenuation decrements increase in the presence of the viscosity of the rod and decrease in the presence of the external viscosity of the Winkler base. Moreover, with an increase in intensive dissipation, aperiodic modes (purely imaginary eigenvalues) arise, starting with the highest eigenforms, in the case of taking into account the viscosity of the rod. By taking into account energy dissipation, the viscoelastic rod model makes it possible to study forced steady-state oscillations at resonances.