Stationary response of the system “Cylindrical shell – viscoelastic filler” to the effect of a moving load

The paper considers the stationary response of the system “cylindrical shell - viscoelastic filler” to the action of a moving load. The filler and the shell were assumed to be viscoelastic. Using the principle of analogy of viscoelastic and elastic problems (elastic-viscoelastic analogy), a stationary solution of the action of mobile loads on a shell with a viscoelastic filler is obtained. It is shown that this principle makes it possible to generalize the class of problems obtained for the case of viscoelastic media. Since the viscoelastic filler has damping properties, the inversion integrals have no singularities on the real axis, and the elements of the determinants in integrals become complex since in this problem in the image space, the Lame coefficients are complex and depend on the speed of the load movement, as well as the Fourier transform parameter. Numerical results are obtained, and an analysis is made.


Introduction
The stationary states of a multilayer viscoelastic base under the influence of stationary mobile loads are considered in [1][2][3].In [3], the dependence between stresses and deformations in the Boltzmann-Voltaire integral form was used.The effect of a mobile stationary load on the elastic floor of the space is considered in [4].When solving the half-space equation (Lame equation), the integral Fourier transform in spatial coordinates is used.The expression of mixing and stresses in the images is obtained.The solution in the given coordinates is obtained by the inverse transform.The effect of mobile stationary bending loads on a non-axisymmetric (or axisymmetric) shell interacting with a filler has been studied in [5,6].The propagation of own waves in a shell with a viscous liquid is considered in [7,8].The problem is solved in the displacement potentials.The fluid motion equation satisfies the Navier-Stokes equations.In general, the problems of torsional and radial transverse vibrations of a cylindrical shell with a viscous liquid were solved in works [9,10].
In this paper, the effectively of a stationary mobile load on a cylindrical shell with a filler is studied.

Problem statement and solution methodology
The integro-differential equations given in the paper is quite general: in particular, two special cases can be obtained from them, when the viscosity (R * = 0) is read, then we get linear stationary problems of cylindrical bodies.
The equation of dynamics of a cylindrical body with a filler takes the form: The stress-strain relationship for viscoelastic materials is adopted in the form of a hereditary Boltzmann-Voltaire integral [11] 2 ( , , ) ( ) instantaneous modulus of elasticity of the material.Contact conditions are as follows: (2) For solving of equations (1) the integral Fourier transform ( ) along the longitudinal axes is applied.Applying the Fourier transform to (1) and (2) equations, then the expression of the stresses (in the images) [12] is obtained:

;
with , u ГГ  they move, remaining unchanged, together with the moving coordinate system.The solution of a stationary problem in a moving coordinate system under the influence of moving loads can be obtained from the above principle 13].Based on the obtained formulas in the images, the stresses and displacements of each point of the mechanical cylinder -filler system are determined, the Lame coefficients are assumed to be complex.To solve the boundary value problems of the above, boundary and initial conditions are required.For this, boundary conditions are used, for example, some parts of the cylindrical surface of displacements and stresses on Г  and u Г , do not change over time: The choice of a particular stationary load depends on the speed range in which resonance is expected to occur, the shape of the inherently deformable element in question, the required accuracy and technical equipment of the calculation, etc. Let's assume that the flow is subsonic, then the expected oscillations will not be of particularly high frequency.In this case, to calculate displacements and stresses you can use the equations: where , ps cc -speeds; ω -frequency.
If it is necessary to investigate the dynamic stress-strain state, then, as a first approximation, the displacement functions can be taken, for example, the forms of the basic tones of purely bending and purely torsional vibrations of the shell.
Generalization of the proposed approach for the construction of mathematical models and the development of a technique for the numerical solution of problems about the oscillation of a shell made of physically linearly hereditarily deformable materials is not difficult [12] APITECH-V-2023 Journal of Physics: Conference Series 2697 (2024) 012004 IOP Publishing doi:10.1088/1742-6596/2697/1/012004 Gkn t vc  = − Using the inverse transform the solution is obtained.For the case of linear differential operators (2) the results are obtained [16] ( ) where  -the inverse of the relaxation time; 12 , the values inverse to the recovery time of uniaxial deformation and shear deformation, respectively; , ee mm  -effective Lame constants [17].
For this model, formula ( 8) is converted as

Results and discussion
Numerical calculations are obtained for an annular concentrated moving load on a shell with a solid filler.The moving speed is assumed to be constant.The problem is solved in dimensionless parameters with such values: 0, 02; 0,3; / 125; / 12. 5;  In Figure 1, the solid lines refer to the Timoshenko-type shell, and the dotted lines -to the Kirchhoff-Love shell.When obtaining numerical solutions of the speeds of movement of the ring load, they take different values.For the curve for the curve for the curve .
Thus, method of calculating the integrals of the circulation allows paying to attention the properties of visco elastiсity of the filler at a stationary moving load is developed.

Conclusions
Thus, the developed method will make it possible to pay attention to the properties of the viscoelasticity of the filler at a stationary moving load.It follows from the calculations that considering the inertia of rotation and transverse shear significantly affects the stress values.

jn-
normal to the boundary of the cylindrical shell; Г  -boundary at which stresses are specified; u Г -the boundary on which the movements are specified.Cylindrical coordinates are taken as coordinates.

c
load movement speed and time are also taken in dimensionless form.In the particular case of the problem under consideration, uniaxial deformation and characterized by varying.The solution to the stationary problem is a cylindrical shell -viscoelastic filler for the action of a moving load is the only one at any speed of movement of the load.The inverse transformations are carried out by the method of integral inversions, which are not special.The values of integral inversions are found by the Fileon method.The change of deflections along the length for different speeds of the ring load is shown in Fig.1.

Figure 1 .
Figure 1.Deflections of the shell with viscoelastic filler under different modes of motion of the ring load.