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Purpose. To extend the approach previously proposed by the authors on the application of exact equations of elasticity theory to problems of dynamics of three-layer cylindrical shells with isotropic filler to one of the possible cases of anisotropy of the middle layer material, namely the situation when the filler is transversely isotropic. To obtain accurate formulas and, based on them, to construct a picture of the stress-strain state in such a composite structure when moving along the outer surface at a constant normal (radial) load speed. Research methods. A mathematical model of the dynamics of a three-layer cylindrical shell has been constructed, where the motion of the supporting layers is described by the equations of thin shell theory, and for a transversely isotropic filler, the dynamic equations of the theory of elasticity of an anisotropic medium in general form are used. When considering the problem in a stationary setting, Galilean transformation is applied, after which the integral Fourier transform in complex form is applied to all sought and given values in the moving coordinate system. To calculate non-proper Fourier integrals, quadrature formulas based on the Filon method for integrating rapidly oscillating functions were developed, which made it possible to efficiently obtain numerical results with a predetermined accuracy. Results. Based on the constructed model, the problem of a moving load that causes a stationary stress-strain state of a layered cylindrical shell under various conditions on the surfaces of the joint between the filler and the supporting layers is considered. In this case, the contact is considered both rigid and sliding, but the lag of the shells from the filler is excluded. The difficulties that arise when solving the equations of motion of a transversely isotropic filler are overcome by introducing a special method using undefined coefficients of potential functions. For all possible boundary conditions, the results are obtained in the form of non-special improper integrals, which are calculated using special quadrature formulas. The distribution patterns of displacements and stresses along both the length and thickness of the filler are shown, a comparison with the results for the corresponding isotropic filler is made, and a mechanical analysis of the results is performed. Scientific novelty. For the first time in such a formulation, when the behaviour of the filler is described by exact equations of the dynamics of an elastic anisotropic body, a solution to the stationary dynamic problem for a three-layer cylindrical shell has been obtained. A comparison was made with the results previously obtained for the case of isotropic filler. A special technique was used to introduce potential functions to find displacements and stresses in the dynamic equations for transversely isotropic materials. Important partial boundary conditions at the boundaries of layer contacts were considered. Practical value. The results obtained with this approach can be used as reference values when constructing simplified models of the dynamic behaviour of three-layer cylindrical shells, in particular those that take into account the anisotropy of the filler. Examples of such materials include so-called ribbed sound-insulating materials.
Published in: New Materials and Technologies in Metallurgy and Mechanical Engineering