Natural oscillations of three-layered viscoelastic flat spherical shells

Viscoelastic sloping multilayer structures with filler have found their application in aircraft construction, construction, design of space structures and other areas. The free vibrations of two flat spherical viscoelastic shells, between which it is filled with a deformable filler, are investigated in the work. The main purpose of the article is to develop a scientific basis for constructing a method for solving the problem, and an algorithm for natural oscillations of spherical shells with a filler. Parametric analysis of the natural frequency of the physical and mechanical parameters of the mechanical system is also carried out. As an example, linear natural oscillations of three-layer structures for shells of free support are considered. The shell materials are isotropic and have identical rheological properties, as well as asymmetric in shell thickness. The viscoelastic properties of the shell and filler materials are subject to the integral Boltzmann-Voltaire relations. The oscillation equations of the filler satisfy the Lame equations of visco-elasticity. And the shell equation is derived on the basis of the Kirchhoff-Love theory. The developed methodology is based on the method of separation of variables, special functions of mathematical physics, the methods of Muller, Gauss and Laplace. It is found that the real and imaginary parts of the natural frequencies increase with increasing stiffness of the shell. Taking into account the viscoelastic properties of the shells and the filler allows to increase the free frequency fluctuations up to 15-20%.

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