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Mathematical modeling of S.P. Timoshenko anisotropic mesh cylindrical panel’s vibrations based on the micropolar theory of elasticity with constrained rotation

DOI 10.18127/j19997493-201904-07


E.Yu. Krylova - Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematical and Computer Modeling, Saratov State University
I.V. Papkova - Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling of the Yuri Gagarin Saratov State Technical University
O.A. Saltykova - Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling of the Yuri Gagarin Saratov State Technical University
A.B. Kirichenko - Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling of the Yuri Gagarin Saratov State Technical University
A.V. Krysko - Dr.Sc. (Eng.), Professor, Head of the Department of Mathematic and Modelling,  Yuri Gagarin State Technical University of Saratov

A mathematical model of oscillations of flexible anisotropic thin shallow flat cylindrical mesh panels is constructed taking into account transverse shear deformations. Classical continuum models do not take into account the effects of scale, therefore, to model the behavior of a cylindrical panel, a micropolar (asymmetric, moment) theory with constrained rotation is used. It is assumed that the fields of displacements and rotations are not independent. An additional independent material length parameter associated with the symmetric rotation gradient tensor is introduced. The equations of motion of an anisotropic micropolar cylindrical panel element, boundary and initial conditions are obtained from the Ostrogradsky-Hamilton energy principle based on the kinematic hypotheses of S.P.Timoshenko. Geometric nonlinearity is taken into account according to the model of Theodor von Karman. The panel consists of n families of densely spaced edges of the same material, which makes it possible to use the continuum model of G.I.Pshenichnov. Thus, the original mesh panel is replaced by a continuous layer. The material of the cylindrical panel is orthotropic, elastic and obeys Hooke's law. A dissipative mechanical system is considered.
By the method of establishing (Feodosiev’s method), a study was made of the influence of the geometry of the grid on the behavior of a flat flexible cylindrical panel S.P. Timoshenko rigidly clamped at the ends of a plane flexible plan under the influence of a normal distributed load. For this, the system of partial differential equations with respect to spatial coordinates is reduced to the system of ordinary differential equations by the Bubnov – Galerkin method in higher approximations. The obtained Cauchy problem in time is solved by the fourth-order Runge-Kutta methods. It has been established that as the distance between the edges of the family of rods decreases, the deflections of the mesh plate approach the continuous deflections. It is shown that taking into account the orthotropic properties of the material leads to an increase in deflections, that is, to a decrease in the bending stiffness of the mesh panel. The question of the influence of the angle of inclination of the ribs on the behavior of the mesh cylindrical panel is investigated. It is shown that when the angle of inclination of the edges of the lattice approaches the coordinate axes, the stiffness of the panel increases.

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