Dmitriev

Several various methods could be applied for description of mass-inertia properties of the finite element; each method leads to a different mass matrix. In particular, one could use the method of lumped masses, various shape functions both matching with (consistent) or distinct from shape functions for a stiffness matrix. Other approaches are also possible. The purpose of this work was to analyze some non-trivial approaches to determination of a mass matrix, such as non-standard shape functions, oscillation modes of a bar element, a method of combining different mass matrices with weighting factors. The article compares five different mass matrices by the example of a bar finite element and identifies inherent errors of these matrices obtained during determination of eigenfrequencies of the fixed bar. A system of equations for the finite element method for all mass matrices was solved analytically in the trigonometric series defined at the nodal points. It was shown that calculation of eigenfrequency based on inconsistent mass matrices could provide higher accuracy in comparison with diagonal and consistent matrices.

In this paper the authors propose improved formulas for calculating coefficients of transfer matrix in the presence of damping. The authors consider a classical damping which means the damping matrix is proportional to the stiffness matrix. These formulas are designed for large finite-element models. Preliminary determination of the lower bound for natural frequencies and oscillation modes is expected. Approximate formulas are used only to calculate coefficients of obtained bending modes. In the revised formula contribution of higher vibration tones is taken into account; corrections for both real and imaginary parts of coefficients of the transfer matrix were also found. Contribution of rejected vibration tones is considered to be static, but within a given range of frequencies in the lower part of the spectrum accuracy is quite acceptable, and the effect of correction is significant.

The author proposes an analytical solution of the contact problem of elastic deformations of a thin plate rolled up into a cylinder and placed into a cylindrical cage of the same radius. In this case, the plate is only partially adjacent to the surface of the cylinder, on the other part there is delamination. For linearization of basic correlations it is assumed that the plate deviates from the surface of the cylindrical cavity to a very little degree; therefore, it is possible to use the equilibrium equation, as well as physical and geometric equations of cylindrical shell. It is considered that the shell bending is cylindrical. The rolling of the plate is taken into account as a pre-stressed state. It is supposed that it is a state of pure bending in plane. As a result of solution, the angle of delamination, displacement and internal power factors are determined.