Free vibrations of variablesection beams taking rotational and frictional forces into account

Introduction. Beams are widely used in the construction industry as load-bearing structures of bridges, overpasses, coverings, slabs, stairs, equipment platforms, etc. In order to fully utilize the bearing capacity of such structures and to reduce the rate of material consumption, beams of variable cross-section along the length can be used. During operation, such structural elements are subjected to various types of vibrations, which determines the relevance of studying various aspects of vibrational motion.

Aim. To apply numerical methods for studying the influence of rotational inertial forces in the presence of viscous frictional forces on free vibrations of variable-section beams. This area of research presents interest, since the calculations are directly related to the determination of frequencies and forms of natural vibrations of structures.

Materials and methods. Free vibrations were described by a homogeneous partial differential equation of hyperbolic type. The methods of variable separation and finite differences were used. A discrete domain in the form of a set of uniform grid nodes and a homogeneous system of algebraic equations were introduced. The system of equations in the matrix-vector form was used.

Results. The spectra of natural frequencies, damping coefficients, and eigenforms of beam vibrations are determined. It is shown that the coefficient matrix has a banded and pentadiagonal form. The matrix elements are functions of the characteristic index. The damping coefficient and the frequency of free vibrations are determined from a system of two nonlinear equations. The solution of the system of equations is found using the method of coordinate descent. An example of calculation of a welded I-beam is considered. Five elements of the spectra of damping coefficients and natural frequencies are calculated.

Conclusions. State-of-the-art MATLAB solvers allows numerical and graphical methods to be combined. In the solved examples, the advantages of these methods were successfully applied to determine the eigenvalues of matrices and eigenfunctions. The high validity and accuracy of the results obtained confirm the simplicity and versatility of the methodology for determining the characteristics of free vibrations of beams of variable cross-section.

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