Abstract
Free compressed beam transverse vibrations of a variable cross section carrying a distributed mass are considered. The mathematical model of oscillations is presented as a boundary value problem from the basic partial differential equation of the fourth order hyperbolic type in spatial coordinate, the second order in time and boundary conditions. The technical theory of the rods’ bending vibrations, based on the Bernoulli hypothesis about the invariance of flat beam cross-sections, has been used. The task is to determine the eigenvalues, eigenmodes of oscillation and the attenuation coefficient (extended Sturm-Liouville problem). It has been argued that solving the problem by the analytical methods is impractical due to the large transformations and calculations volume. The variables separation methods, finite differences and computer technology have been used. An algorithm for solving the problem, implemented in the Matlab software environment in the form of high-precision graphical and analytical calculations, has been developed. On a concrete example of a beam, verification of the proposed mathematical model has been demonstrated. The practical conclusions have been made.
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