Free and forced vibrations of beams under distributed load

Introduction. Many years of experience in design and operation of structures prove that the reliability of structures cannot be ensured by strength calculations alone. In addition to strength and stiffness of structures, their vibrations are often required to be considered. These types of calculations appear highly comprehensive, involving a large number of different factors to be taken into account. At present, as structures are becoming more and more complex, great attention is paid to seismic design, which is especially relevant for buildings subjected to high seismic loads. Engineering structures experience increased loads associated with the above-mentioned and other reasons. Free and forced vibrations of various elastic structures obtain a significant relevance among researchers as manifested by numerous publications on this issue.

Aim. To present a complete mathematical problem for the two most common methods of end restraint, to determine natural frequency spectrum and eigenforms of beam vibrations.

Materials and methods. The present study involves a beam of variable cross-section made of homogeneous material, subjected to transverse distributed load, which makes bending vibrations. Free and forced vibrations are described by differential equations. The homogeneous equation is solved first, then – the nonhomogeneous equation. The study involves application of D’Alembert’s principle, variable separation method, and test checks.

Results. The authors obtained a fourth-order partial differential equation with constant coefficients and determined a natural frequency spectrum and eigenforms of vibrations. A high accuracy of the obtained results enables the characteristics of free and forced vibrations of beams to be determined in a shorter way with fewer calculations. Notably, the amplitude and eigenform of forced vibrations of beams appear dependent on the proximity of the disturbing frequency to the eigenvalues and the phase change of components in vector disturbance process.

Conclusions. The authors advanced hypotheses about a dependence between the damping and linear viscous friction coefficients as well as about the constancy of damping coefficient for all eigenvalues.

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