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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Free Vibrations of Cantilever Bars with Linear and Nonlinear Variable Cross-Section

Discontinuity, Nonlinearity, and Complexity 6(4) (2017) 489--501 | DOI:10.5890/DNC.2017.12.007

Jacek Jaworski; Olga Szlachetka

Department of Civil Engineering, Faculty of Civil and Environmental Engineering, Warsaw University of Life Sciences - SGGW, Nowoursynowska 166, 02-787 Warsaw, Poland

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The topic of this study is the first mode of natural transverse vibrations of isotropic, homogeneous and elastic bars (columns or beams) with clamped bottom and free head. The columns of the first group are shaped as truncated solid cones or as tubes with linearly variable wall thickness and with different inclination of lateral faces, from cylinder to cone. The columns of the second group were shaped in similar way, but the generatrices of the solids of revolution were curvilinear ?in the shape of a parabola. The first frequency of free vibrations was determined using the Rayleigh method. The deflection line of the column axis during the vibration was assumed in form of the bending line of the column axis subjected to a uniform load. Resulting frequencies (or periods) were compared with these obtained with the use of FEM (ANSYS) and a good compliance of results was observed. As the expression for the energy of an elementary slice of material was integrated over the length of the rod, the formula for the frequency was obtained in form of an integral equation. In some cases an exact solution of integral equation was obtained, however in other cases only a numerical solution was possible.


The authors would like to express their sincere thanks to Jan Grudzińki, Ph.D. Eng., for his help in the FEM calculations in the ANSYS.


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