Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

Abstract

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section

“…In previous publications, beams with fractal cross section have been studied, specifically with Sierpinski's carpet-like, where the beam also is a fractal [20]. However, this manuscript is devoted to analyze the response of Euclidean beams with a fractal cross section, whose fractal geometry is Koch snowflake-like [26]. It is well-known that the Hausdorff dimension of the classical Koch snowflake is given by d H = 2 log 2/ log 3.…”

“…Other authors [25] proposed a Koch Snowflake shaped microstrip terahertz antenna for superwideband spatial diversity application and discovered that interport isolation for the MIMO antenna is less than -23 dB across the wideband of operation, whereas the values of the MIMO performance parameters are best. Rostami et al [26] analyzed a Timoshenko beam, also with a Koch snowflake cross-section, using different boundary conditions for variable properties, and it was found out that beams with fractal geometry have the highest natural frequency compared with the ones with Euclidean geometry. It is worth highlighting that natural frequencies (eigenvalues) and the shape mode (eigenfunctions) problem in a fractal region defined by the Koch snowflake geometry were numerically solved in [27,28] In these fractal models, however, there is not a relationship that describes the influence of fractal parameters as the Hausdorff dimension and the iteration numbers of the Koch snowflake .…”