PSI - Issue 63

Lenka Koubova / Procedia Structural Integrity 63 (2024) 35–42

42

4. Conclusions The structure may vibrate under dynamic load. At certain frequencies, natural/resonant frequencies, it vibrates in so-called mode shapes. Every structure has an infinite number of these mode shapes in which it can vibrate. Any arbitrarily complex vibration of a structure can be decomposed into a possibly infinite series of the structure's mode shapes. This paper presents the numerical solution of natural frequencies and mode shape of any planar bar construction. The solution is based on the method of stiffness constants. The bisection method is used when searching for the zero determinant of the created matrix; thus, we obtain the values of the natural frequencies, and subsequently, the mode shapes of the structure can be determined. The presented procedure was programmed using the commonly available spreadsheet editor Microsoft Excel (using the macro programming language Visual Basic for Applications). A simple beam solution is presented and compared with another solution. Next, the mode shapes of the parabolic arc are solved. The procedure presented in the paper can be used for the solution of any planar bar structure, possibly also for other structures for which we have already determined the stiffness matrix and the mass matrix. Acknowledgements Financial support from VSB-Technical University of Ostrava by means of the Czech Ministry of Education, Youth, and Sports through the institutional support for conceptual development of science, research, and innovations is gratefully acknowledged. References Attia, S., Mohareb, M., Martens, M., Adeeb, S., 2024. Finite element analysis for free vibration of pipes conveying fluids–physical significance of complex mode shapes. Thin-Walled Structures, Vol. 200, ISSN 02638231, https://doi.org/10.1016/j.tws.2024.111894. Brøns, M., Thomsen, J., J., 2019. Experimental testing of Timoshenko predictions of supercritical natural frequencies and mode shapes for free free beams, Journal of Sound and Vibration, Vol. 459, ISSN 0022460X, https://doi.org/10.1016/j.jsv.2019.114856. Moheuddin, M., Uddin, J., Kowsher, M., 2019. A New Study to Find Out the Best Computational Method for Solving the Nonlinear Equation, Applied Mathematics and Sciences an International Journal (MathSJ), Vol. 6, ISSN 23496223, https://doi.org/10.5121/mathsj.2019.6302. Nachum, S., Altus, E., 2007. Natural frequencies and mode shapes of deterministic and stochastic non-homogeneous rods and beams, Journal of Sound and Vibration, Vol. 302, ISSN 0022460X, https://doi.org/10.1016/j.jsv.2006.12.021. Nicoletti, R., 2020. On the natural frequencies of simply supported beams curved in mode shapes, Journal of Sound and Vibration, Vol. 485, ISSN 0022460X, https://doi.org/10.1016/j.jsv.2020.115597. Silva, G. A. L., Nicoletti, R., 2017. Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration, Vol. 397, ISSN 0022460X, https://doi.org/10.1016/j.jsv.2017.02.053.