# IWPDF2023

Damping in Functionally Graded Shafts under Torsion: an Analytical Approach

V. Rizov ∗

Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, Sofia 1046, Bulgaria

∗ v_rizov_fhe@uacg.bg

Keywords: energy damping, torsion, functionally graded material.

The present paper addresses the question for the damping of the energy in functionally graded shafts loaded in cyclic torsion. In particular, a shaft of circular crosssection is researched here. One of the motives for conducting this research is that the previous energy dumping studies deal mainly with beams of rectangular cross-section subjected to bending [1, 2]. Another motive is that the functionally graded materials are commonly used in modern engineering for manufacturing of various components and members of constructions and machines [3]. Very often the functionally graded members and components are subjected to cyclic loading. Under such loading conditions, the question for analysis of dumping is of prime significance. The analysis of dumping is closely related to the problems of safety and reliability of the engineering constructions, machines and equipments. The shaft that is researched in the present paper is rigidly fixed in both ends. A torsion moment is applied near the mid-span. A stress-strain relationship of the type of Ramberg Osgood is used in the analysis of the dumping. The three material parameters of the stress strain relationship are distributed continuously along the cross-section radius since the material is functionally graded in radial direction. The distribution is treated by using exponential functions. The analysis of dumping goes through researching of the parameters of the stressed and strained state of the shaft with considering of the material non-linearity. In the context of analyzing of the stressed and strained state of the shaft, relevant equations of equilibrium are compiled and solved. After that, the distribution of the parameters of the unit damping energy along the radius of the shaft cross-section is derived. The total dumping energy in the shaft is found-out by integrating of unit damping energy. The solution derived offers very good possibilities for obtaining of numerical results with purpose of clarifying how the total dumping energy is influenced by different factors such as distribution of material parameters along the radius of the cross-section, geometry of the functionally graded shaft, boundary conditions and parameters of the external loading. References [1] Dowling, N. (2007). Mechanical Behavior of Materials, Pearson. [2] Rizov, V. I. (2021). On the Analysis of Damping in Elastic-plastic Inhomogeneous Beams. Materials Science Forum, 1046, 59-64. [3] Mahamood, R. C., Akinlabi, E. T. (2017). Functionally Graded Materials, Springer.

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