Torsional vibrations of a rod with an axial geometric in homogeneity

The dynamics torsional vibrations of a rod with an axial torsional rigidity inhomogeneity. Rods of different configurations are widely used for simulating stress-strain state in case of static and dynamic load to the objects of mechanical engineering, construction, biomechanics, etc. This work’s objective is creating of a general approach to building mathematical models of torsional vibrations of rods of variable section. As an object we have considered an elastic rod, the torsional rigidity of which changes according to the power law from the longitudinal coordinate. The dynamic process is described by a wave equation, and is solved using the Fourier method. To make the solving of the boundary-value problems more convenient, special functions based on recurrence relations for the Bessel functions are introduced. Taking into account the orthogonal property of weighted eigenfunctions, an expression for the norm square is obtained. As an example a case of vibrations is considered with sudden application of load to one end of the rod, with the other end of the rod being rigidly fixed. The free end is supposed to be under local inertial load. Expressions were obtained for the torsion angles and torques in the cross-sections of the rod. A comparison was performed for the obtained results of calculation in relative values with a simplified single-mass model of the weightless rod.