A mathematical model of physically nonlinear torsional vibrations of a circular elastic rod

A mathematical model of non-stationary torsional vibrations of a circular elastic rod is developed taking into account the Kauderer nonlinear law of elasticity. To solve this problem, the nonlinear equation of motion of an elastic body with torsional vibrations of a rod is reduced to two linear Bessel equations (homogeneous and inhomogeneous) in transformations. Considering general solutions of the obtained equations with zero initial and given boundary conditions on the surface of the rod, a refined physically nonlinear equation of torsional vibrations of the rod made of homogeneous and isotropic material is derived. In particular, this equation may be used to obtain some well-known classical oscillation equations. An algorithm is proposed that allows one to determine the stress-strain state of the points along an arbitrary cross-section of the rod in terms of space and time coordinates using the field of the desired functions. Some special cases resulting from the obtained results are analyzed. In particular, by reducing the expressions of Bessel functions in the form of power series to the first few terms, an approximate equation of the circular rod oscillations is derived. Comparative analysis of findings and available data of other authors shows that the obtained equation generalizes the well-known classical linear equation and nonlinear equations of G. Kauderer and Professor I.G. Filippov. Based on the proposed equation and formulas for stresses and displacement, the applied problem of physically nonlinear torsional vibrations of a circular elastic rod under end and surface loads is solved.