Consideration of supports stiffness at torsional and longitudinal vibrations of the rod

Abstract

The study of the impact of the stiffness of the supports on the value of the first natural frequency of torsional and longitudinal free vibrations of the spring-hinged rod was carried out. In both cases of vibrations the mathematical formulation of the problem of determining the first natural frequency of vibrations reduces to compiling and solving the corresponding differential equation under given boundary conditions. The solution of this complex mathematical problem is performed by the numerical method in a specially created program for many combinations of supports stiffness. To obtain an analytical solution, a quadratic approximation of the obtained values was performed, which made it possible to obtain simple dependencies convenient for engineering applied calculations. The quadratic approximation also allowed solving the inverse problem of determining the required stiffness of supports for the required value of the first natural frequency of vibrations. The obtained solutions were compared with known solutions, which showed that the approximation error is less than 5% if the stiffness of the rod supports of the same order. The obtained results can be used in engineering applied calculations of natural frequencies of torsional and longitudinal vibrations of rods with elastic supports.