Determining Frequencies of Free Longitudinal Vibrations of Rods by Analytical and Numerical Methods

Abstract

The longitudinal vibrations of a steel rod of constant cross-section along the length are considered. The left end of the rod is pinched, a concentrated mass is attached to the right end. The mathematical model consists of a hyperbolic partial differential equation and boundary conditions. The D'Alembert's principle is used. Free oscillations are undamped and harmonic without initial conditions. The purpose of solving the problem is to determine the frequencies of free oscillations. Analytical and numerical-graphical methods of solving the problem are used. In the first case, the oscillation frequency is determined from the transcendental equation. In the second case, the basic equation and boundary conditions are replaced by a system of algebraic equations. The required oscillation frequencies are defined as the eigenvalues of a square matrix. At the final stage of determining the frequencies of free oscillations, a numerical-graphical method is used, implemented in the environment of the Matlab computing complex. A concrete example of the solution is given. Conclusions are drawn.