Forced Longitudinal Oscillations of a Rod with a Mass at the End

Abstract

Forced longitudinal vibrations of a steel rod of constant cross-section along its length are considered. The left end of the rod is pinched, a concentrated mass is attached to the right end. The source of the oscillations is the harmonic force acting along the axis. The mathematical model consists of a hyperbolic partial differential equation and boundary conditions. The D'Alembert's principle is used. Forced oscillations are undamped and harmonic without initial conditions. The amplitudes of forced oscillations are determined, their dependences on the frequency of the driving force are analyzed. Analytical and numerical-graphical methods of solving the problem are used. In the first case, by traditional methods, the problem is reduced to a transcendental equation, from which the amplitude of the oscillations is determined. In the second case, the basic equation and boundary conditions are replaced by a system of algebraic equations. The desired oscillation amplitudes are determined from the matrix-vector equation. At the final stage of determining the amplitudes, a numerical-graphical method is used, implemented in the environment of the Matlab computing complex. Examples are given. Conclusions are drawn.