Eigenvalues in problem of free vibrations of rods with variable cross-section

Abstract

The article considers free longitudinal vibrations of a steel rod of variable cross-section along the length. The left end of the rod is pinched, and the mass and the coil spring are concentrated at the right end. The basic equation of the mathematical model of vibrations is compiled using the d'Alembert principle. The result is a model including a hyperbolic partial differential equation and boundary conditions corresponding to the support conditions for the left and right ends of the rod. Free vibrations are undamped and harmonic without initial conditions. The purpose of solving the problem is to determine the frequencies of free vibrations. The use of the finite difference method for solving the problem is substantiated. The domain of continuous variation of the displacement function argument along the rod axis is replaced by a uniform grid function. Therefore, the mathematical model turns into a system of algebraic equations. The desired vibration frequencies are determined as eigenvalues of a square matrix. At the final stage of determining the free vibrations frequencies, a numerical-graphical method implemented in the environment of the Matlab computer complex is used. A specific example is given. The analysis results in the conclusions important for practical applications.