Exact solutions and numerical simulation of longitudinal vibration of the Rayleigh-Love rods with variable cross-sections

Abstract

Exact solutions of equations of longitudinal vibration of conical and exponential rod are analyzed for the Rayleigh-Love model. These solutions are used as reference results for checking accuracy of the method of lines. It is shown that the method of lines generates solutions, which are very close to those that are predicted by the exact theory. It is also shown that the accuracy of the method of lines is improved with increasing the number of intervals on the rod. Reliability of numerical methods is very important for obtaining approximate solutions of physical and technical problems. In the present paper we consider the Rayleigh-Love model of longitudinal vibrations of rods with conical and exponential cross-sections. It is shown that exact solution of the problem of longitudinal vibration of the conical rod is obtained in Legendre spherical functions and the corresponding solution for the rod of exponential cross-section is expressed in the Gauss hypergeometric functions. General solution of these problems is expressed in terms of the Green function. For numerical solution of the problem we use the method of lines. By means of this method the partial differential equations describing the dynamics of the Rayleigh-Love rod are reduced to a system of ordinary differential equations. For checking of accuracy of the numerical solution we chose special initial conditions, namely we assume that initial longitudinal displacements of the rod are proportional to one of eigenfunction of the system and initial velocities are zero. In this case vibrations of every point of the rod are harmonic and their amplitudes are equal to the initial displacements. Periods of these vibrations, obtained by the method of lines, are estimated and compared with the theoretically predicted eigenvalues of the rod, thus giving us estimations of accuracy of the numerical procedures.