The Effect of Rotatory and Lateral Inertia on Flexural Vibration of Prismatic Bars

Abstract

The problem of the flexural vibration of prismatic bars neglecting the rotatory and lateral inertia is treated in various texts on sound and vibrations. The effect of neglecting these two terms introduces no appreciable error when the cross section of the bar is small in comparison to its length. For bars of large cross section the error introduced in neglecting these terms is sufficient to warrant attention. In his text on sound Rayleigh offers an approximate solution introducing a correction term for the rotatory inertia, under certain boundary conditions. In a more recent paper Mason gives an exact solution for the frequency of vibration, where the rotatory and lateral inertia are taken into account. The disadvantage of this solution lies in the fact that his Eq. (14) for the eigenvalues of the frequency is extremely complicated, thereby throwing up a barrier to further analysis. Also no mention is made by previous investigators of the effect of the rotatory and lateral inertia on the node positions or the shape of the deflection curve. It is the purpose of this paper to present a simplified exact solution and to determine the effect of the two terms mentioned on the position of the nodes and the relative amplitude of vibration. Curves applicable to the solution of any type of constant cross section are included.