A comparison of shortening of the projection to axial elasticity

In this paper, we consider the combined bending and axial vibrations of beams and we analyze the magnitudes of the different effects that contribute to the motion. Specifically, we consider the transverse elasticity, the axial elasticity, and the shortening of the projection. The goal of our study is to ascertain the circumstances under which the axial elasticity is larger than the shortening of the projection and vice versa. We show that for the majority of cases the shortening of the projection has a larger amplitude than the axial stretch. We also develop a general closed form solution for the response when a force is applied at a free end.