Beck's rod with shear,compressibility and pulsating force

In-plane motion of an elastic rod, subjected to a compressive follower force is analyzed. Equations of motions are derived for the case when deformations are not small. It is assumed that the compressive force has a periodic component, so that a parametric instability is possible. The stability bound is estimated analytically and determined numerically. In deriving the differential equations of motion a generalized constitutive equation taking into account compressibility of the rod axis and the influence of shear stresses is used.