Equations of oscillations of rods of variable composition

Abstract

We obtain differential equations for the general case of longitudinal, torsional, and transverse oscillations of rods to some parts of which masses are being added or detached. We solve certain special problems concerning the oscillations of such rods of variable composition. In deriving generalized equations of oscillations of rods of variable composition we employ the assumption of planar sections, the assumption of small deformations, and other customary simplifications. We also employ the simplifying assumption of close action; i.e., we assume that the masses being detached and added interact with the rod only at the instant of direct contact. Forces of internal nonelastic resistance are not taken into account. We assume also that in the undeformed state the elastic axis is rectilinear and that the centers of gravity of cross sections are not displaced from their initial positions relative to the cross sections. There may be a change of mass per unit length of the rod both on account of a change in density as well as on account of a change in area of a cross section, the latter being understood to be the union of the initial area of the cross section and the areas of the parts being added and detached. In addition, with the rod there may be associated particles of variable mass distributed continuously or discretely along the length of the rod. We assume that these particles do not interact among themselves but only with the rod.