For convenience, a beam without any attachments is called “bare” beam, and the beam carrying concentrated elements (CEs) is called “loaded” beam. One of predominant differences between an unconstrained beam (UB) and a constrained beam (CB) is that the free oscillations of an UB consist of the “rigid-body” motions and “elastic” vibrations, while those of a CB consist of the “elastic” vibrations only. In practice, most beams are the CBs, thus, the literature regarding “rigid-body” motions are rare. However, an elastically supported beam is a kind of UB so that its natural frequencies and mode shapes for both the “rigid-body” motions and “elastic” vibrations are required by the engineers. For the last reason, with the effects of rigid-body (heave and pitch) motions considered, this paper presents a modified mode-superposition method (MMSM) to study the topic being not yet to appear in the existing literature: free vibration characteristics of the free–free (F–F) loaded double-tapered beams (DTBs). In which, based on the normal mode shapes for both “rigid-body” motions and “elastic” vibrations of the F–F bare DTB, the partial differential equation of motion for the F–F loaded DTB was transformed into the matrix equation so that the free vibration characteristics of the F–F loaded DTB can be easily obtained. Numerical examples reveal that the MMSM can provide not only the heave and pitch frequencies for the “rigid-body” motions but also the satisfactory natural frequencies and mode shapes for the “elastic” vibrations of the F–F loaded DTB. However, this is not true for the conventional mode-superposition method (CMSM). Good agreements between the numerical results obtained from the MMSM and the corresponding ones obtained from the finite element method (FEM) confirm the reliability of the presented theories and developed computer programs for this paper.
