INFLUENCE OF RIGID-BODY MOTIONS ON FREE VIBRATION CHARACTERISTICS OF A FREE-FREE BEAM CARRYING ARBITRARY CONCENTRATED ELEMENTS

In this paper, a beam without any attachments is called “bare” beam, and the beam attached by any concentrated elements (CEs) is called “loaded” beam. One of the predominant differences between the “constrained bare beam (CBB)”, such as a clamped-free (C-F) beam, and the “unconstrained bare beam (UBB)”, such as a free-free (F-F) beam, is that all natural frequencies of the CBB are greater than zero and associated with the “elastic” vibrations of the CBB, however, the natural frequencies associated with “rigid-body” motions of the UBB are equal to zero and those associated with “elastic” vibrations of the UBB are greater than zero. For a “constrained” beam, the superposition of the lowest ( ) normal mode shapes for the “elastic” vibrations of the CBB and the consideration of effects of the attached CEs can give the lowest natural frequencies and mode shapes of the associated loaded beam with satisfactory accuracy. However, for an “unconstrained” (F-F) beam, the last approach is not available unless the “rigid-body” motions of the UBB are also taken into account. For the last reason, this paper aims at presenting a modified mode-superposition method (MMSM) with natural frequencies and normal mode shapes for both the “rigid-body” motions and the “elastic” vibrations of a F-F bare beam considered, so that the free vibration characteristics of the associated F-F loaded beam (carrying any CEs) can be easily obtained. It was found that all numerical results obtained from the MMSM are in good agreements with those obtained from the finite element method (FEM) or the theory for a single-degree-of-freedom (SDOF) spring-mass system.