Free vibration analysis of a free-free Timoshenko beam carrying multiple concentrated elements with effect of rigid-body motions considered

A modified mode-superposition method (MMSM) was presented in this paper to solve the free vibration problem of the "unconstrained" beams that the conventional mode- superposition method (CMSM) cannot easily tackle. For convenience, a beam without any attachments is called the "bare" beam, and the beam carrying any concentrated elements (CEs) is called the "loaded" beam, in this paper. Furthermore, the mode-superposition method (MSM) with only the "elastic" modes of the bare beam considered is called the CMSM, and the MSM with both the "elastic" and the "rigid-body" modes considered is called the MMSM. From the existing literature, one finds that the CMSM is one of the effective approaches for the free vibration analysis of a "constrained" bare beam (such as the C-C, P-P, or C-F beam) carrying various CEs, where "C, P and F" denote the abbreviations of "clamped, pinned and free", respectively. However, the CMSM is not available for that of an "unconstrained" bare beam (such as the F-F beam) carrying various CEs. For this reason, this paper presented a MMSM to solve the title problem so that one can easily obtain the natural frequencies and mode shapes for both the "rigid-body" motions and the "elastic" vibrations of the F-F loaded Timoshenko beam. The main difference between a "constrained" bare beam and an "unconstrained" one is that the free vibration responses of latter consist of the "rigid-body" motions and those of the former do not, so that the coupling effect between the "rigid-body" motions and the "elastic" vibrations" of the F-F loaded beam are not considered by using the CMSM. To confirm the correctness of the presented theory and the developed computer program for this paper, all numerical results obtained from the MMSM are compared with those obtained from the finite element method (FEM) and good agreement is achieved. Numerical examples reveal that the CMSM can provide neither any information regarding the "rigid-body" motions nor the accurate natural frequencies and mode shapes for the "elastic" vibrations of a F-F loaded beam, and all the above-mentioned drawbacks of the CMSM have been improved by the presented MMSM. (C) 2018 Elsevier Ltd. All rights reserved.