Free vibration analysis of a uniform multi-span beam carrying multiple spring–mass systems

The literature regarding the free vibration analysis of single-span beams carrying a number of spring–mass systems is plenty, but that of multi-span beams carrying multiple spring–mass systems is fewer. Thus, this paper aims at determining the “exact” solutions for the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple spring–mass systems. Firstly, the coefficient matrices for an intermediate pinned support, an intermediate spring–mass system, left-end support and right-end support of a uniform beam are derived. Next, the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. In this paper, the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made, thus, the last solutions are the exact ones. The effects of attached spring–mass systems on the free vibration characteristics of the 1–4-span beams are studied.