A Wave-Based Analytical Solution to Free Vibrations in a Combined Euler–Bernoulli Beam/Frame and a Two-Degree-of-Freedom Spring–Mass System

Abstract

In this paper, natural frequencies and modeshapes of a transversely vibrating Euler–Bernoulli beam carrying a discrete two-degree-of-freedom (2DOF) spring–mass system are obtained from a wave vibration point of view in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external forces applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam and 2DOF spring–mass system, the vibrating discrete spring–mass system injects waves into the distributed beam through the spring forces at the two spring attached points. Assembling these wave relations in the beam provides an analytical solution to vibrations of the combined system. Accuracy of the proposed wave analysis approach is validated through comparisons to available results. This wave-based approach is further extended to analyze vibrations in a planar portal frame that carries a discrete 2DOF spring–mass system, where in addition to the transverse motion, the axial motion must be included due to the coupling effect at the angled joint of the frame. The wave vibration approach is seen to provide a systematic and concise technique for solving vibration problems in combined distributed and discrete systems.