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

[+] Author and Article Information
Carole Mei

Department of Mechanical Engineering The University of Michigan - Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA

1Corresponding author.

ASME doi:10.1115/1.4039961 History: Received July 18, 2017; Revised April 09, 2018


In this paper, natural frequencies and mode shapes of a transversely vibrating Euler-Bernoulli beam carrying a discrete two degrees-of-freedom 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 two degrees-of-freedom 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 two degree-of-freedom 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 combine distributed and discrete systems.

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