Numerical Analysis of the Vibration of Slender Beams

Abstract

The use of rotating beam-like structures is very common in engineering systems including power generation machines, automotive systems, aircraft structures, energy harvesting systems, and many more. The dynamic response analysis of these systems is of utmost importance for proper prediction of their performance and life. Among different outcomes of the dynamic response analysis, one of the major parameters is the frequency of the free vibration. Because of realistic loads and couplings between different degrees-of-freedom of motion, for majority of the cases, there is no analytical solution available. Hence, the governing equations need to be solved numerically. There are several numerical approaches available to solve for the coupled frequency of free vibration of rotating beams. In this paper, an overview of the different numerical methods is presented for coupled, free vibration analysis of slender, rotating beam-like structures. Three different degrees-of-freedom including out-of-plane bending, in-plane bending, and torsional deformations are considered for the most general case. At first, the eigen value problem of the coupled, mathematical model of free vibration of the beam is presented analytically. Following that, the use of different numerical methods is presented with relevant examples and corresponding beam parameters. Finally, the implementation of the finite difference method is presented to compare the corresponding results with that obtained by other methods.