Explicit Formulations of Mass and Stiffness Matrices for Vibration Analysis of Tapered Elements

Abstract

This paper presents explicit formulations to calculate the mass and stiffness matrices used in the finite element method for vibration analysis in rod, beam, and shaft tapered elements. The cross-sections of these elements are considered to vary linearly. The analyses of two types of cross-sections are carried out; the circular cross-section and the rectangular cross-section. The shape functions are formulated for each type of elements. The presented formulations are straightforward to use in a piecewise form. As an acceptable alternative, the obtained expressions are benchmarked by replacing the tapered member with a sufficient number of rigidly connected uniform elements. The comparison is made between the obtained modal parameters, mainly the natural frequencies of the analyzed element. The obtained results are accurate and confirm the reliability of the presented formulations.