Good Practices in Finite Element Method with a Frequency Analysis Example

Abstract

The finite element method (FEM) allows engineers to solve different types of problems (solid mechanics, heat transfer, vibration, electromagnetic, acoustic, etc.) and is nowadays often included in the curricula of undergrad engineering programs. This method consists of discretizing a continuum into smaller elements which properties are defined in matrix form. Then the element matrices are used to assemble the global matrices which represent the properties of the whole structure. Frequency analysis is a very relevant topic during the design of new products and it is usually carried out during the design of any component of a structure subjected to dynamic loads. Even though resonance problems have been known for a long time, this problem continues to appear in many structures such as in the London Millennium Bridge in the U.K. in year 2000. The objective of this work is to improve the understanding of both FEM and frequency analysis. Thus, after students carry out the labs included in this work, students must have mastered the basics of the finite element method and have a strong understanding of a frequency analysis (solution of the generalized eigenvalue problem to compute the natural frequencies and modes of vibration of a structure). In this work natural frequencies and modes of vibration of beams are obtained using: a) the Rayleigh-Ritz method; b) implementing a FEM code in matlab; c) Using a commercial FEM code; and d) experimental work. While implementing b) students must use wire elements, shell elements as well as solid elements. This helps students to understand when it may be more convenient to use each type of element. Written reports are used to assess students’ work. The content of the reports include how the FEA results converge as the number of degrees of freedom increases, how stiffness and mass distribution of the beam influence the natural frequencies of the beam. Students must also compare their FEA results using a commercial code with the other types of solutions implemented earlier (Rayleigh-Ritz method, FEM code in Matlab and experimental test). ABET outcomes a, b, g, and k are assessed. Results show that students enjoy working with commercial FEM codes and experimental work. This is shown with clearly higher lab reports’ grades -usually higher than 90%, while the average of exams’ grades were around 75% with a standard deviation around 15%.