MATLAB Implementation of Finite Element Method for Cantilever Beam Analysis

This article presents a comprehensive approach to developing finite element programs using MATLAB, specifically focusing on cantilever beam analysis. The implementation features triangular mesh generation algorithms for domain discretization, which provides flexible geometric modeling capabilities. The computational core employs eigenvalue solvers to determine the natural frequencies and extract the corresponding vibration modes through modal analysis techniques. Key implementation aspects include: - Mesh generation using Delaunay triangulation algorithms - Stiffness matrix assembly through element-by-element computation - Mass matrix formulation consistent with triangular elements - Eigenvalue extraction using MATLAB's eigs() function for sparse systems - Mode shape visualization through surface plotting and deformation animation The program calculates the fundamental frequency and first three modal patterns, enabling deeper understanding of cantilever beam vibrational characteristics. Results are presented through MATLAB's advanced visualization tools, including deformed shape overlays and animated mode demonstrations. This implementation serves as both an educational resource for finite element concepts and a practical template for structural dynamics analysis. The code structure incorporates modular design principles, separating mesh generation, matrix assembly, eigenvalue computation, and visualization into distinct functional units. This organization facilitates code maintenance and adaptation to different structural configurations. The article demonstrates how MATLAB's computational power combined with finite element methodology can effectively solve complex engineering vibration problems.