Computational Analysis of Cantilever Beam Frequencies and Vibration Modes

To calculate the natural frequencies and vibration modes of a cantilever beam, engineers must account for multiple physical parameters including beam length, cross-sectional dimensions, and material properties such as Young's modulus and density. The vibration characteristics are significantly influenced by boundary conditions, particularly the fixed-free configuration typical of cantilever structures. Implementation typically involves solving the Euler-Bernoulli beam equation using numerical methods like the Finite Element Method (FEM), where the global stiffness and mass matrices are assembled from element matrices. Key computational steps include eigenvalue extraction using algorithms like the QR iteration method to obtain natural frequencies (eigenvalues) and mode shapes (eigenvectors). The code implementation would typically involve defining material properties, discretizing the beam into elements, applying fixed boundary conditions by constraining nodal displacements, and solving the generalized eigenvalue problem [K]{φ} = ω²[M]{φ}. Additional considerations include handling different loading conditions through modal superposition techniques and validating results against analytical solutions for simple cases.