Axial Free Vibration of Euler-Bernoulli Beams via Ritz Method

This study used analytical and numerical methods to investigate Euler-Bernoulli beams’ free axial vibration behavior with homogeneous and linearly elastic properties. The equations derived within the framework of Hamilton’s principle were simplified through a non-dimensionalization approach and the system parameters were consolidated into a single dimensionless stiffness ratio. The analytical solution enabled the computation of natural frequencies by determining the roots of a transcendental equation. The approximate solutions obtained via the Ritz method were compared with the analytical results to assess the method’s convergence properties. The results indicate that increasing the number of parameters significantly enhances the agreement of the approximate solutions derived from the Ritz method with the analytical results, thereby validating the method’s applicability and effectiveness in structural dynamics analyses.