Análise modal teórica de vigas: estudo dos modelos contínuo e discreto de Ritz

Abstract

In structural engineering, beams are widely used, whether in buildings, bridges, and even aircraft structures. When these elements are very slender and flexible, they become more susceptible to mechanical vibrations. Therefore, a more accurate study of these elements becomes essential to evaluate their performance. Faced with this problem, theoretical modal analysis emerges as a tool to determine modes of vibration and natural frequencies of vibration. Thus, this work’s main objective is to study the discrete Ritz and continuous models, aiming at the theoretical modal analysis of Euler Bernoulli beams. In the continuous beam model, the differential equation of motion was obtained using Hamilton’s principle, which was solved analytically using the variable separation technique, thus obtaining exact answers to the free vibration problem. The same beam model was discretized using the Ritz method, obtaining the natural frequencies and approximate vibrating modes for the first four vibrating modes. In order to evaluate the two mathematical models, a theoretical modal analysis of the beams tested by Yoshioka (2016) was carried out. The comparative analysis between mathematical and experimental methods demonstrated a relative error of less than 3,2% for natural frequencies. The vibration modes were compared using MAC (Modal Assurance Criterion), obtaining correlations greater than 99% between the three analysis methods.