Computation of dynamic deflection in thin elastic beam via symmetries

The deflection profiles governed by Euler Bernoulli's fourth-order equations under varied applied loads are investigated in this research. This study provides essential insights for engineers designing aircraft components, bridges, and similar structures, ensuring system safety and efficiency. The investigation emphasizes critical factors such as amplitude and frequency, load history, and material properties. Initially, conservation laws of the equations with applied loads are derived by expressing them in the Euler-Lagrange form, where the resultant conservation laws satisfy the divergence expression. The association between symmetries and conservation laws is demonstrated, followed by the application of double reduction theory, which reduces both the variables and the order of the equation. Graphical representations of the outcomes illustrate the impact of load variations on the beam's deflection profiles. These visual aids facilitate a deeper understanding of the influence of different loading conditions. A comparison between varying loads is presented, showcasing the impact of these variations on structural behavior. The findings are crucial for enhancing structural design and ensuring safety under varied loading conditions, showcasing the novelties in the analytical approach and the practical applications of the derived results.