H1H¹‐Galerkin Mixed Finite Element Method for the Vibration Equation of Damped Beam With Time Fractional

In this paper, the ‐Galerkin mixed finite element method (MFEM) is used to solve the time‐fraction‐order damped beam vibration equations with simple support at both ends. Compared with the standard finite element method (FEM), the ‐Galerkin MFEM can calculate the deflection, bending moment and other parameters of the beam more directly, so it is more suitable for solving the beam vibration equations, which are high‐order partial differential equations. The Caputo fractional derivative is approximated using the L1 formula, and a fully discrete numerical scheme is established by linear backward Euler method. The stability of the proposed scheme, as well as the existence, uniqueness, and convergence of the numerical solution, are rigorously analyzed. Numerical examples are provided to verify the theoretical results, with real‐world beam data used to investigate how variations in the damping coefficient and the order of the fractional derivative affect the beam's vibrational behavior. Numerical simulations demonstrate that the amplitude of the beam decays more rapidly with larger damping coefficients. Moreover, as the order of the fractional derivative increases, the decay rate of the beam vibration first increases and then decreases, while the peak of the curve gradually shifts to the right.