Numerical analysis and energy decay of Timoshenko linear beam system

Abstract This paper presents a comprehensive study of the Timoshenko linear beam model, with a focus on the numerical approximation and energy decay properties of the system. We consider a system of partial differential equations that describes the dynamics of a Timoshenko beam, incorporating mass, inertia, and damping effects. The model was subject to clamped and free boundary conditions, and the initial conditions were specified. We propose a finite element method combined with an implicit Euler scheme to approximate the solution of the system. The discrete energy associated with the numerical scheme was analyzed, and we prove that it decays over time under certain conditions. Numerical simulations were conducted to validate the theoretical results and demonstrate the effectiveness of the proposed method in capturing the energy decay behavior of the Timoshenko beam.