A Finite Difference Scheme For Solving The Timoshenko Beam Equations With Boundary Feedback

In recent years, the boundary control problem of flexible structure has attracted much attention with the rapid development of high technology such as space science and flexible robots, and a series of important results have been obtained. It is well known that the transversal vibration of an elastic beam can be described by the Euler-Bernoulli beam equation if the cross-section dimension of the beam is negligible in comparison with its length. If the cross-section dimension is not negligible, then it is necessary to consider the effect of the rotatory inertia, and if the deflection due to shear is not negligible either, then the transversal vibration must be described by the so-called Timoshenko beam equation:where ρ, I_ ρ, EI, K, l are mass density, moment of mass inertia, rigidity coefficient, shear modulus of elasticity, and length of the beam, respectively. We consider the general case and denote by w(x,t) and φ(x,t) the transversal displacement and rotational angle of the beam, respectively. In general, the Timoshenko beam model is more complicated, but more precise. The analytic solution of the Timoshenko beam equation can be difficult to obtain in general due to the complicated boundary conditions. In this paper, we present a finite difference scheme for the Timoshenko beam by the method of reduction of order and prove the unique solvability, unconditional stability and second order convergence in L_∞ norm by the discrete energy method. Numerical results demonstrate the theoretical results.