A difference scheme for solving the Timoshenko beam equations with tip body

Abstract

In this article, a Timoshenko beam with tip body and boundary damping is considered. A linearized three-level difference scheme of the Timoshenko beam equations on uniform meshes is derived by the method of reduction of order. The unique solvability, unconditional stability and convergence of the difference scheme are proved. The convergence order in maximum norm is of order two in both space and time. A numerical example is presented to demonstrate the theoretical results.

Uniform stability for the Timoshenko beam with tip load

“…A numerical approach to the problem, using the finite element method, was undertaken by Zietsman et al [29] who from their empirical studies could not conclude uniform stabilization, neither in the case of non-zero rotary inertia, I m , of the tip load, nor when I m is neglected. Their FEM approach was followed most recently by a finite difference approach due to F. Li et al [17] in which unique solvability, unconditional stability (to the initial values and inhomogeneous terms) and convergence of their difference scheme of the problem, which includes the rotary inertia of the tip load, are proved.…”

A numerical solution technique for a Timoshenko beam-column with generalized end conditions

“…In section 2, the difference scheme is derived by the method of reduction of order [6,7]. some numerical results are presented in section 3.…”

Boundary stabilization of a Bresse‐type system

“…Their finite element method approach was followed most recently by a finite difference approach due to Li et al , in which unique solvability, unconditional stability (to the initial values and inhomogeneous terms), and convergence of their difference scheme of the problem, which includes the rotary inertia of the tip load, are proved.…”