Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry

Abstract

The aim of this paper is to analyze a mixed finite element method for computing the vibration modes of a Timoshenko curved rod with arbitrary geometry. Optimal order error estimates are proved for displacements, rotations and shear stresses of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are essentially independent of the thickness of the rod, which leads to the conclusion that the method is locking free. Numerical tests are reported in order to assess the performance of the method.

A Locking-Free FEM in Active Vibration Control of a Timoshenko Beam

“…This eigenvalue problem has been recently analyzed in [15], as a particular case of a more general problem, and it is proved that the spectrum consists of a sequence of finite multiplicity real eigenvalues converging to infinite.…”

“…We prove existence and uniqueness of the solution for general terms in the Sobolev space H −1 . Using the results presented in [15], we obtain estimates that do not degenerate with the thickness of the beam. It includes, in addition, local W 2,∞ a priori estimates for the state equations with source terms in L 2 .…”

“…The first mathematical piece of work dealing with numerical locking and how to avoid it is the paper by Arnold [3], in which he proves that locking arises because of the shear term, and proposes and analyzes a locking-free method based on a mixed formulation. Recently, this proposed method has been used and analyzed when it is applied to the problem of free vibrations of a general curved rod (see [15]), which covers the Timoshenko beam case.…”

Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

“…In particular, they showed that the standard inf-sup and ellipticity in the kernel conditions, which ensure convergence for the mixed formulation of source problems, are not enough to attain the same goal in the corresponding eigenvalue problem. Among the existing techniques to solve the vibration problem of Timoshenko beams, we can mention [21] where a mixed formulation in terms of displacement, rotation and shear stress has been proposed and analyzed for Timoshenko rods (which are of course applicable to Timoshenko beams).…”

A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

“…In this paper, we present a rigorous thorough analysis of a low order finite element method to compute the buckling coefficients and modes of a non-homogeneous Timoshenko beam, the method was introduced for source problems on homogeneous beams by Arnold in [1], and was recently analyzed for the vibration problem of a rod in [8] (which covers the vibration problem of the Timoshenko beam).…”