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

Abstract

In this paper we analyze the numerical approximation of an active vibration control problem of a Timoshenko beam. In order to avoid locking, we focus on the finite element method used to compute the beam vibration, to minimize it. Optimal order error estimates are obtained for the control variable, which is the amplitude of secondary forces modeled as Dirac's delta distributions. These estimates are valid with constants that do not depend on the thickness of the beam. In order to assess the performance of the method, numerical tests are reported.

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

“…Moreover, λ 0 is positive. In fact, by taking appropriate test functions, it is easy to check that any solution of (19)- (20) satisfies…”

“…The Timoshenko theory to date is one of the most used models to approximate the deformation of a thin or moderately thick elastic beam [5,9,12,18,20,26,30]. It is well understood that standard finite elements applied to this model lead to wrong results when the thickness of the beam is small due to the so called locking phenomenon.…”

Numerical approximation of the LQR problem in a strongly damped wave equation

“…The so-called LQR (Linear-quadratic regulator) problem constitutes a cornerstone of the modern linear control theory. Studied originally in a finitedimensional context, the LQR problem became also a subject of interest in the framework of control theory for partial differential equations (see [2,12]), which are related to several applications, including the control of parabolic systems like the heat equation (see [5]), the active control of noise (see [6,17]), and the active control of flexible structures (see [11,13,18]), among others.…”

Reduced-order LQG control of a Timoshenko beam model

“…This locking-free Galerkin approximation generates a finite-dimensional sequence state space representations (A h , B h , C h ) for which the solution the optimal control problem converge to the solution of the abstract problem (10). This has been extensively discussed in previous works [3,13,9].…”

“…For more details concerning the locking-free discretization of the Timoshenko beam model in the context of optimal control, we refer to [10]. The approximation of the operator B is given by…”