A Fully Galerkin Method for the Recovery of Stiffness and Damping Parameters in Euler-Bernoulli Beam Models

Abstract

In the modeling and control of large flexible structures, one is often required to numerically recover one or more material parameters given data measurements at various points. Although these structures are in general very complex, in many cases the essential features can be developed by considering a fixed Euler-Bernoulli beam which is assumed to have KelvinVoigt damping. In this paper, a fully Sinc-Galerkin method is presented for the numerical recovery of the stiffness parameter EI and the damping parameter cDI in the state space model

$$\begin{array}{*{20}{c}} {\mathcal{L}(EI,{c_D}I)u = f(x,t),}&{0 < x < 1, t > 0} \\ {u(0,t) = u(1,t) = 0,}&{t > 0} \\ {\frac{{\partial u}}{{\partial x}}(0,t) = \frac{{\partial u}}{{\partial x}}(1,t) = 0,}&{t > 0} \\ {u(x,0) = \frac{{\partial u}}{{\partial t}}(x,0) = 0,}&{0 \leqslant x \leqslant 1} \end{array}$$

(1.1)

with

$$L\left( {E\operatorname{I} ,c_D I} \right)u \equiv \frac{{\partial ^2 u}} {{\partial t^2 }} + \frac{{\partial ^2 }} {{\partial x^2 }}\left( {EI\left( x \right)\frac{{\partial ^2 u}} {{\partial x^2 }} + c_D I\left( x \right)\frac{{\partial ^3 u}} {{\partial x^2 \partial t}}} \right),$$

, given measurements of the data at the points $\left\{ {\left( {x_p ,t_q } \right)} \right\}\begin{array}{*{20}c} {q = 1, \ldots ,n_q } \\ {p = 1, \ldots ,n_p } \\ \end{array}$ in the domain (0,1) × ℝ⁺. From physical considerations, it is reasonable to let the admissible parameter set Q be defined by

$$Q = \left\{ {(EI,{c_D}I) \in \prod\limits_{k = 1}^2 {{H^2}(0,1):EI(x) \geqslant E{I_0} > 0,{c_D}I(x) \geqslant 0} } \right\}$$

(see [5]). With this definition, the existence of a unique soLution u to the forward problem can be obtained on any fixed time interval [0,τ],τ > 0, for f sufficiently smooth.