Identification of an unknown shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection

Abstract

In this paper, a novel mathematical model and new approach is proposed for identification of an unknown shear force in a system governed by the general form Euler–Bernoulli beam equation , subject to the boundary conditions u(0,t) = ux(0,t) = 0, , , from available boundary observation (measured output data), namely, the measured deflection at x = l. The approach is based on weak solution theory for PDEs, Tikhonov regularization combined with the adjoint method. A uniqueness result for the problem under consideration is proved. The Neumann-to-Dirichlet operator corresponding to the inverse problem is introduced. It is shown that this operator is injective, compact and Lipschitz continuous. The last property allows us to prove an existence of a quasi-solution of the inverse problem. Fréchet differentiability of the Tikhonov functional is also proved. In the case when Tr = 0, an implicit formula for the Fréchet gradient of this functional is derived by making use of the unique solution to corresponding adjoint problem. Furthermore, a class of admissible shear forces in which the Fréchet gradient of the Tikhonov functional is Lipschitz continuous, is derived. Numerical examples with random noisy measured outputs are presented to illustrate the validity and effectiveness of the proposed approach.

Parametric inverse impulse response based on reduced order modeling and randomized excitations

“…Moreover, note that Eq. (11) makes the (physically reasonable) hypothesis that h À1 j ðxÞ is well defined for all x in the range of interest [52].…”

Inverse Problems of Identifying the Unknown Transverse Shear Force in the Euler-Bernoulli Beam with Kelvin-Voigt Damping

“…We study the inverse problems (1)-( 2) and ( 1), (8) as a minimization problems for the Tikhonov functionals J 1α (g) and J 2α (g) on the set G 1 , G 3 respectively. In the absence of the internal damping term (κ(x)u xxt ) xx in (1), the inverse problems of (1) with measurements ( 2) and ( 8) were studied respectively in [26] and [27].…”

“…Next, let us review some of the recent papers on the Euler-Bernoulli equation with external damping. In the paper [26], the authors determines the unknown transverse shear force by using measured boundary deflection u(ℓ, t), and in the article [27], they consider the same inverse problem based on measured bending moment−r(0)u xx (0, t). When the temporal load G(t) = 1 and G(t) = e −ηt , η > 0 the effect of the damping parameter in the unique determination of spatial load in the Euler-Bernoulli beam equation from final time measured data has been explored in [2] by applying the singular value decomposition.…”

“…Indeed, for the existence and uniqueness of the solution to the direct problem (1), we need to develop appropriate identities to handle these types of mixed boundary conditions for deriving priori estimates for the direct problem and proving the regularity of solutions. This makes the current paper different from [26], [27] where only external damping µ(x)u t is considered, and so the balancing effect of the Kelvin-Voigt damping term on the boundary conditions is also not needed on those papers. On the other hand, Kelvin-Voigt damping also has some sort of regularizing effect in proving the solutions of the direct problem.…”

“…On the other hand, Kelvin-Voigt damping also has some sort of regularizing effect in proving the solutions of the direct problem. For instance, in [26], [27], the authors require higher regularity like g ∈ H 2 (0, T ) to prove u t ∈ L 2 (0, T ; V 2 1 (0, ℓ)), u tt ∈ L 2 (0, T ; L 2 (0, ℓ)), whereas in this paper we prove those estimates with g ∈ H 1 (0, T ) by coupling with appropriate identities and Sobolev embedding theorems. We also prove the existence of solutions to the inverse problems (1)-( 2) and ( 1), (8) when the transverse shear force g(t) belongs to the admissible inputs G 1 ⊂ H 1 (0, T ).…”