Identification of an unknown shear force in a cantilever Euler–Bernoulli beam from measured boundary bending moment

Abstract

Abstract An inverse problem of identifying an unknown shear force g ⁢ ( t ) {g(t)} on the inaccessible boundary x = l {x=l} in a system governed by the general form Euler–Bernoulli beam equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , l ) × ( 0 , T ) \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}=0,\quad(x,t)\in(0,l)\times(0,T) subject to the boundary conditions u ⁢ ( 0 , t ) = u x ⁢ ( 0 , t ) = 0 , u x ⁢ x ⁢ ( l , t ) | x = l = 0 , - ( r ⁢ ( x ) ⁢ u x ⁢ x ⁢ ( x , t ) ) x | x = l = g ⁢ ( t ) , u(0,t)=u_{x}(0,t)=0,\quad u_{xx}(l,t)|_{x=l}=0,\quad-(r(x)u_{xx}(x,t))_{x}|_{x% =l}=g(t), is studied. The bending moment 𝙼 ⁢ ( t ) :- - r ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ ( 0 , t ) {\mathtt{M}(t)\coloneq-r(0)u_{xx}(0,t)} given at the accessible boundary x = 0 {x=0} is assumed to be a measured output. The Neumann-to-Neumann operator Φ [ ⋅ ] : 𝒢 ⊂ H p ( 0 , l ) ↦ L 2 ( 0 , T ) , ( Φ g ) ( t ) :- - r ( 0 ) u x ⁢ x ( 0 , t g ) \Phi[\,{\cdot}\,]\colon\mathcal{G}\subset H^{p}(0,l)\mapsto L^{2}(0,T),\quad(% \Phi g)(t)\coloneq-r(0)u_{xx}(0,t_{g}) corresponding to this inverse problem is shown to be compact ( p = 3 {p=3} ) and Lipschitz continuous ( p = 2 {p=2} ). These properties allow us to prove the existence of a solution of the minimization problem for the Tikhonov functional J ⁢ ( g ) :- ∥ Φ ⁢ g - 𝙼 ∥ L 2 ⁢ ( 0 , T ) 2 {J(g)\coloneq\lVert\Phi g-\mathtt{M}\rVert^{2}_{L^{2}(0,T)}} . It is proved that this functional is Fréchet differentiable. Furthermore, an explicit formula for the Fréchet gradient of this functional is derived by making use of the unique solution to corresponding adjoint problem. A numerical method based on Hermitian finite elements and conjugate gradient algorithm is developed for the solution of the inverse boundary value problem. Numerical examples with random noisy measured outputs are presented to illustrate the validity and effectiveness of the proposed approach.