Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs

Abstract

Abstract In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ⁢ ( x ) > 0 \rho(x)>0 and flexural rigidity r ⁢ ( x ) > 0 r(x)>0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = 0 \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}=0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) (x,t)\in(0,\ell)\times(0,T) , subject to boundary conditions u ⁢ ( 0 , t ) = u x ⁢ ( 0 , t ) = 0 u(0,t)=u_{x}(0,t)=0 , u x ⁢ x ⁢ ( ℓ , t ) = 0 u_{xx}(\ell,t)=0 , - ( r ⁢ ( x ) ⁢ u x ⁢ x ⁢ ( x , t ) ) x | x = ℓ = g ⁢ ( t ) -(r(x)u_{xx}(x,t))_{x}|_{x=\ell}=g(t) , from the available measured boundary deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) \nu(t):=u(\ell,t) and rotation θ ⁢ ( t ) := u x ⁢ ( ℓ , t ) \theta(t):=u_{x}(\ell,t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 L^{2} -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam.