Method for reconstructing the distribution of unknown spatio-temporal loads in a structure based on viscoelasticity in the Euler-Bernoulli beam coupling

In this paper, a novel mathematical model and new approach is proposed for the inverse source problem of recovering the unknown spatial-temporal load F(x,t) in the simply supported non-homogeneous Euler-Bernoulli beam governed by the equation p (x) utt + μ(x) ut + (r(x) uxx)xx + kLu = F(x,t), (x,t) ∈ (0,ℓ) × (0, T), resting on a viscoelastic foundation, is studied. It is assumed that the rotation at the left boundary θ(t): = ux(0, t), t ∈ (0, T), and also the deflection u_Т (t):= u (x, T), x ∈ (0, ℓ) at the final time T > 0, are given as measured outputs. The Tikhonov functional J(F)≔(1/2)||θ-u_x (0,∙) 〖||〗_(L^2 (0,T))^2+(1/2)||u_T-u(∙,T) 〖||〗_(L^2 (0,l))^2 is introduce to reformulate the inverse problem as a minimization problem for the Tikhonov functional. An explicit gradient formula for this functional is derived. Based on this formula a conjugate gradient algorithm is developed for the considered inverse problem. This algorithm allows to recover the unknown spatial-temporal load with high accuracy, from noise free as well as from random noisy measured outputs.