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

Abstract

In this paper, a novel mathematical model and new approach is proposed for the inverse sourceproblem of recovering the unknown spatial-temporal load F(x,t) in the simply supported non-homogeneousEuler-Bernoulli beam governed by the equation p(x)u_tt + μ(x)u_t + (r(x)u_xx)_xx+k_Lu = 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 boundaryθ(t): = u_x(0, t), t∈ (0,T),and also the deflection  (t):=u(x, T), x∈ (0, ℓ)at the final time T > 0, are givenas measured outputs. The Tikhonov functional is introduce to reformulate the inverse problem as a minimization problem for the Tikhonov functional. Anexplicit gradient formula for this functional is derived. Based on this formula a conjugate gradient algorithm isdeveloped for the considered inverse problem. This algorithm allows to recover the unknown spatial-temporalload with high accuracy, from noise free as well as from random noisy measured outputs.