Uniqueness results for the Timoshenko beam model and identification of forces

We study an inverse problem for the Timoshenko beam model, which describes the transverse displacement u and cross-sectional rotation φ of an elastic beam, accounting for shear and rotary inertia. For a beam of length $ L > 0 $, with Young's modulus E, shear modulus G, density ρ, cross-sectional area A, moment of inertia I, and shear correction factor κ, the system under a source g(t)(f₁(x), f₂(x)) reads:$ {equation} \{ {aligned} {} & ρ A ∂^2 u/∂ t^2 - ∂/∂ ξ[AGκ(∂ u/∂ ξ - φ)] = g(t) f_1, \\ & ρ I ∂^2 φ/∂ t^2 - ∂/∂ ξ(EI∂ φ/∂ ξ) - κ AG(∂ u/∂ ξ - φ) = g(t) f_2. \ {aligned} . {equation} $We prove that, although the model involves both $ u $ and $ φ $, for known $ g ∈ C^1 $ the observation of $ u(t, x) $ on $ (0, T]×Ω_w $, where $ ∅ ≠ Ω_w ⊂ (0, L) $ is an open set, uniquely determines $ (f_1, f_2)∈ H⁻¹(Ω)^2 $ – an advantage in practice, for rotation is very difficult to measure. Numerical results show that, while the Timoshenko model is more accurate, the simpler Euler-Bernoulli model still yields satisfactory reconstructions beyond its formal range of validity.