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 $ \varphi $ 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 $ \rho $, cross-sectional area $ A $, moment of inertia $ I $, and shear correction factor $ \kappa $, the system under a source $ g(t)(f_1(x), f_2(x)) $ reads:

$ \begin{equation} \left\{ \begin{aligned} {} & \rho A \frac{\partial^2 u}{\partial t^2} - \frac{\partial}{\partial \xi}\left[AG\kappa\left(\frac{\partial u}{\partial \xi} - \varphi\right)\right] = g(t) f_1, \nonumber \\ & \rho I \frac{\partial^2 \varphi}{\partial t^2} - \frac{\partial}{\partial \xi}\left(EI\frac{\partial \varphi}{\partial \xi}\right) - \kappa AG\left(\frac{\partial u}{\partial \xi} - \varphi\right) = g(t) f_2. \nonumber\ \end{aligned} \right. \end{equation} $

We prove that, although the model involves both $ u $ and $ \varphi $, for known $ g \in \mathcal{C}^1 $ the observation of $ u(t, x) $ on $ (0, T]\times\Omega_w $, where $ \emptyset \neq \Omega_w \subset (0, L) $ is an open set, uniquely determines $ (f_1, f_2)\in H^{-1}(\Omega)^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.