Designed short-time observations for rotational damping recovery in Timoshenko beams

We study the recovery of spatial rotational damping in a clamped Timoshenko beam. The coefficient belongs to a prescribed bounded finite-dimensional family. One fixed spatial average of rotation is recorded in separately labelled experiments with known initial rotational velocities. We construct these velocities so that the leading coefficient-to-data map is an invertible Gram matrix. Testing the equations against the smooth observation kernel gives a uniform parameter-Lipschitz remainder without differentiating the damping coefficient. A weighted temporal moment then yields global injectivity, Lipschitz stability and a globally convergent projected fixed-point reconstruction for sufficiently short observation times. Upper and deterministic two-point lower bounds give the same noise-amplification order $T^{-5/2}$ for $L^2$ time-history errors. A second sensitivity estimate supplies a separate sufficient condition for strong convexity and global projected-gradient convergence. We derive continuous and exact discrete adjoints, prove finite-time convergence of a conforming finite-element midpoint scheme, and propagate forward error into coefficient error. Reproducible synthetic experiments examine symmetry, gradient accuracy, noise and discretization bias.