Dynamic versus quasi-static response of a cantilevered beam rotated harmonically
We investigate a cantilevered elastic beam subjected to harmonic rotational motion. In the rotating frame, the beam experiences centrifugal and Euler fictitious forces, with negligible Coriolis effects. We validate a reduced-order \textit{elastica} model through precision experiments on slender beams rotating with a controlled sinusoidal angular velocity. Systematically exploring the parameter space, we identify regimes where inertial effects are negligible, enabling a quasi-static treatment despite harmonic driving. We characterize the transition to dynamic response using two dimensionless parameters, the Euler and centrifugal numbers, which compare centrifugal and Euler forces to bending forces. Counterintuitively, the quasi-static regime expands as rotational speed increases: faster rotation produces less dynamic response. The critical Euler number separating these regimes remains constant at low centrifugal numbers but follows square-root scaling at higher rotation rates, a transition driven by centrifugal stiffening. Our results establish the conditions under which quasi-static approximations remain valid for rotating flexible beams under harmonic driving.