Manifold based control for a 1-DOF beam with one-sided spring using feedback linearization

Vibration reduction in a harmonically excited 1-DOF beam with one-sided spring is realized by forcing the system from a long-term stable ½ subharmonic response towards a coexisting unstable harmonic response of lower vibration amplitude using feedback linearization. The control effort can be kept small once the unstable harmonic response is stabilized, because this response is a natural solution of the uncontrolled system. To reduce control effort, the approximated stable manifold is used to determine the desired trajectory for control. The stable manifold is approximated using the stable eigenvectors of the monodromy matrix. Due to the local validity of the approximation, a two-stage control strategy is implemented. In the first stage, the system state is controlled directly towards the unstable harmonic response, to reach the region where the stable manifold can be approximated accurately by the stable eigenvectors. In the second stage, the system state is controlled towards the stable eigenvectors, and evolves towards the unstable harmonic response with hardly any control effort