Optimal resonant control of flexible structures

When introducing a resonant controller for a particular vibration mode in a structure this mode splits into two. A design principle is developed for resonant control based on equal damping of these two modes. First the design principle is developed for control of a system with a single degree of freedom, and then it is extended to multi-mode structures. A root locus analysis of the controlled single-mode structure identifies the equal modal damping property as a condition on the linear and cubic terms of the characteristic equation. Particular solutions for filtered displacement feedback and filtered acceleration feedback are developed by combining the root locus analysis with optimal properties of the displacement amplification frequency curve. The results are then extended to multi-mode structures by including a quasi-static representation of the background modes in the equations for the damped mode. Applications to multi-degree-of-freedom systems are illustrated by idealized models of a piezoelectric sensor–actuator device on a beam and an accelerometer–actuator device on a cable. In both cases near-ideal response characteristics are obtained, when including the quasi-static correction of the modal properties.