Parametric control of vibration in flexible and axially-moving material systems

The purpose of this study is to develop methods of asymptotically stabilizing vibration in flexible and axially-moving material systems with parametric control actuators.;The first contribution of this study is a bound on the response of Hill's equation, the fundamental equation of parametrically excited systems. It is shown that the response of Hill's equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. The exponential decay caused by viscous damping is calculated to balance the maximal exponential growth from the parametric excitation.;The second contribution is a method of controlling vibration in discrete systems, or n modes of a distributed system, with parametric excitation. The excitation is varied according to a quadratic, state feedback, modal control law. To eliminate the requisite full state sensing, an observer is designed that estimates the states from a minimum number of linear measurements. Using the exponential bound on the response of Hill's equation, observer-based feedback is shown to be asymptotically stabilizing for a sufficiently small control bound. Perturbation methods are applied to a controlled mode to predict the effects of the control gains on the transient and forced response of the system. The perturbation results also demonstrate that forced spillover can parametrically destabilize an uncontrolled mode. A minimum modal damping and maximum control bound are calculated that eliminate this instability.;The final contribution is a method of parametrically damping the transverse vibration of a nonconservative, distributed, axially-moving beam. A modified Hamilton's principle is used to generate the nonlinearly coupled, transverse and longitudinal equations of motion. A novel technique produces Lyapunov functionals for the pinned, axially-moving string and the clamped, axially-moving beam. It is shown that an axial damper control law parametrically damps transverse vibration, and that the linearized, longitudinal vibration is also stabilized by an axial damper. Hyperstability theory and a modified Lyapunov functional generalize the axial damper control law to strictly positive real derivative feedback in parallel with proportional feedback. A simulation of six modes of an axially-moving, clamped beam demonstrates asymptotically stable response. (Abstract shortened by UMI.).