In this paper we develop a nonlinear control strategy based on modal coupling using the center manifold theory. As an example we use the technique for vibration suppression of a flexible beam. The controller in this case is a mass-spring-dashpot mechanism which is free to slide along the beam. The equations of the plant/controller are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes. As a result, the performance of the system can not be improved by linear control theory or by most conventional nonlinear control techniques.
We use the normal forms method to simplify the center manifold equations and derive a relation which includes all system parameters. We then show that there exists a set of optimal controller parameters (feedback gains, controller damping and frequency) which maximized the energy dissipation. Finally we consider the stability and design issues, and use numerical simulation to verify the results.