Stabilization of a translating tensioned beam through a pointwise control force

A spectral analysis determining asymptotically the distribution of eigenvalues of a constrained, translating, tensioned beam in closed form is the subject of this paper. The constraint is modeled by a spring-mass-dashpot subsystem that is located at any position within the span of the beam. It can represent a feedback controller with a collocated sensor and actuator. The necessary and sufficient condition that ensures a uniform stability margin for all the modes of vibration is determined. Influences of system parameters on the distribution of eigenvalues are identified. The analytical predictions are validated by numerical analyses. The constraint location maximizing the stability margin of the distributed model is predicted through a combined analytical and numerical approach. The implications and utility of the results are illustrated. The methodology developed can be extended to predict stability margins and optimize control parameters for controlled translating beams with other types of boundary conditions and controller structures.