Multiple Mode Spatial Vibration Reduction in Flexible Beams Using $H$₂- and $H$_∞-Modified Positive Position Feedback

This paper develops $H$₂ modified positive position feedback (⁠$H$₂-MPPF) and $H$_∞-MPPF controllers for spatial vibration suppression of flexible structures in multimode condition. Resonant vibrations in a clamped–clamped (c–c) and a cantilever beam are aimed to be spatially suppressed using minimum number of piezoelectric patches. These two types of beams are selected since they are more frequently used in macro- and microscale structures. The shape functions of the beams are extracted using the assumed-modes approach. Then, they are implemented in the controller design via spatial $H$₂ and $H$_∞ norms. The controllers are then evaluated experimentally. Vibrations of multiple points on the beams are concurrently measured using a laser vibrometer. According to the results of the c–c beam, vibration amplitude is reduced to less than half for the entire beam using both $H$₂- and $H$_∞-MPPF controllers. For the cantilever beam, vibration amplitude is suppressed to a higher level using the $H$₂-MPPF controller compared to the $H$_∞-MPPF method. Results show that the designed controllers can effectively use one piezoelectric actuator to efficiently perform spatial vibration control on the entire length of the beams with different boundary conditions.