Distributed transducers, collocation, and smart structural control

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Abstract
Collocated sensors and actuators offer significant robustness and design benefits in distributed parameter vibration control. It is shown that sensors and actuators are collocated in this sense when they are both physically coincident and their spatial derivative orders are equal. These requirements are expressed more generally as an integral equation that constrains the transducers' "modified" spatial distributions. Distributed parameter collocation is demonstrated by a transducer placement study for a nondimensional cantilever beam. The first configuration uses two disthbuted piezoceramic actuators and two accelerometer sensors. This combination typically leads to input/output frequency response functions having slow nonminimum phase transmission zeros and poor singular value frequency response characteristics, with high sensitivity to placement. The problem is exacerbated when the accelerometers and distributed actuators are "collocated". The second configuration uses two distributed piezoelectric actuators, now combined with two distributed strain gage sensors. When these equivalent-order distributed transducers are collocated the resulting frequency response matrices are strictly minimum phase, and have desirable singular value characteristics determined solely by the actuator placement's coupling into the beam's modal curvatures. These characteristics are maintained for small strain sensor apertures centered over larger actuator apertures, including some misalignment, for low frequencies. A generalization is developed to facilitate the collocation of transducers having dissimilar spatial derivative order.

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