Spatially Convolving Wave Propagation Sensors for Structural Control-Part I: Analytical Development for One-Dimensional Structures

A novel method for sensing actual disturbance information which can propagate along one-dimensional structural waveguides is developed. The motivation for this work arises from the inability to realize most active wave control designs using output information from physical measurements such as deflection, slope, curvature and internal shear force. Often the use of actual physical measurements leads to instability and suboptimal performance of certain active wave control loops. Thus, there is a desire to develop wave-propagation filters which can extract magnitude and direction of structural disturbances along dispersive and non-dispersive members. In this work a distributed wave sensing scheme is reported which exploits the characterization of 1-dimensional structural motion in terms of traveling waves. This form of the solution is then combined with distributed sensors to convolve future and past measurements into a single temporal signal to overcome the problem of causality and spatial aliasing inherent in discrete sensor approaches. Thus, by imposing specific shapes to spatially convolving sensors, it is possible to combine the output of these sensors with point measurements to observe directional propagating wave components. Because this method requires some approximation of the spatial domain, there will be errors due to spatial discretization and truncation. This work addresses these issues and presents analytical results for one-dimensional structural elements.