Shaped Modal Sensors for Uncertain Dynamical Systems

Abstract

The idea of using modal sensors and actuators for beam and plate type structures has been a subject of intense interest for many years. Using modal sensors in active control reduces problems of spillover, where high frequency unmodelled modes affect the stability of the closed loop system. This paper is concerned with distributed sensors made of piezoelectric material to measure the response of beam and plate structures. The design of modal sensors for beam structures is well established. For example, a modal sensor for a beam type structure may be obtained by varying the sensor width along the length of the beam. If the sensor covers the whole beam the shape of the sensor may be derived using the mode shape orthogonality property. Friswell considered modal sensors that cover only part of the beam, segmented modal sensors for multiple modes, and the effect of geometric tolerances during manufacture on the quality of the sensors. Friswell parameterized the width of the sensor for beam structures using the finite element method and uses the underlying shape functions to approximate the transducer shape. For plate structures constant thickness sensors are difficult to design, although methods that parameterize sensor boundary, or using topology optimization, have been suggested. Currently modal sensors and actuators are designed using deterministic models of the structure, and assuming that they may be manufactured perfectly. The performance of the transducers is critically dependent on these assumptions and this paper investigates the effect of modelling uncertainty and manufacturing errors. The origin of these uncertainties and their form are discussed, and their effect is modeled and propagated through the system to determine the errors induced on the modal outputs. Simulated examples are used to demonstrate the issues raised.