Shaped Modal Sensors for Linear Stochastic Beams

Modal sensors and actuators using distributed piezoelectric material have a wide range of applications, for example in vibration control and piezoelectric transformers. The design of these transducers usually ignores any uncertainty and variability in the host structure, which can have a significant effect on their performance. This article investigates the design of shaped piezoelectric sensors for beam structures that are robust with respect to uncertainties in the system. The modal transducers are defined using a discrete approximation to the equations of motion for linear stochastic systems and their shapes are represented using the underlying finite element shape functions. The optimal shape design has been coupled with the stochastic finite element method to consider parametric uncertainty described using random fields, using a first-order perturbation-based approach to obtain the second-order covariance of the modal matrix. The numerical results for linear elastic beam structures showed that the shape of the sensors of the stochastic system can differ significantly from the corresponding deterministic system. However, sensors with shapes designed using a smoothness criterion also perform very well for structures with uncertainty.