Poster Title

Creating a computational tool to simulate vibration control for piezoelectric devices

Grade Level at Time of Presentation

Junior

Major

Mathematics

Institution

Western Kentucky University

KY House District #

CD2

KY Senate District #

CD2

Department

Mathematics

Abstract

Piezoelectric materials have the unique ability to convert electrical energy to mechanical vibrations and vice versa. This project takes a stab to develop a reliable computational tool to simulate the vibration control of a novel “partial differential equation” model for a piezoelectric device, which is designed by integrating electric conducting piezoelectric layers constraining a viscoelastic layer to provide an active and lightweight intelligent structure. Controlling unwanted vibrations on piezoelectric devices (or harvesting energy from ambient vibrations) through piezoelectric layers has been the major focus in cutting-edge engineering applications such as ultrasonic welders and inchworms. The corresponding mathematical models for piezoelectric devices are either heuristic or mathematically oversimplified differential equations. Moreover, their “unjustified” approximated reductions consider only the first several vibrations on these devices. In this project, a novel partial differential equation model, accounting for all vibrational modes, is analyzed to provide new insights for a cost-efficient sensor feedback design. Therefore, the sensor feedback signals are not allowed to be contaminated by the residual modes. Our primary goal is to develop reproducible computational tools by an emerging stable approximation technique, so-called filtered Finite Difference Method, which is proved to provide faster and reliable computation. Filtering in the approximation is necessary since the spurious vibrations, due to the blind application of the Finite Difference Method, provide a false stability result. To see the efficiency of the algorithm, we compare the approximation to the one obtained by the Finite Element Method based on the Galerkin's approximation, which is a common technique being used in the engineering literature.

The mathematical techniques and computational tools developed in this project are essential to provide new insights into the active controlling of piezoelectric devices. Improving the efficiency of active controlling enables us to take better advantage of piezoelectric technology change since one-time design and fabrication may be unavoidable for many applications such as cardiac pacemakers or NASA/commercially-operated inflatable space antennas. Our state-of-the-art partial differential equation model and its stable approximations will be adaptable for a large class of piezoelectric devices.

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Creating a computational tool to simulate vibration control for piezoelectric devices

Piezoelectric materials have the unique ability to convert electrical energy to mechanical vibrations and vice versa. This project takes a stab to develop a reliable computational tool to simulate the vibration control of a novel “partial differential equation” model for a piezoelectric device, which is designed by integrating electric conducting piezoelectric layers constraining a viscoelastic layer to provide an active and lightweight intelligent structure. Controlling unwanted vibrations on piezoelectric devices (or harvesting energy from ambient vibrations) through piezoelectric layers has been the major focus in cutting-edge engineering applications such as ultrasonic welders and inchworms. The corresponding mathematical models for piezoelectric devices are either heuristic or mathematically oversimplified differential equations. Moreover, their “unjustified” approximated reductions consider only the first several vibrations on these devices. In this project, a novel partial differential equation model, accounting for all vibrational modes, is analyzed to provide new insights for a cost-efficient sensor feedback design. Therefore, the sensor feedback signals are not allowed to be contaminated by the residual modes. Our primary goal is to develop reproducible computational tools by an emerging stable approximation technique, so-called filtered Finite Difference Method, which is proved to provide faster and reliable computation. Filtering in the approximation is necessary since the spurious vibrations, due to the blind application of the Finite Difference Method, provide a false stability result. To see the efficiency of the algorithm, we compare the approximation to the one obtained by the Finite Element Method based on the Galerkin's approximation, which is a common technique being used in the engineering literature.

The mathematical techniques and computational tools developed in this project are essential to provide new insights into the active controlling of piezoelectric devices. Improving the efficiency of active controlling enables us to take better advantage of piezoelectric technology change since one-time design and fabrication may be unavoidable for many applications such as cardiac pacemakers or NASA/commercially-operated inflatable space antennas. Our state-of-the-art partial differential equation model and its stable approximations will be adaptable for a large class of piezoelectric devices.