Dynamics of a Longitudinal Piezoelectric Beam
An interactive view of coupled mechanical motion, electrical response, sensing, and actuation.
Overview
This Demonstration visualizes the longitudinal motion of a piezoelectric beam and its coupled electrical response at the electrodes. The live controls choose initial position and velocity profiles, step through time, set the initial compression or extension, and tune the feedback-sensor amplifier and viscous damping; the gauges track voltage, total energy, sensor output, and actuator response.
Mathematical model
Writing $\mathbf{u}=(v,p)^{\mathsf T}$ for longitudinal displacement and electrode charge, a strongly coupled piezoelectric model has a mass matrix on the left and a stiffness-coupling matrix on the right. The off-diagonal terms express how strain changes the electrical state and voltage-driven actuation changes the mechanical state, so feedback must reduce the coupled energy rather than treating the two wave types independently:
\[\begin{bmatrix} \rho & 0 \\ 0 & \mu \end{bmatrix} \mathbf{u}_{tt} = \begin{bmatrix} \alpha & -\gamma\beta \\ -\gamma\beta & \beta \end{bmatrix} \mathbf{u}_{xx}.\]Research project
This Demonstration is part of Piezoelectric Beam Stabilization, which develops boundary-feedback amplifiers for rapid, robust energy decay. It provides the basic dynamic picture behind the project’s analysis of mechanical and electrical feedback channels.
Reference
Walterman, Jacob, Ahmet Kaan Aydin, Samuel Leveridge, and Ahmet Özkan Özer. 2023. Dynamics of a Longitudinal Piezoelectric Beam. Wolfram Demonstrations Project.