Stabilization of a Smart Beam with a Tip Mass
An interactive observer-controller model for a magnetizable piezoelectric beam with a dynamic tip load.
Overview
This Demonstration examines a magnetizable piezoelectric beam with a mass dynamically coupled to its free tip. The live controls step through time, change the tip mass, and tune three controller amplifiers and four observer amplifiers; the gauges show velocity and total-current tracking errors alongside controller velocity and the auxiliary $\eta$-dynamics.
Mathematical model
For displacement $w$ and electrode charge $p$, the beam is represented by coupled mechanical and electromagnetic wave equations. At the tip, the mass contributes the inertial term $m w_{tt}(L,t)$, while voltage and strain feedback provide the boundary control; an observer reconstructs the state from left-end measurements and drives its error toward zero:
\[\begin{bmatrix} \rho & 0 \\ 0 & \mu \end{bmatrix} \begin{bmatrix} w \\ p \end{bmatrix}_{tt} = \begin{bmatrix} \alpha & -\gamma\beta \\ -\gamma\beta & \beta \end{bmatrix} \begin{bmatrix} w \\ p \end{bmatrix}_{xx}.\]Research project
This Demonstration is part of Piezoelectric Beam Stabilization, extending boundary-feedback design to a beam with a dynamic tip load. It shows why a payload changes both the boundary equations and the observer-controller architecture needed for rapid stabilization.
Reference
Walterman, Jacob, Ahmet Kaan Aydin, and Ahmet Özkan Özer. 2024. Stabilization of a Smart Beam with a Tip Mass. Wolfram Demonstrations Project.