Boundary-Feedback Control of Vibrations on a String with and without Filtering
An interactive comparison of boundary damping, filtering, and order-reduced numerical models for a vibrating string.
Overview
This Demonstration follows a string clamped at one end and damped at the other. The live controls choose ParametricNDSolve or NDSolve, finite differences, finite elements, or order reduction; select the initial condition and node count; and tune the viscous, filtering, and boundary-damping gains, end time, and normal-mode displacement and velocity.
Mathematical model
The string displacement $w(x,t)$ satisfies a boundary-controlled wave equation. Here $k_1$ is distributed viscous damping and $k_3$ is the boundary-feedback gain; filtered finite-difference and finite-element schemes add viscosity proportional to $h^2w_{xxt}$, whereas the order-reduced method remains stable without that added filter:
\[\begin{aligned} w_{tt} + k_1 w_t - w_{xx} &= 0, \\ w(0,t) &= 0, \\ w_x(L,t) + k_3 w_t(L,t) &= 0. \end{aligned}\]Research project
This Demonstration is part of Observability-Preserving Discretizations. It provides a simple model of the high-frequency observability problem later studied for more strongly coupled beam systems, showing why numerical convergence alone is not enough for reliable boundary control.
Reference
Poynter, Matthew, Logan Stewart, Ahmet Kaan Aydin, and Ahmet Özkan Özer. 2022. Boundary-Feedback Control of Vibrations on a String with and without Filtering. Wolfram Demonstrations Project.