Feedback Sensor Design for a Cantilevered Three-Layer Sandwich Beam
A visual sensor-design experiment for a layered beam clamped at one end and free at the other.
Overview
This Demonstration models a cantilevered three-layer sandwich beam whose tip-velocity measurement is fed back to a boundary actuator. The live controls choose the layer count and solver; vary the initial form, mesh size, damping, position, velocity, and time; and adjust the three layer thicknesses, elastic constants, Poisson ratios, and core shear modulus.
Mathematical model
With $w(x,t)$ denoting bending and $v(x,t)$ the core shear angle, the Mead–Marcus equations are coupled through the core. At the free tip, $w_{xxx}(L,t)-d_3w_t(L,t)$ supplies the velocity-feedback damping, while the order-reduced finite-difference discretization preserves uniform observability as $h\to0$ without the high-frequency filter required by standard schemes:
\[\begin{aligned} w_{tt} + w_{xxxx} - B v_x &= 0, \\ -C v_{xx} + P v + B w_{xxx} &= 0. \end{aligned}\]Research project
This Demonstration is part of Observability-Preserving Discretizations, specifically its order-reduced sensor design for cantilevered Mead–Marcus sandwich beams. It illustrates how a discretization can be built around the sensing and stabilization properties needed by the controller.
Reference
Aydin, Ahmet Kaan, Matthew Poynter, and Ahmet Özkan Özer. 2022. Feedback Sensor Design for a Cantilevered Three-Layer Sandwich Beam. Wolfram Demonstrations Project.