Modeling and semigroup formulation of charge or current-controlled active constrained layer (ACL) beams; electrostatic, quasi-static, and fully-dynamic assumptions

TL;DR

This paper develops a comprehensive mathematical model for three-layer ACL beams with piezoelectric, stiff, and viscoelastic layers, considering electrostatic, quasi-static, and dynamic effects, and analyzes charge and current control cases.

Contribution

It introduces two PDE models for charge and current-controlled ACL beams, incorporating magnetic effects and semigroup formulations for well-posedness analysis.

Findings

Two PDE models for ACL beams under different control modes.

Semigroup formulations establish well-posedness.

Inclusion of magnetic effects in dynamic assumptions.

Abstract

A three-layer active constrained layer (ACL) beam model, consisting of a piezoelectric elastic layer, a stiff layer, and a constrained viscoelastic layer, is obtained for cantilevered boundary conditions by using the reduced Rao-Nakra sandwich beam assumptions through a consistent variational approach. The Rao-Nakra sandwich beam assumptions keeps the longitudinal and rotational inertia terms. We consider electrostatic, quasi-static and fully dynamic assumptions due to Maxwell's equations. For that reason, we first include all magnetic effects for the piezoelectric layer. Two PDE models are obtained; one for the charge-controlled case and one for the current-controlled case. These two cases are considered separately since the underlying control operators are very different in nature. For both cases, the semigroup formulations are presented, and the corresponding Cauchy problems are shown to be well- posed in the natural energy space.