Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams

Abstract

Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piezoelectric beam with and without the electromagnetic interactions (fully dynamic and electrostatic). This modeling strategy is extended for the three-layer piezoelectric smart composites by adopting the two widely-accepted sandwich beam theories. For both single-layer and three-layer models, the resulting infinite dimensional equations of motion can be formulated in the state-space form. It is observed that the fully dynamic nonlinear models are unbounded boundary control systems (same in linear theory) ẏ(t) = (A + N)y(t) + Bu(t), the electrostatic nonlinear models are unbounded bilinear control systems ẏ(t) = (A + N)y(t) + (B 1 + B 2 y)u(t) in sharp contrast to the linear theory. Finally, we propose B * −type feedback controllers to stabilize the single piezoelectric beam models. The filtered semi-discrete Finite Difference approximations is adopted to illustrate the findings.

Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results

“…On the other hand, consideration of a remedial damping injection (by a mechanical feedback controller) to the piezoelectric layer of the fully dynamic models is under consideration. Numerical results confirm that mechanical feedback controllers have a stronger effect to suppress vibrations [27]. Together with the effect of shear damping, the investigation of the optimal decay rates to tune up the damping parameters and the feedback gains is the topic of future research.…”

“…Let (F, G) ∈ H 2 L (0, L) × H 1 (0, L). If we multiply (27) by F and (28) by G, then integrate by parts, we obtain…”

“…where c k ij are elastic stiffness coefficients of each layer, and γ ij and ε ij , are piezoelectric and permittivity coefficients for the piezoelectric layer. Refer to [27] for sample piezoelectric constants. The displacement field, strains, and the constitutive relationships for each layer are given in Table 1.…”