Stability analysis of piezoelectric beams

Abstract

Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract-Piezoelectric materials are used in many engineering application. When modeling piezoelectric materials the standard assumption is that the electromagnetic field which is used to actuate the piezoelectric material is quasi static. In this paper we show that although the assumption of a quasi static electrical field is valid when one is interested in simulating a piezoelectric material, this assumption renders the system non stabilizable in terms of control. We also show that this issue is caused by the assumption of a quasi static electrical field and therefore can be avoided by modeling a dynamical electrical field.

On control of voltage-actuated piezoelectric beam: A Krasovskii passivity-based approach

“…obtained by the Legendre transformation on the Lagrangian (12). In (19), the non-energetic coupling terms are absent as expected.…”

“…We use, to the best of our knowledge, a new treatment of the electromagnetic domain that does not require the use of a gauge function. We show that the new model is the more extensive version which includes the fully electromagnetic coupling of the quasi-static electromagnetic model in [19] and [8]. We consider both the Euler-Bernoulli and Timoshenko beam theories for the mechanical domain and investigate the well-posedness of the derived piezoelectric composite models.…”

“…The use of the combined Lagrangian makes the derivation of the fully dynamic electromagnetic current-controlled models (18) and (20) possible. Furthermore, a physical interpretation can be given to the coupling components present in the Lagrangian which are not present in the total energies (19) and (21), through the existence of so-called traditors [20]. In the case of the novel current actuated piezoelectric composites, we are dealing with a gyrator type traditor with gyrating constant γ present in the combined Lagrangian L k and also in ( 16) and (17), used for deriving the mathematical models.…”

“…One system is derived as a purely current-actuated system with a quasi-static electric field assumption through interconnection, and the other is a current-through-the-boundary system with a fully dynamic electromagnetic field assumption. The approximations of the fully dynamic piezoelectric beam [8], using the mixed finite element method [18], has been shown in [19] to satisfy a necessary condition for asymptotic stabilizability, whereas the quasi-static systems lack this condition.…”

“…The purely current actuated piezoelectric fully dynamical models from [16] and [15] are derived with the use of the electric and magnetic potentials, related to the electric field by use of Gauss's magnetic law (5d). The purely current actuated quasi-static piezoelectric model from [8] and [19] is derived by imposing an algebraic constraint on the electric field, reducing the electromagnetic coupling to the quasi-static electric field assumption using different quantities to express the reduced electromagnetic domain. The coupled dynamics are obtained through the passive interconnection of the quasi-static dynamical equation and the mechanical beam equations in a similar fashion as the fully dynamical current-through-the-boundary model in [8], which uses electric flux and charge to describe the electromagnetic dynamics.…”