Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability Results for Novel Serially-connected Magnetizable Piezoelectric and Elastic Smart-System Designs

Mohammad Akil, Serge Nicaise, Ahmet Özkan Özer, Virginie Régnier

Abstract

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected Elastic-Piezoelectric-Elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected Piezoelectric-Elastic design with a local damping acting on the elastic part only. Unlike the existing literature, piezoelectric layers are considered magnetizable, and therefore, a fully-dynamic PDE model, retaining interactions of electromagnetic fields (due to Maxwell's equations) with the mechanical vibrations, is considered. The design (i) is shown to have exponentially stable solutions. However, the nature of the stability of solutions of the design (ii), whether it is polynomial or exponential, is dependent entirely upon the arithmetic nature of a quotient involving all physical parameters. Furthermore, a polynomial decay rate is provided in terms of a measure of irrationality of the quotient. Note that this type of result is totally new (see Theorem 3.6 and Condition $\rm{\mathbf{(H_{Pol})}}$). The main tool used throughout the paper is the multipliers technique which requires an adaptive selection of cut-off functions together with a particular attention to the sharpness of the estimates to optimize the results.

Stability Results for Novel Serially-connected Magnetizable Piezoelectric and Elastic Smart-System Designs

Abstract

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected Elastic-Piezoelectric-Elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected Piezoelectric-Elastic design with a local damping acting on the elastic part only. Unlike the existing literature, piezoelectric layers are considered magnetizable, and therefore, a fully-dynamic PDE model, retaining interactions of electromagnetic fields (due to Maxwell's equations) with the mechanical vibrations, is considered. The design (i) is shown to have exponentially stable solutions. However, the nature of the stability of solutions of the design (ii), whether it is polynomial or exponential, is dependent entirely upon the arithmetic nature of a quotient involving all physical parameters. Furthermore, a polynomial decay rate is provided in terms of a measure of irrationality of the quotient. Note that this type of result is totally new (see Theorem 3.6 and Condition ). The main tool used throughout the paper is the multipliers technique which requires an adaptive selection of cut-off functions together with a particular attention to the sharpness of the estimates to optimize the results.
Paper Structure (12 sections, 25 theorems, 239 equations, 2 figures, 3 tables)

This paper contains 12 sections, 25 theorems, 239 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Contributions and Main Results
  3. Stability results for the \ref{['EPE']} system
  4. Well-Posedness
  5. Strong Stability
  6. Exponential Stability
  7. Stability results for the system \ref{['PE']}
  8. Well-Posedness
  9. Strong Stability
  10. Exponential and Polynomial Stability Results
  11. Illustration of the hypothesis $\rm{\mathbf{(H_{Pol})}}$
  12. Conclusions and open problems

Key Result

Theorem 1.1

If LD-P holds, the $C_0-$semigroup of contractions $(e^{tA_{EPE}})_{t\geq 0}$ is exponentially stable, i.e. there exists $M\geq 1$ and $\omega >0$ such that where $\mathcal{H}$ and $\mathcal{A}_{EPE}$ are defined in defH and defAEPE, respectively.

Figures (2)

  • Figure 1: Serially-connected Elastic-Piezoelectric-Elastic transmission system clamped at both ends. The piezoelectric material itself is an elastic material covered by electrodes at their top and bottom surfaces, and connected to an external electric circuit. As the elastic layers stretches or shrinks, the piezoelectric beam stretches or shrinks as well, and therefore, charges separate and line up in the vertical direction, and electric field (voltage) is induced in the electrodes. The overall motions on the system are considered to be only longitudinal.
  • Figure 2: Serially connected Elastic-Piezoelectric transmission line clamped at both ends.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 23 more