Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model
Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model
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控制能量收集模型动力学的非自共算子特征值的渐近表示
DOI：
10.1007/s00245-016-9347-3
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发表时间：
2016
影响因子：
1.8
通讯作者：
M. Shubov
中科院分区：
文献类型：
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作者：
M. Shubov
我们考虑一个众所周知的压电式能量收集器模型。收割机被设计成梁，在其顶面上附着一层压电瓷层(单晶片结构)。一对薄的理想导电电极覆盖着压电层的顶面和底面。这些电极连接到阻性负载。该模型由一个由两个方程组成的系统来管理。第一个是梁的横向振动的欧拉-伯努利模型方程，第二个是电路的基尔霍夫定律。由于正、逆压电效应，这两个方程是耦合的。梁方程的边界条件是无固支的。我们将系统表示为Hilbert空间中的单算子发展方程。该系统的动力学生成器是一个具有紧预解的非自伴算子。我们的主要结果是该生成器的特征值的显式渐近公式，即我们对电负载(非短路)系统进行了模态分析。我们证明了谱分裂成一个无限的稳定本征值序列，该序列靠近左半平面上的一条垂直线，并且可能有有限个不稳定本征值。本文是一系列三部作品中的第一部。在第二章中，我们将证明动力学生成元的广义本征向量在能量空间中形成Riesz基(而且是Bari基)。在第三篇文章中，我们将应用前两个结果来控制该模型的问题。
We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.