 Award ID(s):
 1810826
 NSFPAR ID:
 10298774
 Date Published:
 Journal Name:
 Asymptotic Analysis
 ISSN:
 09217134
 Page Range / eLocation ID:
 1 to 38
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully nondissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$plane given by the equation $\Re \nu = \alpha + (1k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1k_1 k_2)/(k_1+k_2)$ is such that $\left K\right \neq 1$ then the following relations are valid: $\Re (\nu _n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left n\right \to \infty$; (iii) when $\beta =0$, $K = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left n\right \to \infty$; (iv) when $\beta =0$, $K = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left n\right ) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left n\right \to \infty$.more » « less

Abstract Asymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on the
torsion control parameter, while of the β‐branch eigenvalue depends on twobending control parameters. 
Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents Kirchhoff’s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a nonselfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order O ( 1 n ) , where n is the number of an eigenvalue. These formulas are derived for the model with variable physical parameters. The second set of the asymptotic formulas has remainder terms of the order O ( 1 n 2 ) , and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the α branch eigenvalues and the θ branch eigenvalues. These eigenvalues being multiplied by “i” produce the set of the vibrational modes of the system. The α branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the θ branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester.more » « less

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[2,2]$ and $H_0$ does not have eigenvalues embedded into $(2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } nV(n)=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+n}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $V(n)\leq \frac{h(n)}{1+n}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

Abstract In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ_{0}to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ_{0}of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ_{0}, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ_{0}, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ_{1}of the boundary
. Under a certain geometric assumption on Γ_{1}, an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.