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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 two bending control parameters. [ABSTRACT FROM AUTHOR]

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term g(x)f(t) on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of f(t) that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of f(t) such that the voltage output is equal to some given function v(t) (with g(x) being given as well). The third control problem deals with an explicit construction of both the force profile, g(x), and the control, f(t), which generate the desired voltage output v(t). Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems. [ABSTRACT FROM AUTHOR]

The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter (⁠|$\alpha ,\beta ,k_1,k_2$|⁠) 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 |$\Re \nu = \alpha + (1-k_1k_2)/\beta$|⁠ ; (ii) when |$\beta = 0$| and the parameter |$K = (1-k_1 k_2)/(k_1+k_2)$| 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 |$\left |n\right |\to \infty$|⁠. [ABSTRACT FROM AUTHOR]

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 non-selfadjoint 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. [ABSTRACT FROM AUTHOR]

The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, L, of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as L on different domains, is introduced. It is proven that each of these operators, as well as L, is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, L, shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The solution of the pole reassignment control problem for a distributed parameter system is presented. The system is a vibrating string with spatially non-homogeneous density and modulus of elasticity coefficients and with both distributed and boundary damping. The string equation and the boundary conditions are reduced to a single linear evolution equation in the Hilbert space of two-component Cauchy data equipped with energy metric. It is the dynamics generator of this equation, whose spectrum is rearranged with a feedback control. The solution is based on the fact that the dynamics generator, its finite-rank perturbations and the adjoints to these operators are Riesz spectral, i.e. their sets of the generalized eigenvectors form (non-orthogonal) Riesz bases in the state space. The approach extends to any system, whose dynamics generator has similar properties. [ABSTRACT FROM PUBLISHER]

Abstract: The paper presents the mathematical results for a double-walled carbon nanotube model with nonconservative boundary conditions, given in the form of two Timoshenko beams coupled through a distributed Van Der Waals force. The system is clamped at the left end and subject to a four-parameter family of dynamical boundary conditions at the right end. We also present the four-branch vibrational spectrum of the system. All the results are purely analytical; they will be complemented by numerical simulations in a forthcoming work. [ABSTRACT FROM AUTHOR]

The paper is devoted to rigorous spectral analysis of recently developed mathematical model of a double-walled carbon nanotube. The model is governed by a system of four partial differential equations describing vibrations of two Timoshenko beams coupled through distributed van der Waals forces. The system is equipped with a four-parameter family of nonconservative boundary conditions. The corresponding initial boundary-value problem has been reduced to an evolution equation in the state space. The dynamics generator is a nonselfadjoint matrix differential operator with purely discrete spectrum. It is shown that the entire spectrum asymptotically splits up into four spectral branches. Asymptotical representation has been derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity. To prove the results, a two-step procedure involving construction of the left and right reflection matrices has been used. [ABSTRACT FROM AUTHOR]

In this paper, the numerical results on two problems originated in aircraft wing modeling have been presented. The first problem is concerned with the approximation to the set of the aeroelastic modes, which are the eigenvalues of a certain boundary-value problem. The affirmative answer is given to the following question: can the leading asymptotical terms in the analytical formulas be used as reasonably accurate description of the aeroelastic modes? The positive answer means that these leading terms can be used by engineers for practical calculations. The second problem is concerned with the flutter phenomena in aircraft wings in a subsonic, incompressible, inviscid air flow. It has been shown numerically that there exists a pair of the aeroelastic modes whose behavior depends on a speed of an air flow. Namely, when the speed increases, the distance between the modes tends to zero, and at some speed that can be treated as the flutter speed these two modes merge into one double mode. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a one-parameter family of the nonself-adjoint operators, which are the dynamics generators for systems governed by the wave equations containing dissipative terms. The equations contain viscous damping terms and are equipped with the boundary conditions involving an arbitrary complex parameter. In the current engineering literature, this type of boundary condition is used to model the action of smart materials (self-sensing/self-straining actuators). In the previous research of the first writer, the aforementioned dynamics generators have been studied analytically and precise asymptotic formulas for the eigenvalues have been derived (the asymptotic when the number of the eigenvalues tends to infinity). The goal of the present paper is to demonstrate that the analytic formulas are not only important theoretically, but also extremely efficient practically. Namely, we show that the leading terms in the asymptotic formulas approximate the actual eigenvalues with excellent accuracy. To justify the results, we use two methods, i.e., the Newton method and the Tchebychev method. First, Newton’s method is applied to the characteristic equation using asymptotic formulas as initial guesses to find the eigenvalues. The convergence of Newton’s method is improved by modifying the asymptotic formula. Second, we use Tchebychev discretization to circumvent the nonlinear characteristic equation and to obtain a finite-dimensional generalized eigenvalue problem that approximates the infinite-dimensional one. Finally, to solve the generalized eigenvalue problem, we use the QT algorithm. [ABSTRACT FROM AUTHOR]