Search

The Euler-Bernoulli beam model with nonconservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and offdiagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped-free self-adjoint problem have been established. [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]

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: ( a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; ( b) IBVP for small vibrations of an ideal filament with a one-parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non-singular and singular; ( c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; ( d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; ( e) IBVP for a coupled Euler-Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); ( f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double-walled carbon nanotube. Problems [ABSTRACT FROM AUTHOR]

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]

Abstract: In this paper, we present a recently developed mathematical model for a short double-wall carbon nanotube. The model is governed by a system of two coupled hyperbolic equations and is reduced to an evolution equation. This equation defines a dissipative semi-group. We show that the semi-group generator is an unbounded nonselfadjoint operator with compact resolvent. Moreover, this operator is a relatively compact perturbation of a certain selfadjoint operator. [Copyright &y& Elsevier]

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]

This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled hyperbolic partial differential equations and a four-parameter family of the boundary conditions modeling the action of self-straining actuators. The aforementioned system of equations of motion together with the boundary conditions is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of boundary parameters, the dynamics generator has a compact inverse. If all four boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonself-adjoint operator in the energy space. We present two main results in this research. As the first one, we calculate the spectral asymptotics of the dynamics generator. We find that the complex spectrum lies in a strip parallel to the real axis, and is asymptotically close to the axis. The latter fact means that the system is stable, but is not uniformly stable. As the second main result, we prove that the set of the root vectors of the dynamics generator forms a Riesz basis in the energy space. The results obtained in the present paper allow us to conclude that the dynamics generator is a Riesz spectral operator in the sense of N. Dunford. [ABSTRACT FROM AUTHOR]

Abstract: This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of self-straining actuators. The aforementioned equations of motion together with a four-parameter family of boundary conditions form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of the boundary parameters, the dynamics generator has a compact inverse. If all boundary parameters are not purely imaginary numbers, then the dynamics generator is a non-self-adjoint operator in the energy space. We calculate the spectral asymptotics of the dynamics generator. We find that the spectrum lies in a strip parallel to the horizontal axis, and is asymptotically close to the horizontal axis––thus the system is stable, but is not uniformly stable. We have also found that the set of the eigenfunctions of the dynamics generator forms an unconditional basis (the Riesz basis) in the state space of the system. The Riesz basis property allows us to solve explicitly boundary and/or distributed controllability problems via the spectral decomposition method. [Copyright &y& Elsevier]

The present paper is devoted to the asymptotic and spectral analysis of a system of coupled Euler-Bernoulli and Timoshenko beams. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The above equations of motion form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. This is a dynamics generator of the semigroup which is our main object of interest in the present paper. We prove that for each set of boundary parameters, the dynamics generator has a compact inverse and this inverse operator belongs to class \documentclass{article}\pagestyle{empty}\usepackage{amsfonts}\begin{document}$\mathfrak{S}_p$\end{document} of compact operators with p > 1. We also show that if both boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonselfadjoint operator in the energy space. However, its inverse operator is a finite-rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof of the fact that the root vectors of the dynamics generator form a complete and minimal set in the energy space. We will use the spectral results in our forthcoming papers to prove that the dynamics generator of the system is a Riesz spectral operator in the sense of Dunford and to use the latter fact for the solution of several boundary and distributed controllability problems via the spectral decomposition method. [ABSTRACT FROM AUTHOR]