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 $\left |n\right |\to \infty$.
Communicated by: P. M. Mariano MSC Classification: 35E15; 47A10; 47A75; 58J45; 74J05The 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, , 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 on different domains, is introduced. It is proven that each of these operators, as well as , 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, , 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. KEYWORDS eigenfunctions, eigenvalues, non-self-adjoint operator, spectral equationThis equation represents a commonly used model for the motion of a straight beam of length L, cross-sectional area A(x), mass density (x), modulus of elasticity of the beam material E(x), and cross-sectional moment of inertia I(x) (EI(x) is the bending stiffness). The model is obtained by using Hooke's law, and the simplifying assumptions that the thickness and width of the beam are small compared with the length and the cross sections of the beam remain plane during deformation. 1,2 We assume that the beam is clamped at the left end, ie, h(0, t) = h x (0, t) = 0 (left-end clamped), 2)Math Meth Appl Sci. 2018;41 4691-4713. wileyonlinelibrary.com/journal/mma
Analytic and numerical results of the Euler-Bernoulli beam model with a twoparameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters (k 1 and k 2 ) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, k 1 , is positive and another one, k 2 , is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k 1 ! 1 þ and k 2 ¼ 0. Third, the existence of double deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.
Asymptotic and spectral results on the initial boundaryvalue 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.
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