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
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Abstract torsion control parameter, while of the β‐branch eigenvalue depends on twobending control parameters. 
The present paper is devoted to mathematical analysis of the model that describes fluid flow moving in a channel with flexible walls, which are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be consumed by the structure (the walls) inducing “traveling wave flutter.” In the problems involving two‐media interactions (fluid/structure), flutter‐like perturbations can occur either in the fluid flowing in the channel with harmonically moving walls, or in the solid structure interacting with the flow. In the present research, it is shown that there are no abrupt (or flutter‐like) changes in the flow velocity profiles. Using the mass conservation law and incompressibility condition, we obtain the initial boundary value problem for the
stream function . The boundary conditions reflect that (i) there is no movement in the vertical direction along the axis of symmetry and (ii) there is no relative movement between the near‐boundary flow and the structure (“no‐slip” condition). The closed form solution is derived for the stream function, which is represented in the form of an infinite functional series. 
null (Ed.)The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a fourparameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial 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 of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.more » « less

Solovjovs, Sergejs (Ed.)In the present paper, we summarize the results of the research devoted to the problem of stability of the fluid flow moving in a channel with flexible walls and interacting with the walls. The walls of the vessel are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be transferred and consumed by the structure (the walls), inducing “traveling wave flutter.” The problem of stability of fluidstructure interaction splits into two parts: (a) stability of fluid flow in the channel with harmonically moving walls and (b) stability of solid structure participating in the energy exchange with the flow. Stability of fluid flow, the main focus of the research, is obtained by solving the initial boundary value problem for the stream function. The main findings of the paper are the following: (i) rigorous formulation of the initial boundary problem for the stream function, ψ x , y , t , induced by the fluidstructure interaction model, which takes into account the axisymmetric pattern of the flow and “noslip” condition near the channel walls; (ii) application of a double integral transformation (the Fourier transformation and Laplace transformation) to both the equation and boundary and initial conditions, which reduces the original partial differential equation to a parameterdependent ordinary differential equation; (iii) derivation of the explicit formula for the Fourier transform of the stream function, ψ ˜ k , y , t ; (iv) evaluation of the inverse Fourier transform of ψ ˜ k , y , t and proving that reconstruction of ψ x , y , t can be obtained through a limiting process in the complex k plane, which allows us to use the Residue theorem and represent the solution in the form of an infinite series of residues. The result of this research is an analytical solution describing blood flowing through a channel with flexible walls that are being perturbed in the form of a traveling wave.more » « less

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 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 on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of that steers an initial state to zero on a time interval [0,
T ]. The second control problem deals with the construction of such that the voltage output is equal to some given function (with being given as well). The third control problem deals with an explicit construction of both the force profile, , and the control, , which generate the desired voltage output . Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.