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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]

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. [ABSTRACT FROM AUTHOR]

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 four-parameter (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. [ABSTRACT FROM AUTHOR]

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 fluid-structure 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 fluid-structure interaction model, which takes into account the axisymmetric pattern of the flow and "no-slip" 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 parameter-dependent 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. [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 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 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]

The present paper is the second one in a series of three works dealing with a mathematical model for an electro-mechanical energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face. A pair of conductive electrodes, covering the top and bottom faces of the piezoceramic layer, are connected to a resistive load. The model is governed by a system of two differential equations. The first of them is the equation of the Euler–Bernoulli beam model subject to actions of an external damping and of an external force. The second equation represents the Kirchhoff’s law for the electric circuit. Both equations are coupled by means of the direct and converse piezoelectric effects. The system is represented as a single operator evolution equation in a Hilbert space (the state space). The dynamics generator of this system is a non-selfadjoint operator with a compact resolvent. In the first paper, the asymptotic formulas for the eigenvalues of the dynamics generator have been derived. In the present paper, the following results have been shown. 1) The set of the generalized eigenvectors of the dynamics generator is complete and minimal in the state space. 2) The set of normalized generalized eigenvectors forms a Riesz basis, which is quadratically close to an orthonormal basis (a Bari basis). 3) The exact controllability problem has been solved via the spectral decomposition method. [ABSTRACT FROM AUTHOR]

The performance of ultra long range artillery projectiles is analyzed. In the past ultra long range artillery was useless due to low accuracy, but now with relatively inexpensive guidance available for long range projectiles, such artillery is very promising. Given the projectiles' initial velocity, mass, density, and shape, the projectile's range and impact velocity are determined. For better results a MatLab program described in the paper has been used. It is shown that both ranges and impact velocities increase with mass, density, and initial velocity. For projectiles weighing 5 kg to 142 kg, and launched from a conventional cannon at initial velocity 1,700 m/s, the ranges are 127 km to 226 km and the impact velocities are 562 m/s to 1,127 m/s. For projectiles weighing 5.3 kg to 14.6 kg, and launched from an electromagnetic rail gun initial velocity 2,500 m/s, the ranges are 359 km to 499 km and the impact velocities are 1,361 m/s to 1,996 m/s. The temperature distribution on the surface of the projectile as a function of the distance from the nose tip is investigated as well. [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]

We present a set of our mathematical results on a model of piezoelectric energy harvester. The model is well known and is described in engineering literature. 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-Bemoulli model for the transverse vibrations of a beam subject to actions of an external (viscous/fluid) damping and of an external force. The second equation 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 with an additional term that is proportional to the voltage on the resistive load and is produced by the converse piezoelectric effect. The circuit equation contains an extra term that depends on the transverse displacement of the beam and is produced by the direct piezoelectric effect. 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. We present precise statements and outline an approach to the proofs of the following results. 1) Asymptotic formulas for the eigenvalues and the generalized eigenvectors of the dynamics generator. 2) Completeness, minimality, and Riesz basis property of the generalized eigenvectors. 3) Solutions by the spectral decomposition method of two control problems: zero controllability problem and a version of output tracking problem. [ABSTRACT FROM AUTHOR]

The paper deals with unstable aeroelastic modes for aircraft wing model in subsonic, incompressible, inviscid air flow. In recent author's papers asymptotic, spectral and stability analysis of the model has been carried out. The model is governed by a system of two coupled integrodifferential equations and a two-parameter family of boundary conditions modelling action of self-straining actuators. The Laplace transform of the solution is given in terms of the 'generalized resolvent operator', which is a meromorphic operator-valued function of the spectral parameter λ, whose poles are called the aeroelastic modes. The residues at these poles are constructed from the corresponding mode shapes. The spectral characteristics of the model are asymptotically close to the ones of a simpler system, which is called the reduced model. For the reduced model, the following result is shown: for each value of subsonic speed, there exists a radius such that all aeroelastic modes located outside the circle of this radius centred at zero are stable. Unstable modes, whose number is always finite, can occur only inside this 'circle of instability'. Explicit estimate of the 'instability radius' in terms of model parameters is given. [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]

In this paper we investigate the Rayleigh beam model with non-dissipative boundary conditions recently considered in the literature. The beam is clamped at the left end and subject to a feedback control type boundary conditions at the right. The components of the 2-dim input vector are shear and moment at the right end and the components of the observation vector are time derivatives of displacement and slope at the right end. The input is related to the observation through a co-diagonal matrix depending on two non-negative control parameters. The paper contains two main results. First we derive the leading term, the second order term, and the remainder in the asymptotic representation of the eigenmodes of the system. It follows from spectral asymptotics that the system has an infinite alternating sequence of stable and unstable eigenmodes. This type of instability has been recently obtained in the literature by a different method. The instability result might be of interest for energy harvesting models where destabilization of a system is desirable. The second result is the fact that the generalized eigenvectors of the dynamics generator of the system form a Riesz basis in the state space equipped with the energy metric. This result implies, in particular, that the system generates a C0-semigroup in the state space. [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 Whittaker–Shannon–Kotelʼnikov (WSK) sampling theorem provides a reconstruction formula for the Paley–Wiener class of bandlimited functions. It is known that the WSK sampling theorem can be obtained from a sampling theorem associated with a linear, self-adjoint, first order boundary-value problem. In the present paper, a generalization of the WSK sampling theorem is extended to the Hardy space of functions in the upper half-plane, . A notion of bandlimitedness in is defined and a sampling theorem for this class of bandlimited functions is obtained. Analogous to the Paley–Wiener class of bandlimited functions, which is the image under the Fourier transformation of functions with compact supports, the class of bandlimited functions in the Hardy space is the image of vector-functions with compact supports under an integral transformation, which realizes the incoming spectral representation in scattering theory. The boundary-value problem that generates the sampling points is of the second order and non-self-adjoint (the Regge problem). Its eigenvalues (the sampling points) are known in scattering theory as resonances and eigenfunctions are related to the resonance states. It is shown that the sampling points are complex numbers in the upper half-plane that are uniformly separated and symmetric with respect to the imaginary axis. One of the novelties of the paper is that it sheds light on a connection between three different subjects that do not seem to have much in common, i.e., sampling theorems, spaces, and the Lax–Phillips scattering theory. [Copyright &y& Elsevier]

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]

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]

The paper is the second in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. In the first paper “Reduction of the boundary-value problem to Possio integral equation in theoretical aeroelasticity”, we reformulated the initial boundary-value problem, involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions, in the form of a singular integral equation, the Possio equation. In our derivation, we have used the ideas and techniques totally different from the ones used by C. Possio in his original work. In the present paper, we show that the integral equation can be split up into two parts in such a way that the first part is not small with respect to a complex parameter entering the equation, while the second part is asymptotically small and tends to zero as the above parameter tends to infinity. In this paper, we justify such a splitting and in the next paper, we will prove solvability of the nonvanishing part of the Possio equation and then prove that asymptotically small terms cannot destroy the aforementioned unique solvability. [ABSTRACT FROM AUTHOR]

The zero controllability problem for the hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model of a finite length, is studied in this paper. The equation of motion is supplied with a one-parameter family of physically meaningful boundary conditions containing damping terms. The control is introduced as a separable forcing term g(x)f(t) in the right-hand side of the equation. A force profile function, g(x), is assumed to be given. To construct the control, f(t), which brings a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method is applied. The necessary and/or sufficient conditions for the exact controllability as well as the explicit formulas for the control laws are given. Approximate controllability is also discussed. The exact controllability results are generalized to the case of multiple eigenvalues of the main dynamics generator. [ABSTRACT FROM PUBLISHER]

The zero controllability problem for the system of two coupled hyperbolic equations which governs the vibrations of the coupled Euler–Bernoulli and Timoshenko beam model is studied in the paper. The system is considered on a finite interval with a two-parameter family of physically meaningful boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi(x)fi(t), i = 1, 2, on the right-hand sides of both equations. The force profile functions gi(x), i = 1, 2, are assumed to be given. To construct the controls fi(t), i = 1, 2, which bring a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method has been applied. The approach, used in the present paper, is based on the results obtained in the recent works by the author and the collaborators. In these works, the detailed asymptotic and spectral analyses of the non-self-adjoint operators generating the dynamics of the coupled beam have been carried out. It has been shown that for each set of the boundary parameters, the aforementioned operator is Riesz spectral, i.e. its generalized eigenvectors form a Riesz basis in the energy space. Explicit asymptotic formulas for the two-branch spectrum have also been derived. Based on these spectral results, the control problem has been reduced to the corresponding moment problem. To solve this moment problem, the asymptotical representation of the spectrum and the Riesz basis property of the generalized eigenvectors have been used. The necessary and/or sufficient conditions for the exact controllability are proven in the paper and the explicit formulas for the control laws are given. The case of the approximate controllability is discussed in the paper as well. [ABSTRACT FROM PUBLISHER]

The present paper is the last part of a three-part survey paper, in which I give a review of several research directions in the area of mathematical analysis of flutter phenomenon. Flutter is known as a structural dynamical instability, which occurs in a solid elastic structure interacting with a flow of gas or fluid and consists of violent vibrations of the structure with rapidly increasing amplitudes. The focus of this paper is a collection of models of fluid-structure interaction, for which precise mathematical formulations are available. My main interest is in the analytical results on such models: the results that can be used to explain flutter and its qualitative and even quantitative treatments. This study does not pretend to be a comprehensive review of an enormous engineering literature on analytical, computational, and experimental aspects of the flutter problem. I present a brief exposition of the results obtained in several selected papers or groups of papers. In this paper, I concentrate on the most well-known cases of flutter, i.e., flutter in aeroelasticity. Namely, I discuss aircraft flutter in historical retrospective and outline some future directions of flutter analysis. The last two sections of the paper are devoted to the precise analytical results obtained in my several recent works on a specific aircraft wing model in a subsonic, inviscid, incompressible airflow. I also mention that in the previous papers (Parts I and II of the survey), I discuss such topics as: (1) bending–torsion vibrations of coupled beams; (2) flutter in transmission lines; (3) flutter in rotating blades; (4) flutter in hard disk drives; (5) flutter in suspension bridges; and (6) flutter of blood vessel walls. [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]

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]

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]

In this study I present a review of several research directions in the area of mathematical analysis of flutter phenomenon. Flutter is known as a structural dynamical instability that occurs in a solid elastic structure interacting with a flow of gas or fluid, and consists of violent vibrations of the structure with rapidly increasing amplitudes. The focus of this review is a collection of models of fluid-structure interaction, for which precise mathematical formulations are available. The main objects of interest are analytical results on such models, which can be used for flutter explanation, its qualitative and even quantitative treatments. This paper does not pretend to be a comprehensive review of the enormous amount of engineering literature on analytical, computational, and experimental aspects of the flutter problem. I present a brief exposition of the results obtained in several selected papers or groups of papers on the following topics: (1) bending-torsion vibrations of coupled beams; (2) flutter in transmission lines; (3) flutter in rotating blades; (4) flutter in hard disk drives; (5) flutter in suspension bridges; and (6) flutter of blood vessel walls. Finally, I concentrate on the most well-known case of flutter, i.e., flutter in aeroelasticity. The last two sections of this review are devoted to the precise analytical results obtained in my several recent papers on a specific aircraft wing model in a subsonic, inviscid, incompressible airflow. [ABSTRACT FROM AUTHOR]

A review of several research directions in the area of mathematical analysis of the flutter phenomenon is presented in the present study. Flutter is known as a structural dynamical instability, which occurs in a solid elastic structure interacting with a flow of gas or fluid and consists of violent vibrations of the structure with rapidly increasing amplitudes. The focus of this review is a collection of models of fluid-structure interactions, for which precise mathematical formulations are available. The main objects of interest are analytical results on such models, which can be used for flutter explanation, qualitative, and even quantitative treatments of the models. This study does not pretend to be a comprehensive review of the enormous engineering literature on analytical, computational, and experimental aspects of the flutter problem. The entire survey provides a brief exposition of the results obtained in several selected papers or groups of papers on the following topics: (1) Bending-torsion vibrations of coupled beams; (2) flutter in transmission lines; (3) flutter in rotating blades; (4) flutter in hard disk drives; (5) flutter in suspension bridges; and (6) flutter of blood vessel walls. The last topic of the review is devoted to the most well-known cases of flutter, i.e., flutter in aeroelasticity. Namely, the precise analytical results obtained in the author’s several recent papers on a specific aircraft wing model in a subsonic, inviscid, incompressible airflow are discussed. [ABSTRACT FROM AUTHOR]

We study the zero controllability problem for the system of two coupled hyperbolic equations which govern the vibrations of the Timoshenko beam with spatially nonhomogeneous coefficients: mass density, flexural rigidity and shear stiffness. The system is considered on a finite interval with a two-parameter family of boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi(x)fi(t), i = 1,2, in the right-hand sides of both equations. We apply the spectral decomposition method to construct the controls fi(t), i = 1,2, which bring a given initial state of the system to zero on the specific time interval [0, T]. The force profile functions, gi(x), i = 1,2, are assumed to be given. Our approach is based on the results obtained in the recent work by the author. In this work, a detailed asymptotic and spectral analysis of the nonselfadjoint operator, which generates the dynamics of the Timoshenko beam with the dissipative boundary conditions, was given. It was shown that the aforementioned operator was Riesz spectral, i.e. its generalized eigenvectors formed a Riesz basis in the energy space. Explicit asymptotic formulas for two branches of the Timoshenko beam spectrum were also presented. Based on these spectral results, we reduce the control problem to the corresponding moment problem. We solve this moment problem using the asymptotics of the spectrum and give explicit formulas for the controls. [ABSTRACT FROM PUBLISHER]

We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems. [ABSTRACT FROM AUTHOR]

Resonances for one-particle three-dimensional Schrödinger operators with Coulomb potential perturbed by a spherically symmetric compactly supported function q(r) are studied. The mth derivative of q(r) has a jump on the boundary of the support (m≥0). Resonances are defined as poles of an analytical continuation of the quadratic form of the resolvent to the second Riemann sheet through the branch cut along the continuous spectrum. It is shown that there exists an infinite set of resonance poles which splits into an infinite sequence of infinite series corresponding to different values of an angular momentum. Resonances in each series have the only point of accumulation at infinity. The main result of the work is an asymptotic formula for resonances in each series. It follows from this formula that high-energy resonances of the above operator are asymptotically close to those of the Schrödinger operator with the same potential q(r) but without Coulomb term. It was shown in one of the previous works of the author that, in contrast with the non-Coulomb case, some perturbations q(r) of the Coulomb potential can produce a sequence of resonances converging to zero. The result of this work together with the above result shows that the Coulomb part of the potential, while dramatically changing the geometry of resonances at low energies, does not destroy their asymptotic behavior at high energies. [ABSTRACT FROM AUTHOR]

We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordiantes from the 3‐dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. The nonconservative boundary conditions are given on the sphere |x|=a. Our main objects of interest are the nonselfadjoint operators in the energy space of 2‐component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and boundary conditions. Our main results are precise asymptotic formulas for the complex eigenvalues and eigenfunctions of these operators and the corresponding nonselfadjoint quadratic operator pencils. Based on the asymptotic results of the present work, we will show in a forthcoming paper that the sets of root vectors of the above operators and of the dynamics generator corresponding to the full 3‐dimensional damped wave equation form Riesz bases in the appropriate energy spaces. Therefore, we will show that all these operators are spectral in the sense of Dunford. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works of Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The unknown functions (the bending and the torsion angle) depend on time and one spatial variable. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution type equation in the state space of the system equipped with the so-called energy metric. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator. The generalized resolvent operator is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function defined on the complex plane having the branch cut along the negative real semi-axis. The poles of the generalized resolvent are precisely the aeroelastic modes, and the residues at these poles are the projectors on the generalized eigenspaces. In this paper, our main object of interest is the dynamics generator of the differential parts of the system. It is a non-selfadjoint operator in the state space with a pure discrete spectrum. In the present paper, we show that the spectrum consists of two branches, and we derive their precise spectral asymptotics. Based on these results, in the next paper we will derive the asymptotics of the aeroelastic modes and approximations for the mode shapes. [ABSTRACT FROM AUTHOR]