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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]

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 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]

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]