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The present paper is the last part of a three-part survey paper, in which 1 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. 1 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, 1 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 11 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. DOI: 10.1061/(ASCE)0893-1321(2006)19:1(1) CE Database subject headings: Flutter; Aeroelasticity; Mathematical models.

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. CE Database subject headings: Eigenvalues; Damping; Vibration; Wave equations.

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. DOI: 10.1061/(ASCE)0893-1321(2004)17:2(56) CE Database subject headings: Mathematical models; Flutter; Beams; Electric transmission lines; Bridges, suspension; Aeroelasticity; Equipment.

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. DOI: 10.1061/(ASCE)0893-1321(2004)17:2(70) CE Database subject headings: Mathematical analysis; Flutter; Electric transmission lines; Bridges, suspension; Aeroelasticity; Beams; Equipment.

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]