Aeroelastic vibrations and stability of plates and shells (de Gruyter Studies in Mathematical Physics, 25)

Back-action of wind onto wings causes vibrations, endangering the whole structure. By careful choices of geometry, materials and damping, hazardous effects on wind engines, planes, turbines and cars can be avoided. This book gives an overview of aerodynamics and mechanics behind these problems and describes a range of mechanical effects. Numerical and analytical methods to study and analyse them are developed and supplemented by Fortran code.--Provided by publisher Preface 8 Contents 10 Introduction 14 Part I Flutter of plates 16 1 Statement of the problem 18 2 Determination of aerodynamic pressure 19 3 Mathematical statement of problems 24 4 Reduction to a problem on a disk 27 5 Test problems 33 6 Rectangular plate 49 6.1 Problem statement and analytical solution 49 6.2 Numerical–analytical solution 51 6.3 Results 54 6.4 Bubnov–Galerkin (B–G) method 55 6.5 Dependence of critical flutter velocity on plate thickness 59 6.6 Dependence of critical flutter velocity on altitude 59 7 Flutter of a rectangular plate of variable stiffness or thickness 61 7.1 Strip with variable cross section 61 7.2 Rectangular plates 65 8 Viscoelastic plates 70 Part II Flutter of shallow shells 74 9 General formulation 76 10 Determination of aerodynamic pressure 79 11 The shallowshell as part of an airfoil 84 12 The shallow shell of revolution 87 13 The conical shell: external flow 91 14 The conical shell: internal flow 95 14.1 Statement of the problem 95 14.2 Determination of dynamic pressure 100 15 Example calculations 104 Part III Numerical methods for non-self-adjoint eigenvalue problems 110 16 Discretization of the Laplace operator 112 16.1 The Sturm–Liouville problem 112 16.2 Interpolation formula for a function of two variables on a disk, and its properties 117 16.3 Discretization of the Laplace operator 121 16.4 Theorem of h-matrices 122 16.5 Construction of h-matrix cells by discretization of Bessel equations 125 16.6 Fast multiplication of h-matrices by vectors using the fast Fourier transform 127 16.7 Symmetrization of the h-matrix 129 17 Discretization of linear equations in mathematical physics with separable variables 131 17.1 General form of equations with separable variables 131 17.2 Further generalization 132 18 Eigenvalues and eigenfunctions of the Laplace operator 135 18.1 The Dirichlet problem 136 18.2 Mixed problem 148 18.3 The Neumann problem 149 18.4 Numerical experiments 153 19 Eigenvalues and eigenfunctions of a biharmonic operator 155 19.1 Boundary-value problem of the first kind 158 19.2 Boundary-value problem of the second kind 158 19.3 Numerical experiments 161 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain 164 20.1 Eigenvalues and eigenvectors of the Laplace operator 164 20.1.1 The Dirichlet problem 171 20.1.2 Mixed problem 171 20.1.3 The Neumann problem 172 20.1.4 Description of the program LAP123C 172 20.2 Program for conformal mapping 177 20.3 Numerical Experiments 179 21 Eigenvalues and eigenfunctions of a biharmonic operator on an arbitrary domain 181 21.1 Eigenvalues and eigenfunctions of a biharmonic operator 181 21.1.1 Boundary-value problem of the first kind 186 21.1.2 Boundary-value problem of the second kind 186 21.1.3 Description of the program BIG12AG 186 21.2 Program for conformal mapping 190 21.3 Numerical experiments 192 22 Error estimates for eigenvalue problems 193 22.1 Localization theorems 193 22.2 A priori error estimate in eigenvalue problems 196 22.3 A posteriori error estimate for eigenvalue problems 198 22.4 Generalization for operator pencil 198 Conclusion 200 Bibliography 202 Back-action of aerodynamics onto structures such as wings cause vibrations and may resonantly couple to them, thus causing instabilities (flutter) and endangering the whole structure. By careful choices of geometry, materials and damping mechanisms, hazardous effects on wind engines, planes, turbines and cars can be avoided. Besides an introduction into the problem of flutter, new formulations of flutter problems are given. Further, novel interpretations of the theory as well as treatises of new mechanical effects such as the oscillation stabilization of fluctuations of flow velocity are presented. Numerical and analytical methods to study them are developed and applied to the analysis of new classes of flutter problems for plates and arbitrary shallow shells. Specific problems discussed in the book in the context of numerical simulations are supplemented by Fortran code examples (available on the website). Accompanying simulation code written in Fortran is available from the website. The series is devoted to the publication of monographs and high-level texts in mathematical physics. They cover topics and methods in fields of current interest, with an emphasis on didactical presentation. The series will enable readers to understand, apply, and develop further, with sufficient rigor, mathematical methods to given problems in physics. For this reason, works with a few authors are preferred over edited volumes. The works in this series are aimed at advanced students and researchers in mathematical and theoretical physics. They can also serve as secondary reading for lectures and seminars at advanced levels. Book jacket

Back-action of aerodynamics onto structures such as wings cause vibrations and may resonantly couple to them, thus causing instabilities (flutter) and endangering the whole structure. By careful choices of geometry, materials and damping mechanisms, hazardous effects on wind engines, planes, turbines and cars can be avoided.

Besides an introduction into the problem of flutter, new formulations of flutter problems are given as well as a treatise of supersonic flutter and of a whole range of mechanical effects. Numerical and analytical methods to study them are developed and applied to the analysis of new classes of flutter problems for plates and shallow shells of arbitrary plane form. Specific problems discussed in the book in the context of numerical simulations are supplemented by Fortran code examples (available on the website).