Vibrations of Hollow Elastic Bodies (eBook)

Honored Science Worker, academician Mekhtiyev Magomed Ferman oghlu graduated from department of Mechanics-Mathematics of Baku State University. He defended his Ph.D. thesis at the chair of elasticity theory of Rostov State University and there. In 1989 he defended his Doctoral thesis in Leningrad (St.Petersburg) State University. In 1966-1991 he occupied various positions in the Institute of Mechanics and Mathematics of National Academy of Sciences of Azerbaijan. He has worked in Baku State University since 1991. In 1994 he became professor. Scientific-research direction of prof. M.F.Mekhtiyev is mathematical methods of Solid Mechanics and Qualitative questions of optimal control. He has published over 120 scientific papers and two monographs in this field. M.F.Mekhtiyev is awarded with Gold medal of Scientific-industrial Chamber of European Union. At present he is Dean of the department of Applied Mathematics and Cybernetics and heads the chair of Mathematical Methods of Applied Analysis. 

Preface 6
References 13
About the Book 14
Contents 15
About the Author 17
1 Asymptotic Analysis of Dynamic Elasticity Problems for a Hollow Cylinder of Finite Length 18
Abstract 18
1.1 Construction of Homogeneous Solutions 18
1.2 Analysis of the Roots of the Dispersion Equation 23
1.3 Construction of Asymptotic Formulas for Displacements and Stresses 36
1.4 Generalized Orthogonality Condition of Homogeneous Solutions: Satisfaction of Boundary Conditions at the Cylinder Ends 44
1.5 Construction of Dynamic Refined Applied Theories of a Hollow Cylinder 54
1.6 Torsional Vibrations of an Isotropic Hollow Cylinder 59
1.7 Elastic Vibrations of a Hollow Cylinder with a Fixed Side Surface 63
1.8 Forced Vibrations of a Hollow Cylinder with Mixed Boundary Conditions on the Side Surface 67
References 74
2 Asymptotic Analysis of Dynamic Elasticity Problem for a Hollow Sphere 76
Abstract 76
2.1 The General Representation of the Solution to the Equations of Axisymmetric Dynamic Elasticity Theory in Spherical Coordinates 76
2.2 Inhomogeneous Solutions 78
2.3 Construction of Homogeneous Solutions 80
2.4 Asymptotic Analysis of the Dispersion Equation 84
2.5 Asymptotic Analysis of Homogeneous Solutions for a Spherical Shell 90
2.6 Dynamical Torsion of a Spherical Layer 99
2.7 Non-axisymmetric Dynamic Problems of Elasticity Theory for a Hollow Sphere 106
References 118
3 Free Vibrations of Isotropic Hollow Cylinder and Closed Hollow Sphere 120
Abstract 120
3.1 Free Vibrations of an Isotropic Hollow Cylinder 121
3.2 Analysis of the Frequency Equation and Vibration Forms of a Cylinder 123
3.3 Axisymmetric Free Vibrations of a Hollow Sphere 136
References 144
4 Asymptotic Analysis of Stress-Strain State of a Truncated Hollow Cone 146
Abstract 146
4.1 Construction of Homogeneous Solutions 146
4.2 Analysis of the Roots of the Characteristic Equation 151
4.3 Analysis of the Stress-Strain State 154
4.4 Reduction to Infinite Systems 160
4.5 Construction of Refined Applied Theories for a Conical Shell 164
4.6 Axisymmetric Problem for a Plate of Variable Thickness 168
4.7 Analysis of the Characteristic Equation for a Plate of Variable Thickness 169
4.8 Analysis of Stress-Strain State of a Plate 170
4.9 Reduction of a Boundary Value Problem for a Plate of Variable Thickness and Infinite Systems at Given Stresses 175
4.10 Construction of Applied Theories for the Plates of Variable Thickness 178
4.11 Investigation of Elastic Equilibrium of a Hollow Cone with a Fixed Side Surface and Mixed Boundary Conditions on the Side Surface 182
4.12 Asymptotic Analysis of the Solutions of Some Axisymmetric Problems for Plates of Variable Thickness 187
4.13 Asymptotic Analysis of the Characteristic Equation 192
4.14 Construction of Asymptotic Formulas for Displacements and Stresses 194
4.15 Kirsch Problem for Plates of Variable Thickness 200
4.16 Torsional Vibrations of a Conical Shell of Variable Thickness 205
References 213
Appendix 214
References 214