In this post, we will see the book Nonconservative Problems Of The Theory Of Elastic Stability by V. V. Bolotin.

About the book

The present book is devoted to the study of the stability of elastic systems under the action of non-conservative forces. It is well known that for such systems the usual methods of the theory of elastic stability, which are based on an examination of forms of equilibrium close to the undisturbed form, are in general no longer applicable. Here we need to use more general methods and more involved means of investigation.
The book contains an introduction and four chapters. The first chapter covers general problems, their formulation and methods of solution. It is based on a paper read by the author at the Third All-Soviet Mathematical Conference in Moscow in 1956. The remaining chapters are devoted to applications. The second chapter considers the stability
of elastic systems under the action of non-conservative forces which during the process of loss of stability behave according to some pre-determined law (so called “follower” forces). The third chapter considers the stability of high-speed rotating elastic rotors under the action of various disturbing forces, for example, forces of internal friction, hydrodynamic and electric forces, etc. The fourth chapter deals with problems of stability of elastic systems in a high-speed gas flow; particular attention is paid to the problem of supersonic flutter of elastic plates and shells. A number of problems are con­sidered in non-linear form, which enables the behavior of the system to be studied after loss of stability. It will be seen that all these problems are of considerable interest in present day mechanical, aeronautical and rocket engineering.

The book was translated from Russian by T. K. Lusher and edited by G. Herrmann and was published in 1963.

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You can get the book here.

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Contents

PREFACE ix
AUTHOR’s PREFACE TO ENGLISH EDITION xi
TRANSLATION EDITOR’S PREFACE TO ENGLISH EDITION xii

INTRODUCTION

1. Evolution of the problem of elastic stability 1
2. Longitudinal bending of an axially compressed bar 5
3. Bar under the action of a “follower” force. Euler’s method 7
4. Stability with respect to small disturbances 9
5. Critical value of “follower” force 11
6. Critical value of “follower” force (continued). System with two degrees of freedom 12
7. Discussion of results. Potential of external forces 15
8. Range of problems covered in the book 18

CHAPTER 1. General Principles

1.1. Introductory remarks 25
1.2. Finite strain 28
1.3. Equilibrium equations and boundary conditions 30
1.4. Geometrical interpretation of results 33
1.5. Relation between stresses and strains 36
1.6. Curvilinear coordinates 36
1.7. Equations of the non-linear theory of elasticity in an arbitrary system of curvilinear coordinates 41
1.8. Formulation of the stability problem. Variational Equations 43
1.9. Various cases of load behavior 47
1.10. Static boundary-value problem 49
1.11. Oscillations about the equilibrium position and Euler’s method 55
1.12. Reduction to a system of ordinary differential equations 58
1.13. Evaluation of coefficients for certain particular systems 62
1.14. Investigation of stability 72
1.15. Example. System with two degrees of freedom 75
1.16. Effect of dissipative forces on stability 79

CHAPTER 2. Stability of equilibrium of elastic systems in the presence of follower forces

2.1. Historical background 86
2.2. Problem of the stability of a bar compressed by a tangential force 90
2.3. Influence of mass distribution 93
2.4. Approximate solution of the problem 95
2.5. Effect of damping on stability 98
2.6. Problem of the stability of a bar compressed by a force with a fixed line of action 100
2.7. Stability of the plane form of bending (derivation of the equations) 104
2.8. Some numerical results 111
2.9. Some further problems 115
2.10. Equations of equilibrium of a bar in compression and torsion 119
2.11. Stability of the rectilinear form of a bar in compression and torsion (Euler’s method). Classification of the boundary conditions 124
2.12. Bar with a concentrated mass at the end. Method of small oscillations 131
2.13. Effect of the distributed mass of the bar and of damping 134

CHAPTER 3. Stability of flexible shafts with controlled speed of revolution

3.1. Introductory remarks 139
3.2. Equations of motion of a flexible shaft 143
3.3. Viscous internal friction. Instability caused by internal friction 146
3.4. Friction independent of velocity 150
3.5. The case of arbitrary dependence of friction on frequency 155
3.6. Generalization of the problem to the case of unequal principal stiffnesses and an infinite number of degrees of freedom 157
3.7. Non-linear problem 161
3.8. Steady asynchronous precession 166
3.9. Examples of amplitude relations 171
3.10. Friction caused by macroscopic thermal diffusion 176
3.11. Effect of frictional forces in the case of components shrunk on a shaft 184
3.12. Instability of rotors due to the effect of an oil layer in the bearings 186
3.13. Instability in centrifuges incompletely filled with liquid 191
3.14. Instability of rotors in a magnetic field 194

CHAPTER 4. Stability of elastic bodies in a gas flow

4.1. Short historical introduction 199
4.2. Flutter of a wing as a non-conservative problem of elastic stability 203
4.3. General formulation of problems of stability of elastic bodies in potential gas flow 208
4.4. Stability of an elastic cylindrical shell in compressible gas flow 214
4.5. Case of an infinitely long shell. Various types of flow 218
4.6. Determination of critical flutter and divergence velocities 223
4.7. Stability of elastic plates in potential gas flow 231
4.8. Determination of aerodynamic forces in the case of high supersonic velocities. Law of plane sections 236
4.9. Stability of elastic plates at high supersonic velocities 242
4.10. Application of Galerkin’s variational method. Effect of damping and of forces in the middle surface 247
4.11. Limits of application of Galerkin’s method. Explanation of a paradox in the problem of membrane flutter 257
4.12. Non-linear problems in the theory of aeroelasticity. Effect of geometric and aerodynamic non-linearities 265
4.13. Derivation of the equations of non-linear flutter of a shallow shell at high supersonic velocities. 274
4.14, Approximate method of solution of the equations 280
4.15. Panel supported over its entire contour 285
4.16. Non-linear flutter of a flat panel. Solution by trigonometric series 290
4.17. Small-parameter method for investigation of non-linear flutter 298
4.18. Analysis of results 306

CONCLUDING REMARKS. Suggested directions for future research 313

AUTHOR INDEX 319

SUBJECT INDEX 321