Dynamic Stability of Bodies Containing Fluid

The dynamics of bodies containing fluids is a subject of long-standing im­ portance in many technical applications. The stability of motion of such bodies, in particular, has been the subject of study by Soviet engineers and applied mathematicians who have brought their fuH powers of analysis to bear on the problem, and have succeeded in developing a very weH-founded body of theory. It is difficult to find a more striking example anywhere of the application of the classical methods of analytical mechanics, together with more modern concepts of stability analysis, in such a comprehensive and elegent form as that presented by Profs. Moiseyev and Rumyantsev. Therefore, it is highly significant that this recent monograph has been trans­ lated and made available to the English-speaking community. H. NORMAN ABRAMSON San Antonio July, 1967 v Foreword During the last 15-20 years, problems of dynamics of rigid bodies with fluid-filled cavities have increasingly attracted the attention of scientists.

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One. Dynamics and Stability of Rigid Bodies Containing a Fluid.- 1. Equations of Motion of a Rigid Body with Fluid Containing Cavities.- 1-1. The Hamilton-Ostrogradskiy Principle.- 1-2. Some Formulas of Kinematics and Vector Analysis.- 1-3. Basic Dynamic Parameters.- 1-4. Derivation of the Equations of Motion of an Unconstrained Rigid Body Containing a Fluid.- 1-5. Lagrange's Equations.- 1-6. Forces Exerted by the Fluid on the Rigid Body.- 1-7. Equations of Motion of the System Relative to the Mass Center.- 1-8. Integrals of the Equations of Motion.- 1-9. Viscous Fluids.- 2. Elementary Cases of Motion of a Rigid Body Containing a Fluid.- 2-1. Irrotational Motion of a Fluid.- 2-2. Motion of a Fluid-Containing Rigid Body. Zhukov skiy's Theorem.- 2-3. Velocity Potentials and Moments of Inertia of Equivalent Bodies for Several Cavity Shapes.- 2-4. Homogeneous Vortex Flow of a Fluid.- 2-5. Problems of Stability of Motion. Lyapunov's and Chetayev's Theorems.- 2-6. Steady Screw Displacement of a Body Containing a Fluid.- 2-7. Uniform Rotation of a Body About a Stationary Point.- 2-8. Permanent Rotation of a Hydrostat Moving by Inertia About a Stationary Point.- 2-9. Stability of Rotation of a Rigid Body with an Ellipsoidal Cavity.- 3. Stability of Motion with Respect to a Part of the Variables of a Rigid Body with Cavities Partially Filled with a Fluid.- 3-1. Statement of the Problem.- 3-2. Application of the Method of Lyapunov's Functions to Problems of Stability with Respect to a Part of the Variables.- 3-3. Stability of Motion of a Rotating Body Filled with a Viscous Fluid.- 3-4. Stability of Constant Inertial Screw Motion of a Body Containing a Fluid.- 3-5. Stability of Circular Motion of an Artificial Satellite Containing a Fluid.- 3-6. Stability of Rotational Motions of a Fluid-Filled Projectile.- 3-7. Stability of the Equilibrium of a Fluid-Containing Pendulum.- 4. Stability of Steady Motion of Rigid Bodies with Fluid-Filled Cavities.- 4-1. Equations of Equilibrium and of Steady Motions.- 4-2. Statement of the Problem of Stability of Steady Motion.- 4-3. Some Theorems Concerning the Stability of Steady Motions.- 4-4. The Problem of the Minimum.- 4-5. Stability of Motion of an Artificial Satellite Moving in a Circular Orbit.- 4-6. Stability of Rotation of a Rigid Body Containing a Fluid About a Fixed Axis.- 4-7. Stability of the Relative Equilibrium of a Physical Pendulum Containing a Fluid.- Two. Vibrations of a Fluid and of the Body Containing the Fluid.- 5. Statement of the Problems of the Theory of Vibrations.- 5-1. Vibrations of a Massive Fluid Contained in a Vessel.- 5-2. Vibration of a Fluid in a Field of Variable-Intensity Mass Forces.- 5-3. The Ritz Method in the Stokes-Zhukovskiy Problem.- 5-4. The Pendulum Problem.- 5-5. Vibration of a Conservative System with Fluid Links.- 5-6. Problem of Torsional Oscillations of a Beam with a Fluid-Containing Cavity.- 5-7. Equations of Torsional-Flexural Oscillations of a Fluid- Containing Beam.- 6. General Properties of Equations of Vibrations of Bodies Containing a Fluid.- 6-1. Vibrations of a Fluid Confined in a Vessel.- 6-2. Equations of Vibrations of a Conservative System with a Fluid Link.- 6-3. General Equations for the Oscillations of a Beam with a Fluid-Containing Cavity.- 6-4. Study of the Positive Definiteness of Operator M.- 6-5. Some Problems of the Theory of Forces Suddenly Applied to a Body Containing a Fluid.- 6-6. Concerning the Oscillations of Bodies Floating in a Reservoir of Limited Dimensions.- 7. Fluid Surface Phenomena and Their Effect on the Motion of a Body Containing a Fluid.- 7-1. Statement of Problems of the Theory of Vibrations.- 7-2. Problem of the Equilibrium Configuration of the Free Surface.- 7-3. The Theory of Small Vibrations.- 1-A. Problems of Dynamics of a Bubble.- 1-5. Asymptotic Behavior for "Shallow Water".- Appendix. Formulating the Problem of the Motion of a Fluid Subjected to Gravity Forces and Surface Tension Forces.- 8. Vibrations of a Viscous Fluid and of a Body Containing a Viscous Fluid.- 8-1. Elementary Problems of the Vibrations of a Viscous Fluid with a Large.- Reynolds Number.- 8-2. Vibration of an Open Vessel Containing a Viscous Fluid.- 8-3. Three-Dimensional Problems of the Theory of Vibrations of Viscous Fluid.- Editor's Supplement to the References.- References.