This book is devoted to analytically approximate methods in the nonlinear dynamics of a rigid body with cavities (containers) partly filled by a liquid. The methods are normally based on the Bateman-Luke variational formalism combined with perturbation theory. The derived approximate equations of spatial motions of the body-liquid mechanical system (these equations are called mathematical models in the title) take the form of a finite-dimensional system of nonlinear ordinary differential equations coupling quasi-velocities of the rigid body motions and generalized coordinates responsible for displacements of the natural sloshing modes. Algorithms for computing the hydrodynamic coefficients in the approximate mathematical models are proposed. Numerical values of these coefficients are listed for some tank shapes and liquid fillings. The mathematical models are also derived for the contained liquid characterized by the Newton-type dissipation. Formulas for hydrodynamic force and moment are derived in terms of the solid body quasi-velocities and the sloshing-related generalized coordinates. For prescribed harmonic excitations of upright circular (annular) cylindrical and/or conical tanks, the steady-state sloshing regimes are theoretically classified; the results are compared with known experimental data.

PrefaceIntroduction1. Governing equations and boundary conditions in the liquid sloshing dynamics1.1. Conservation laws and basic hydrodynamic equations1.2. Links between the stress tensors and the strain rate1.3. Governing equations for the contained liquid motions1.4. Boundary and initial conditions1.5. The free-surface sloshing problem in a curvilinear coordinate system2. The Bateman-Luke variational principle and associated projective methods in the dynamics of a rigid tank partly filled by a liquid2.1. Variational formulation of sloshing problem in a motionless container2.2. Projective (multimodal) methods for sloshing problem in a motionless container with upright walls in a neighborhood of the free surface2.3. Generalizing the multimodal methods for complex tank shapes2.4. Variational formulation and modal equations in a linear approximation2.5. The Bateman-Luke variational formulation of sloshing problem for prescribed spatial tank motions2.6. Projective approximate (multimodal) method in sloshing problem for prescribed spatial tank motions2.7. Variational formulation of the Stokes-Joukowski potentials problem2.8. Hydrodynamic force and moment52.9. The Bateman-Luke variational formulation and related projective (multimodal) method for the coupled `rigid tank-contained liquid' dynamics3. Nonlinear approximate (modal) equations for sloshing a rigid upright circular (annular) cylindrical tank3.1. Nonlinear approximate modal equations for sloshing in an upright cylindrical tank of circular and annular cross-sections3.2. Generalizing the nonlinear modal equations for a dissipative incompressible liquid within the framework of the Newton dissipation hypothesis3.3. Computing the hydrodynamic coefficients of the nonlinear modal equations when the tank has a uniform axisymmetric bottom3.4. Computing the hydrodynamic coefficients of the nonlinear modal equations for an upright cylindrical tank of elliptical cross-section3.5. The Stokes-Joukowski potentials for an upright annular cylindrical tank3.6. Expressions for the inertia tensor and other hydrodynamic parameters3.7. Scalar-form equations for the tank-liquid dynamics in particular cases3.8. Nonlinear approximate equations of weakly-perturbed motions of the tank-liquid mechanical system in the case of an upright circular cylindrical tank4. Nonlinear approximate modal equations for sloshing in non-cylindrical axisymmetric containers4.1. Natural sloshing modes for a conical tank4.2. Nonlinear modal equations for sloshing in a conical tank4.3. A single-dimensional modal equation for sloshing in a spherical tank5. Nonlinear approximate modal equations of the tank-liquid dynamics derived by utilizing the perturbation theory65.1. Reducing the nonlinear sloshing problem to a series of linear boundary value problemsPrefaceIntroduction1. Governing equations and boundary conditions in the liquid sloshing dynamics1.1. Conservation laws and basic hydrodynamic equations1.2. Links between the stress tensors and the strain rate1.3. Governing equations for the contained liquid motions1.4. Boundary and initial conditions1.5. The free-surface sloshing problem in a curvilinear coordinate system2. The Bateman-Luke variational principle and associated projective methods in the dynamics of a rigid tank partly filled by a liquid2.1. Variational formulation of sloshing problem in a motionless container2.2. Projective (multimodal) methods for sloshing problem in a motionless container with upright walls in a neighborhood of the free surface2.3. Generalizing the multimodal methods for complex tank shapes2.4. Variational formulation and modal equations in a linear approximation2.5. The Bateman-Luke variational formulation of sloshing problem for prescribed spatial tank motions2.6. Projective approximate (multimodal) method in sloshing problem for prescribed spatial tank motions2.7. Variational formulation of the Stokes-Joukowski potentials problem2.8. Hydrodynamic force and moment52.9. The Bateman-Luke variational formulation and related projective (multimodal) method for the coupled `rigid tank-contained liquid' dynamics3. Nonlinear approximate (modal) equations for sloshing a rigid upright circular (annular) cylindrical tank3.1. Nonlinear approximate modal equations for sloshing in an upright cylindrical tank of circular and annular cross-sections3.2. Generalizing the nonlinear modal equations for a dissipative incompressible liquid within the framework of the Newton dissipation hypothesis3.3. Computing the hydrodynamic coefficients of the nonlinear modal equations when the tank has a uniform axisymmetric bottom3.4. Computing the hydrodynamic coefficients of the nonlinear modal equations for an upright cylindrical tank of elliptical cross-section3.5. The Stokes-Joukowski potentials for an upright annular cylindrical tank3.6. Expressions for the inertia tensor and other hydrodynamic parameters3.7. Scalar-form equations for the tank-liquid dynamics in particular cases3.8. Nonlinear approximate equations of weakly-perturbed motions of the tank-liquid mechanical system in the case of an upright circular cylindrical tank4. Nonlinear approximate modal equations for sloshing in non-cylindrical axisymmetric containers4.1. Natural sloshing modes for a conical tank4.2. Nonlinear modal equations for sloshing in a conical tank4.3. A single-dimensional modal equation for sloshing in a spherical tank5. Nonlinear approximate modal equations of the tank-liquid dynamics derived by utilizing the perturbation theory65.1. Reducing the nonlinear sloshing problem to a series of linear boundary value problems5.2. Vector-form equations for the tank-liquid dynamics5.3. Basic boundary value problems for axisymmetric tanks having an upright walls in the vicinity of the free surface5.4. Scalar-form equations for the tank-liquid dynamics in the case of axisymmetric tanks with upright walls in the vicinity of the free surface; the hydrodynamic coefficients in particular cases5.5. Solving the basic boundary value problems and computing the hydrodynamic coefficients in the case of an upright cylindrical tank of circular and annular cross-sections5.6. Generalizing the perturbation technique for complex axisymmetric tank shapes6. Equivalent mechanical systems in the nonlinear dynamics of a rigid tank partly filled by a liquid6.1. Nonlinear approximate modal equations for sloshing in an upright annular cylindrical tank due to translatory tank motions6.2. Equivalent mechanical systems for the coupled tank-liquid dynamics in the case of an upright annular cylindrical tank7. Forced finite-amplitude liquid sloshing in a mobile tank7.1. The hydrodynamic instability of nonlinear steady-state liquid sloshing in an upright annular cylindrical tank due to harmonic translatory tank excitations7.2. Studying the hydrodynamic instability of forced and parametric steady-state sloshing by using a simplified two-dimensional modal system7.3. Constructing an approximate periodic solution of (7.2.1)7.4. The Bubnov-Galerkin method for finding a time-periodic solution of (7.1.3)7.5. Stability and instability of the time-periodic solution of (7.1.3)7.6. Forced liquid sloshing due to harmonic angular tank excitations7.7. Response curves7.8. Damping effect7.9. Hydrodynamic force and moment7.10. Nearly steady-state sloshing regimesBibliography