Multiscale Multibody Dynamics: Motion Formalism Implementation

Description

This book presents a novel theory of multibody dynamics with distinct features, including unified continuum theory, multiscale modeling technology of multibody system, and motion formalism implementation. All these features together with the introductions of fundamental concepts of vector, dual vector, tensor, dual tensor, recursive descriptions of joints, and the higher-order implicit solvers formulate the scope of the book’s content. In this book, a multibody system is defined as a set consisted of flexible and rigid bodies which are connected by any kinds of joints or constraints to achieve the desired motion. Generally, the motion of multibody system includes the translation and rotation; it is more efficient to describe the motion by using the dual vector or dual tensor directly instead of defining two types of variables, the translation and rotation separately. Furthermore, this book addresses the detail of motion formalism and its finite element implementation of the solid, shell-like, and beam-like structures. It also introduces the fundamental concepts of mechanics, such as the definition of vector, dual vector, tensor, and dual tensor, briefly. Without following the Einstein summation convention, the first- and second-order tensor operations in this book are depicted by linear algebraic operation symbols of row array, column array, and two-dimensional matrix, making these operations easier to understand. In addition, for the integral of governing equations of motion, a set of ordinary differential equations for the finite element-based discrete system, the book discussed the implementation of implicit solvers in detail and introduced the well-developed RADAU IIA algorithms based on post-error estimation to make the contents of the book complete. The intended readers of this book are senior engineers and graduate students in related engineering fields. Preface Contents Part I Preliminary of Motion and Deformation 1 Vector and Tensor 1.1 Vector 1.1.1 Law of Vector Addition 1.1.2 Scalar Multiplication 1.1.3 Cartesian Basis 1.1.4 The Scalar Product 1.1.5 The Vector Product 1.1.6 Scalar Triple Product of Vectors 1.1.7 Vector Triple Product of Vectors 1.1.8 The Tensor Product 1.1.9 The Equation of a Line 1.1.10 The Equation of a Plane 1.2 Dual Vector 1.2.1 Law of Dual Vector Addition 1.2.2 Scalar Multiplication of Dual Vector 1.2.3 The Scalar Product of Dual Vectors 1.2.4 The Vector Product of Dual Vectors 1.2.5 Tensor Product of Dual Vectors 1.2.6 Scalar Triple Product of Dual Vectors 1.2.7 Vector Triple Product of Dual Vectors 1.3 Tensor 1.3.1 Second-order Tensors 1.3.2 Tensors Calculus 1.3.3 Rotation Tensor of a Curve 1.3.4 Curvature Tensor of a Curve 1.3.5 The First Metric Tensor of a Surface 1.3.6 The Second Metric Tensor of a Surface 1.3.7 Rotation Tensor of a Surface 1.3.8 Curvature Tensor of a Surface 1.3.9 Metric Tensor of a Three-Dimensional Mapping 1.3.10 Curvature Tensor of a Three-Dimensional Mapping 1.4 Motion Tensor References 2 Motion and Deformation 2.1 Material Coordinate 2.2 Motion 2.2.1 Kinematics of a Material Particle 2.2.2 Geometric Description of Rotation 2.2.3 Composition of Rotation Tensor 2.2.4 Change of Basis Operations 2.2.5 Derivatives of Rotation Operations 2.2.6 Vectorial Parameterization of Rotation Tensor 2.2.7 Geometric Description of Motion 2.2.8 Composition of Motion Tensors 2.2.9 Derivatives of Motion Operations 2.2.10 Vectorial Parameterization of Motion Tensor 2.3 Deformation References Part II Unified Theory of Cosserat Continuum 3 Cosserat Continuum 3.1 General Cosserat Continuum Theory 3.1.1 Kinematics of Cosserat Continuum 3.1.2 Strain Tensor of Cosserat Continuum 3.1.3 Stress Tensor of Cosserat Continuum 3.1.4 Constitutive Laws for Cosserat Continuum 3.1.5 Variation of Strain Energy of Cosserat Continuum 3.1.6 Virtual Work of External Forces for Cosserat Continuum 3.1.7 Governing Equations in Motion Formalism 3.1.8 Kinematic Energy of Cosserat Continuum 3.1.9 Extended to Dynamic Problem 3.2 General Shell-Like Theory of 5 DOFS 3.2.1 Kinematics of Shell 3.2.2 Strain Tensor of Shell 3.2.3 Constitutive Laws for Shell 3.2.4 Variation of Strain Energy of Shell 3.2.5 Virtual Work of External Forces for a Shell 3.2.6 Governing Equations of Shell in Motion Formalism 3.2.7 Kinematic Energy of Shell 3.2.8 Extended to Shell Dynamic Problem 3.3 General Shell-Like Theory of 6 DOFS 3.3.1 Modification of Material Properties for a Shell 3.3.2 Variation of Strain Energy and Virtual Work of External Forces 3.3.3 Governing Equations of Shell with 6 DOFs in Motion Formalism 3.3.4 Kinematic Energy of Shell with 6 DOFs 3.4 General Beam-Like Theory 3.4.1 Kinematics of Beam 3.4.2 Strain Tensor of Beam 3.4.3 Constitutive Laws for Beam 3.4.4 Variation of Strain Energy of Beam 3.4.5 Virtual Work of External Forces for Beam 3.4.6 Governing Equations of Beam in Motion Formalism 3.4.7 Kinematic Energy of Beam 3.4.8 Extended to Beam Dynamic Problem 3.5 General Rigid Body Theory 3.5.1 Kinematics of Rigid Body 3.5.2 Work Done by External Forces for Rigid Body 3.5.3 Kinematic Energy of Rigid Body 3.5.4 Governing Equations of Motion in Motion Formalism 3.6 General Cauchy Continuum Theory 3.6.1 Kinematics of Cauchy Continuum 3.6.2 Strain Tensor of Cauchy Continuum 3.6.3 Constitutive Laws for Cauchy Continuum 3.6.4 Variation of Strain Energy of Cauchy Continuum 3.6.5 Nonlinear Strain Tensor of Cauchy Continuum 3.6.6 Variation of Nonlinear Strain Energy of Cauchy Continuum 3.6.7 Virtual Work of External Forces for Cauchy Continuum 3.6.8 Governing Equations of Cauchy Continuum 3.6.9 Kinematic Energy of Cauchy Continuum 3.6.10 Extended to Cauchy Continuum Dynamic Problem 3.7 General Membrane Theory 3.7.1 Kinematics of Membrane 3.7.2 Strain Tensor of Membrane 3.7.3 Constitutive Laws for Membrane 3.7.4 Variation of Strain Energy of Membrane 3.7.5 Virtual Work of External Forces for a Membrane 3.7.6 Governing Equations of Membrane 3.7.7 Kinematic Energy of Membrane 3.7.8 Extended to Membrane Dynamic Problem 3.8 General Cable Theory 3.8.1 Kinematics of Cable 3.8.2 Strain Tensor of Cable 3.8.3 Constitutive Laws for Cable 3.8.4 Variation of Strain Energy of Cable 3.8.5 Virtual Work of External Forces for Cable 3.8.6 Governing Equations of Cable 3.8.7 Kinematic Energy of Cable 3.8.8 Extended to Cable Dynamic Problem 3.9 Summary of General Cosserat Continuum Theory References Part III Multiscale Modeling Technology of Multibody System 4 Multiscale Multibody Dynamics 4.1 Multiscale Cosserat Continuum Theory 4.1.1 Kinematics of Multiscale Cosserat Continuum 4.1.2 Strain Tensor of Multiscale Cosserat Continuum 4.1.3 Constitutive Laws for Multiscale Cosserat Continua 4.1.4 Variation of Strain Energy of Multiscale Cosserat Continuum 4.1.5 Virtual Work of External Forces for Multiscale Cosserat Continuum 4.1.6 Governing Equations of Multiscale Cosserat Continuum in Motion Formalism 4.1.7 Extended to Multiscale Dynamic Problem 4.2 Multiscale Shell-Like Theory 4.2.1 Kinematics of Multiscale Shell 4.2.2 Strain Tensor of Multiscale Shell 4.2.3 Constitutive Laws for Multiscale Shell 4.2.4 Variation of Strain Energy of Multiscale Shell 4.2.5 Virtual Work of External Forces for Multiscale Shell 4.2.6 Governing Equations of Multiscale Shell in Motion Formalism 4.2.7 Extended to Multiscale Shell Dynamic Problem 4.3 A Special Multiscale Shell 4.3.1 Kinematics of Multiscale Shell with 5 DOFS 4.3.2 Strain Tensor of Multiscale Shell with 5 DOFS 4.3.3 Constitutive Laws for Multiscale Shell with 5 DOFS 4.3.4 Variation of Strain Energy of Multiscale Shell with 5 DOFS 4.3.5 Virtual Work of External Forces for Multiscale Shell with 5 DOFS 4.3.6 Governing Equations of Multiscale Shell with 5 DOFS in Motion Formalism 4.4 Multiscale Beam-Like Theory 4.4.1 Kinematics of Multiscale Beam 4.4.2 Strain Tensor of Multiscale Beam 4.4.3 Constitutive Laws for Multiscale Beam 4.4.4 Variation of Strain Energy of Multiscale Beam 4.4.5 Virtual Work of External Forces for Multiscale Beam 4.4.6 Governing Equations of Multiscale Beam in Motion Formalism 4.4.7 Extended to Multiscale Beam Dynamic Problem 4.5 A Special Formula of Multiscale Beam 4.5.1 Kinematics of a Special Multiscale Beam 4.5.2 Strain Tensor of a Special Multiscale Beam 4.5.3 Constitutive Laws for a Special Multiscale Beam 4.6 Modal Superelement Based Multiscale Theory 4.6.1 Herting's Transformation 4.6.2 Kinematics of Modal Superelement 4.6.3 Linearized Strain Energy of Modal Superelement 4.6.4 Linearized Kinetic Energy of Modal Superelement 4.7 Summary of Multiscale Multibody Dynamics References 5 Recursive Formulas of Joints 5.1 Motion Formalism of Six Lower Pair Joints 5.2 Motion Formalism of Prismatic Joint 5.3 Motion Formalism of Screw Joint 5.4 Motion Formalism of Cylindrical Joint 5.5 Motion Formalism of Revolute Joint 5.6 Motion Formalism of Spherical Joint 5.7 Motion Formalism of Planar Joint 5.8 Motion Formalism of Rigid Connection 5.9 Motion Formalism of Rigid Rotation References Part IV Implicit Solver Based on Radau IIIA Algorithms 6 Implicit Stiff Solvers with Post-error Estimation 6.1 Linearized Governing Equations of Motion for a Beam 6.2 Mixed Formula of Governing Equations of Motion for a Beam 6.3 Dynamics of Constrained Flexible Multibody System 6.4 Implementation of 2-Stage Radau IIA Algorithm 6.4.1 Solving Nonlinear Algebraic Equations 6.4.2 Solving Block Triangular Equations 6.5 Implementation of 3-Stage Radau IIA Algorithm 6.5.1 Solving Nonlinear Algebraic Equations 6.6 Step Size Selection 6.7 Validation of 2- and 3-Stage Radau IIA Algorithms 6.7.1 Beam Actuated by a Tip Crank 6.7.2 The Windmill Resonance Problem 6.8 Validation of Finite Element Models for Cosserat Continuum 6.8.1 Pure Bending of a Cantilevered Beam 6.8.2 Response of a Twisted Plate 6.8.3 Eigenmode Analysis of Horten IV Flying Wing 6.8.4 Bio-inspired Flight Flapping Wing 6.9 Application of Cosserat Continuum to Static Aeroelasticity 6.9.1 Aeroelastic Coupling Procedure 6.9.2 Solving Reynolds Averaged Navier-Stokes Equations 6.9.3 Interface Mappings 6.9.4 Static Aeroelastic Analysis of Aircraft Wind-Tunnel Model 6.10 Application of Cosserat Continuum to Buffeting 6.10.1 Prediction Procedure of Buffeting Response 6.10.2 Buffeting Onset Identification and Buffeting Loads Prediction 6.10.3 CFD/CSD Data Interface 6.10.4 Buffeting Response of M6 Wing References Index