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Elastic Multibody Dynamics A Direct Ritz Approach

Elastic Multibody Dynamics A Direct Ritz Approach 1. Background This textbook is an introduction to and exploration of a number of core topics in the ?eld of applied mechanics. Mechanics, in both its theoretical and applied contexts, is, like all scienti?c endeavors, a human construct. It re?ects the personalities, thoughts, errors, and successes of its creators. We therefore provide some personal information about each of these individuals when their names arise for the ?rst time in this book. This should enable the reader to piece together a cultural-historical picture of the ?eld s origins and development. This does not mean that we are writing history. Nevertheless, some remarks putting individuals and ideas in context are necessary in order to make clear what we are speaking about – and what we are not speaking about. At the end of the 19th century, technical universities were established eve- where in Europe in an almost euphoric manner. But the practice of technical mechanics itself, as one of the basics of technical development, was in a desolate state, due largely to the refusal of its practitioners to recognize the in?uence of kinetics on motion. They were correct to the extend that then current mechanical systems moved with small velocities where kinetics does not play a signi?cant role. But they had failed to keep up with developments in the science underlying their craft and were unable to keep pace with the speeds of such systems as the steam engine.; This text introduces and explores many core topics in the field of applied mechanics. On the basis of Lagrange’s Principle, it presents a Central Equation of Dynamics that yields a unified view on existing methods in dynamics. ; 1. Background This textbook is an introduction to and exploration of a number of core topics in the ?eld of applied mechanics. Mechanics, in both its theoretical and applied contexts, is, like all scienti?c endeavors, a human construct. It re?ects the personalities, thoughts, errors, and successes of its creators. We therefore provide some personal information about each of these individuals when their names arise for the ?rst time in this book. This should enable the reader to piece together a cultural-historical picture of the ?eld s origins and development. This does not mean that we are writing history. Nevertheless, some remarks putting individuals and ideas in context are necessary in order to make clear what we are speaking about – and what we are not speaking about. At the end of the 19th century, technical universities were established eve- where in Europe in an almost euphoric manner. But the practice of technical mechanics itself, as one of the basics of technical development, was in a desolate state, due largely to the refusal of its practitioners to recognize the in?uence of kinetics on motion. They were correct to the extend that then current mechanical systems moved with small velocities where kinetics does not play a signi?cant role. But they had failed to keep up with developments in the science underlying their craft and were unable to keep pace with the speeds of such systems as the steam engine.; 1. INTRODUCTION; 1.1 Background; 1.2 Contents; 2. AXIOMS AND PRINCIPLES; 2.1 Axioms; 2.2 Principles – the 'Differential' Form; 2.3 Minimal Representation; 2.3.1 Virtual Displacements and Variations; 2.3.2 Minimal Coordinates and Minimal Velocities; 2.3.3 The Transitivity Equation; 2.4 The Central Equation of Dynamics; 2.5 Principles – the 'Minimal' Form; 2.6 Rheonomic and Non-holonomic Constraints; 2.7 Conclusions; 3. KINEMATICS; 3.1 Translation and Rotation; 3.1.1 Rotation Axis and Rotation Angle; 3.1.2 Transformation Matrices; 3.1.2.1 Rotation Vector Representation; 3.1.2.2 Cardan Angle Representation; 3.1.2.3 Euler Angle Representation; 3.1.3 Comparison; 3.2 Velocities; 3.2.1 Angular Velocity; 3.2.1.1 General Properties; 3.2.1.2 Rotation Vector Representation; 3.2.1.3 Cardan Angle Representation; 3.2.1.4 Euler Angle Representation; 3.3 State Space; 3.3.1 Kinematic Differential Equations; 3.3.1.1 Rotation Vector Representation; 3.3.1.2 Cardan Angle Representation; 3.3.1.3 Euler Angle Representation; 3.3.2 Summary Rotations; 3.4 Accelerations; 3.5 Topology – the Kinematic Chain; 3.6 Discussion; 4. RIGID MULTIBODY SYSTEMS; 4.1 Modeling aspects; 4.1.1 On Mass Point Dynamics; 4.1.2 The Rigidity Condition; 4.2 Multibody Systems; 4.2.1 Kinetic Energy; 4.2.2 Potentials; 4.2.2.1 Gravitation; 4.2.2.2 Springs; 4.2.3 Rayleigh’s Function; 4.2.4 Transitivity Equation; 4.2.5 The Projection Equation; 4.3 The Triangle of Methods; 4.3.1 Analytical Methods; 4.3.2 Synthetic Procedure(s); 4.3.3 Analytical vs. Synthetic Method(s); 4.4 Subsystems; 4.4.1 Basic Element: The Rigid Body; 4.4.1.1 Spatial Motion; 4.4.1.2 Plane Motion; 4.4.2 Subsystem Assemblage; 4.4.2.1 Absolute Velocities; 4.4.2.2 Relative Velocities; 4.4.2.3 Prismatic Joint/Revolute Joint – Spatial Motion; 4.4.3 Synthesis; 4.4.3.1 Minimal Representation; 4.4.3.2 Recursive Representation; 4.5 Constraints; 4.5.1 Inner Constraints; 4.5.2 Additional Constraints; 4.5.2.1 Jacobi Equation;4.5.2.2 Minimal Representation; 4.5.2.3 Recursive Representation; 4.5.2.4 Constraint Stabilization; 4.6 Segmentation: Elastic Body Representation; 4.6.1 Chain and Thread (Plane Motion); 4.6.2 Chain, Thread, and Beam; 4.7 Conclusion; 5. ELASTIC MULTIBODY SYSTEMS – THE PARTIAL DIFFERENTIAL EQUATIONS; 5.1 Elastic Potential; 5.1.1 Linear Elasticity; 5.1.2 Inner Constraints, Classification of Elastic Bodies; 5.1.3 Disk and Plate; 5.1.4 Bea; 5.2 Kinetic Energy; 5.3 Checking Procedures; 5.3.1 HAMILTON’s Principle and the Analytical Methods; 5.3.2 Projection Equation; 5.4 Single Elastic Body – Small Motion Amplitudes; 5.4.1 Beams; 5.4.2 Shells and Plates; 5.5 Single Body – Gross Motion; 5.5.1 The Elastic Rotor; 5.5.2 The Helicopter Blade (1); 5.6 Dynamical Stiffening; 5.6.1 The CAUCHY Stress Tensor; 5.6.2 The TREFFTZ (or 2nd Piola-Kirchhoff) Stress Tensor; 5.6.3 Second-Order Beam Displacement Fields; 5.6.4 Dynamical Stiffening Matrix; 5.6.5 The Helicopter Blade (2); 5.7 Multibody Systems – Gross Motion; 5.7.1 The Kinematic Chain; 5.7.2 Minimal Velocities; 5.7.3 Motion Equations; 5.7.3.1 Dynamical Stiffening; 5.7.3.2 Equations of Motion; 5.7.4 Boundary Conditions; 5.8 Conclusion; 6. ELASTIC MULTIBODY SYSTEMS – THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS; 6.1 Galerkin Method; 6.1.1 Direct Galerkin Method; 6.1.2 Extended Galerkin Method; 6.2 (Direct) Ritz Method; 6.3 Rayleigh Quotient; 6.4 Single Elastic Body – Small Motion Amplitudes; 6.4.1 Plate; 6.4.1.1 Equations of motion; 6.4.1.2 Basics;; 6.4.1.3 Shape Functions: Spatial Separation Approach; 6.4.1.4 Expansion in Terms of Beam Functions; 6.4.1.5 Convergence and Solution; 6.4.2 Torsional Shaft; 6.4.2.1 Eigenfunctions; 6.4.2.2 Motion Equations; 6.4.2.3 Shape Functions; 6.4.3 Change-Over Gear; 6.5 Single Elastic Body – Gross Motion; 6.5.1 The Elastic Rotor; 6.5.1.1 Rheonomic Constraint; 6.5.1.2 Choice of Shape Functions – Prolate Rotor ( = 0); 6.5.1.3 Choice of Shape Functions – Oblate; From the reviews: "This textbook is an introduction to and exploration of a number of core topics in the field of applied mechanics. The book consists of introduction, eight chapters and references. … The book can be used by mechanical engineers, scientists and graduate students." (Irina Alexandrovna Bolgrabskaya, Zentralblatt MATH, Vol. 1147, 2008) ; This textbook is an introduction to and exploration of a number of core topics in the field of applied mechanics: On the basis of Lagrange's Principle, a Central Equation of Dynamics is presented which yields a unified view on existing methods. From these, the Projection Equation is selected for the derivation of the motion equations of holonomic and of non-holonomic systems. The method is applied to rigid multibody systems where the rigid body is defined such that, by relaxation of the rigidity constraints, one can directly proceed to elastic bodies. A decomposition into subsystems leads to a minimal representation and to a recursive representation, respectively, of the equations of motion. Applied to elastic multibody systems one obtains, along with the use of spatial operators, a straight-on procedure for the interconnected partial and ordinary differential equations and the corresponding boundary conditions. The spatial operators are eventually applied to a RITZ series for approximation. The resulting equations then appear in the same structure as in rigid multibody systems. The main emphasis is laid on methodical as well as on (graduate level) educational aspects. The text is accompanied by a large number of examples and applications, e.g., from rotor dynamics and robotics. The mathematical prerequisites are subsumed in a short excursion into stability and control. ; Unified view on methods in dynamics Straight-on generic procedure for rigid and elastic multibody systems Clearly structured mathematical representation Easy physical insight and interpretation Simple algorithms for direct computer use http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Elastic Multibody Dynamics A Direct Ritz Approach

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Publisher
Springer Netherlands
Copyright
Copyright © Springer Basel AG
DOI
10.1007/978-1-4020-8680-9
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See Book on Publisher Site

Abstract

1. Background This textbook is an introduction to and exploration of a number of core topics in the ?eld of applied mechanics. Mechanics, in both its theoretical and applied contexts, is, like all scienti?c endeavors, a human construct. It re?ects the personalities, thoughts, errors, and successes of its creators. We therefore provide some personal information about each of these individuals when their names arise for the ?rst time in this book. This should enable the reader to piece together a cultural-historical picture of the ?eld s origins and development. This does not mean that we are writing history. Nevertheless, some remarks putting individuals and ideas in context are necessary in order to make clear what we are speaking about – and what we are not speaking about. At the end of the 19th century, technical universities were established eve- where in Europe in an almost euphoric manner. But the practice of technical mechanics itself, as one of the basics of technical development, was in a desolate state, due largely to the refusal of its practitioners to recognize the in?uence of kinetics on motion. They were correct to the extend that then current mechanical systems moved with small velocities where kinetics does not play a signi?cant role. But they had failed to keep up with developments in the science underlying their craft and were unable to keep pace with the speeds of such systems as the steam engine.; This text introduces and explores many core topics in the field of applied mechanics. On the basis of Lagrange’s Principle, it presents a Central Equation of Dynamics that yields a unified view on existing methods in dynamics. ; 1. Background This textbook is an introduction to and exploration of a number of core topics in the ?eld of applied mechanics. Mechanics, in both its theoretical and applied contexts, is, like all scienti?c endeavors, a human construct. It re?ects the personalities, thoughts, errors, and successes of its creators. We therefore provide some personal information about each of these individuals when their names arise for the ?rst time in this book. This should enable the reader to piece together a cultural-historical picture of the ?eld s origins and development. This does not mean that we are writing history. Nevertheless, some remarks putting individuals and ideas in context are necessary in order to make clear what we are speaking about – and what we are not speaking about. At the end of the 19th century, technical universities were established eve- where in Europe in an almost euphoric manner. But the practice of technical mechanics itself, as one of the basics of technical development, was in a desolate state, due largely to the refusal of its practitioners to recognize the in?uence of kinetics on motion. They were correct to the extend that then current mechanical systems moved with small velocities where kinetics does not play a signi?cant role. But they had failed to keep up with developments in the science underlying their craft and were unable to keep pace with the speeds of such systems as the steam engine.; 1. INTRODUCTION; 1.1 Background; 1.2 Contents; 2. AXIOMS AND PRINCIPLES; 2.1 Axioms; 2.2 Principles – the 'Differential' Form; 2.3 Minimal Representation; 2.3.1 Virtual Displacements and Variations; 2.3.2 Minimal Coordinates and Minimal Velocities; 2.3.3 The Transitivity Equation; 2.4 The Central Equation of Dynamics; 2.5 Principles – the 'Minimal' Form; 2.6 Rheonomic and Non-holonomic Constraints; 2.7 Conclusions; 3. KINEMATICS; 3.1 Translation and Rotation; 3.1.1 Rotation Axis and Rotation Angle; 3.1.2 Transformation Matrices; 3.1.2.1 Rotation Vector Representation; 3.1.2.2 Cardan Angle Representation; 3.1.2.3 Euler Angle Representation; 3.1.3 Comparison; 3.2 Velocities; 3.2.1 Angular Velocity; 3.2.1.1 General Properties; 3.2.1.2 Rotation Vector Representation; 3.2.1.3 Cardan Angle Representation; 3.2.1.4 Euler Angle Representation; 3.3 State Space; 3.3.1 Kinematic Differential Equations; 3.3.1.1 Rotation Vector Representation; 3.3.1.2 Cardan Angle Representation; 3.3.1.3 Euler Angle Representation; 3.3.2 Summary Rotations; 3.4 Accelerations; 3.5 Topology – the Kinematic Chain; 3.6 Discussion; 4. RIGID MULTIBODY SYSTEMS; 4.1 Modeling aspects; 4.1.1 On Mass Point Dynamics; 4.1.2 The Rigidity Condition; 4.2 Multibody Systems; 4.2.1 Kinetic Energy; 4.2.2 Potentials; 4.2.2.1 Gravitation; 4.2.2.2 Springs; 4.2.3 Rayleigh’s Function; 4.2.4 Transitivity Equation; 4.2.5 The Projection Equation; 4.3 The Triangle of Methods; 4.3.1 Analytical Methods; 4.3.2 Synthetic Procedure(s); 4.3.3 Analytical vs. Synthetic Method(s); 4.4 Subsystems; 4.4.1 Basic Element: The Rigid Body; 4.4.1.1 Spatial Motion; 4.4.1.2 Plane Motion; 4.4.2 Subsystem Assemblage; 4.4.2.1 Absolute Velocities; 4.4.2.2 Relative Velocities; 4.4.2.3 Prismatic Joint/Revolute Joint – Spatial Motion; 4.4.3 Synthesis; 4.4.3.1 Minimal Representation; 4.4.3.2 Recursive Representation; 4.5 Constraints; 4.5.1 Inner Constraints; 4.5.2 Additional Constraints; 4.5.2.1 Jacobi Equation;4.5.2.2 Minimal Representation; 4.5.2.3 Recursive Representation; 4.5.2.4 Constraint Stabilization; 4.6 Segmentation: Elastic Body Representation; 4.6.1 Chain and Thread (Plane Motion); 4.6.2 Chain, Thread, and Beam; 4.7 Conclusion; 5. ELASTIC MULTIBODY SYSTEMS – THE PARTIAL DIFFERENTIAL EQUATIONS; 5.1 Elastic Potential; 5.1.1 Linear Elasticity; 5.1.2 Inner Constraints, Classification of Elastic Bodies; 5.1.3 Disk and Plate; 5.1.4 Bea; 5.2 Kinetic Energy; 5.3 Checking Procedures; 5.3.1 HAMILTON’s Principle and the Analytical Methods; 5.3.2 Projection Equation; 5.4 Single Elastic Body – Small Motion Amplitudes; 5.4.1 Beams; 5.4.2 Shells and Plates; 5.5 Single Body – Gross Motion; 5.5.1 The Elastic Rotor; 5.5.2 The Helicopter Blade (1); 5.6 Dynamical Stiffening; 5.6.1 The CAUCHY Stress Tensor; 5.6.2 The TREFFTZ (or 2nd Piola-Kirchhoff) Stress Tensor; 5.6.3 Second-Order Beam Displacement Fields; 5.6.4 Dynamical Stiffening Matrix; 5.6.5 The Helicopter Blade (2); 5.7 Multibody Systems – Gross Motion; 5.7.1 The Kinematic Chain; 5.7.2 Minimal Velocities; 5.7.3 Motion Equations; 5.7.3.1 Dynamical Stiffening; 5.7.3.2 Equations of Motion; 5.7.4 Boundary Conditions; 5.8 Conclusion; 6. ELASTIC MULTIBODY SYSTEMS – THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS; 6.1 Galerkin Method; 6.1.1 Direct Galerkin Method; 6.1.2 Extended Galerkin Method; 6.2 (Direct) Ritz Method; 6.3 Rayleigh Quotient; 6.4 Single Elastic Body – Small Motion Amplitudes; 6.4.1 Plate; 6.4.1.1 Equations of motion; 6.4.1.2 Basics;; 6.4.1.3 Shape Functions: Spatial Separation Approach; 6.4.1.4 Expansion in Terms of Beam Functions; 6.4.1.5 Convergence and Solution; 6.4.2 Torsional Shaft; 6.4.2.1 Eigenfunctions; 6.4.2.2 Motion Equations; 6.4.2.3 Shape Functions; 6.4.3 Change-Over Gear; 6.5 Single Elastic Body – Gross Motion; 6.5.1 The Elastic Rotor; 6.5.1.1 Rheonomic Constraint; 6.5.1.2 Choice of Shape Functions – Prolate Rotor ( = 0); 6.5.1.3 Choice of Shape Functions – Oblate; From the reviews: "This textbook is an introduction to and exploration of a number of core topics in the field of applied mechanics. The book consists of introduction, eight chapters and references. … The book can be used by mechanical engineers, scientists and graduate students." (Irina Alexandrovna Bolgrabskaya, Zentralblatt MATH, Vol. 1147, 2008) ; This textbook is an introduction to and exploration of a number of core topics in the field of applied mechanics: On the basis of Lagrange's Principle, a Central Equation of Dynamics is presented which yields a unified view on existing methods. From these, the Projection Equation is selected for the derivation of the motion equations of holonomic and of non-holonomic systems. The method is applied to rigid multibody systems where the rigid body is defined such that, by relaxation of the rigidity constraints, one can directly proceed to elastic bodies. A decomposition into subsystems leads to a minimal representation and to a recursive representation, respectively, of the equations of motion. Applied to elastic multibody systems one obtains, along with the use of spatial operators, a straight-on procedure for the interconnected partial and ordinary differential equations and the corresponding boundary conditions. The spatial operators are eventually applied to a RITZ series for approximation. The resulting equations then appear in the same structure as in rigid multibody systems. The main emphasis is laid on methodical as well as on (graduate level) educational aspects. The text is accompanied by a large number of examples and applications, e.g., from rotor dynamics and robotics. The mathematical prerequisites are subsumed in a short excursion into stability and control. ; Unified view on methods in dynamics Straight-on generic procedure for rigid and elastic multibody systems Clearly structured mathematical representation Easy physical insight and interpretation Simple algorithms for direct computer use

Published: Jun 19, 2008

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