Elastic Multibody Dynamics: A Direct Ritz Approach by Hartmut BremerElastic Multibody Dynamics: A Direct Ritz Approach by Hartmut Bremer

Elastic Multibody Dynamics: A Direct Ritz Approach

byHartmut Bremer

Paperback | November 25, 2010

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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.
Title:Elastic Multibody Dynamics: A Direct Ritz ApproachFormat:PaperbackDimensions:464 pages, 9.25 × 6.1 × 0 inPublished:November 25, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048179505

ISBN - 13:9789048179503

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Table of Contents

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 Rotor ( = 0); 6.5.1.4 Configuration Space and State Space ( 6= 0); 6.5.1.5 The Laval- (or Jeffcott-) Rotor; 6.5.1.6 Rotor with Fixed Point; 6.5.1.7 Elastic Rotor Properties; 6.6 Gross Motion - Dynamical Stiffening (Ritz Approach); 6.6.1 Rotating Beam - One-Link Elastic Robot; 6.6.1.1 Mass Matrix; 6.6.1.2 Restoring Matrix; 6.6.1.3 Equations of Motion; 6.6.2 Translating Beam - Elastic TT-Robot; 6.6.2.1 Mass Matrix; 6.6.2.2 Restoring Matrix; 6.6.2.3 Equations of Motion; 6.6.2.4 Simplified System; 6.7 The Mass Matrix Reconsidered (Ritz Approach); 6.8 The G-Matrix Reconsidered (Ritz Approach); 6.9 Conclusions;7. ELASTIC MULTIBODY SYSTEMS - ORDINARY DIFFERENTIAL EQUATIONS; 7.1 Summary Procedure; 7.1.1 Rigid Multibody Systems; 7.1.2 Elastic Multibody Systems; 7.2 Mixed Rigid-Elastic Multibody Systems; 7.3 Applications; 7.3.1 Prismatic Joint - The Telescoping Arm; 7.3.1.1 On Mass Distribution: Tip Body Influence; 7.3.1.2 Subsystem Equations; 7.3.1.3 The Kinematic Chain; 7.3.2 Revolute Joint; 7.3.2.1 Subsystem Equations; 7.3.2.2 The Kinematic Chain; 7.3.3 Spatial Motion; 7.3.4 Plane Motion;7.4 Plane Motion - Recalculation; 7.4.1 Minimal Velocities and Projection; 7.4.2 Subsystem Matrices; 7.4.3 Dynamical Stiffening; 7.4.4 The Kinematic Chain; 7.5 Reduced Number of Shape Functions: Controlled Systems; 7.6 Remark on Controlled Systems;8. A SHORT EXCURSION INTO STABILITY AND CONTROL; 8.1 Optimality; 8.1.1 Results from Classical Optimization Theory; 8.1.2 Riccati- (or LQR-) Control; 8.1.3 Control Parameter Optimization; 8.2 Stability; 8.3 Linear Time-Invariant Systems; 8.3.1 Fundamental (or Transition) Matrix; 8.3.2 Theorem of Cayley and Hamilton; 8.3.3 Stability Theorem for Mechanical Systems; 8.4 Stabilization of Mechanical Systems; 8.5 Observers; 8.5.1 Basic Notation; 8.5.2 Complete State Observer for Control; 8.5.3 Disturbance Suppression ("High Gain Observer"); 8.5.4 Disturbance Observation; 8.6 Decentralized Control; 8.7 On Control Input Variables;References; List of Symbols; Index

Editorial Reviews

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)