Kinematic and Dynamic Simulation of Multibody Systems
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Kinematic and Dynamic Simulation of Multibody Systems

The Real-Time Challenge
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Eduardo Bayo
690 g
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1 Introduction and Basic Concepts.- 1.1 Computer Methods for Multibody Systems.- 1.2 Basic Concepts.- 1.2.1 Multibody Systems and Joints.- 1.2.2 Dependent and Independent Coordinates.- 1.2.3 Symbolic vs. Numerical Formulations.- 1.3 Types of Problems.- 1.3.1 Kinematic Problems.- 1.3.2 Dynamic Problems.- 1.3.3 Other Problems: Synthesis or Design.- 1.4 Summary.- References.- 2 Dependent Coordinates and Related Constraint Equations.- 2.1 Planar Multibody Systems.- 2.1.1 Relative Coordinates.- 2.1.2 Reference Point Coordinates.- 2.1.3 Natural Coordinates.- 2.1.4 Mixed and Two-Stage Coordinates.- 2.2 Spatial Multibody Systems.- 2.2.1 Relative Coordinates.- 2.2.2 Reference Point Coordinates.- 2.2.3 Natural Coordinates.- Rigid Body Constraints.- Joint Constraints.- 2.2.4 Mixed Coordinates.- 2.3 Comparison Between Reference Point and Natural Coordinates.- 2.4 Concluding Remarks.- References.- Problems.- 3 Kinematic Analysis.- 3.1 Initial Position Problem.- 3.2 Velocity and Acceleration Analysis.- 3.2.1 Velocity Analysis.- 3.2.2 Acceleration Analysis.- 3.3 Finite Displacement Analysis.- 3.3.1 Newton-Raphson Iteration.- 3.3.2 Improved Initial Approximation.- 3.3.3 Modified Newton-Raphson Iteration.- 3.3.4 Kinematic Simulation.- 3.4 Redundant Constraints.- 3.5 Subspace of Allowable Motions.- 3.5.1 Scleronomous Systems.- 3.5.2 Rheonomous Systems.- 3.5.3 Calculation of Matrix R: Projection Methods.- 3.5.4 Orthogonalization Methods.- 3.6 Multibody Systems with Non-Holonomic Joints.- 3.6.1 Wheel Element in the Planar Case: First Method.- 3.6.2 Wheel Element in the Planar Case: Second Method.- 3.6.3 Wheel Element in the Three-Dimensional Case.- References.- Problems.- 4 Dynamic Analysis. Mass Matrices and External Forces.- 4.1 Background on Analytical Dynamics.- 4.1.1 Principle of Virtual Displacements.- 4.1.2 Hamilton's Principle.- 4.1.3 Lagrange's Equations.- 4.1.4 Virtual Power.- 4.1.5 Canonical Equations.- 4.2 Inertial Forces. Mass Matrix.- 4.2.1 Mass Matrix of Planar Bodies.- 4.2.2 Mass Matrix of Three Dimensional Bodies.- 4.2.3 Kinetic Energy of an Element.- 4.3 External Forces.- 4.3.1 Concentrated Forces and Torques.- 4.3.2 Forces Exerted by Springs.- 4.3.3 Forces Induced by Known Acceleration Fields.- References.- Problems.- 5 Dynamic Analysis. Equations of Motion.- 5.1 Formulations in Dependent Coordinates.- 5.1.1 Method of the Lagrange Multipliers.- 5.1.2 Method Based on the Projection Matrix R.- 5.1.3 Stabilization of the Constraint Equations.- Integration of a Mixed System of Differential and Algebraic Equations.- Baumgarte Stabilization.- 5.1.4 Penalty Formulations.- 5.2 Formulations in Independent Coordinates.- 5.2.1 Determination of Independent Coordinates.- 5.2.2 Extraction Methods (Coordinate Partitioning).- 5.2.3 Methods Based on the Projection Matrix R.- 5.2.4 Comparative Remarks.- 5.3 Formulations Based on Velocity Transformations.- 5.3.1 Open-Chain Multibody Systems.- Definition of Base Body Motion.- Different Joints in 3-D Multibody Systems.- 5.3.2 Closed-Chain Multibody Systems.- 5.4 Formulations Based on the Canonical Equations.- 5.4.1 Lagrange Multiplier Formulation.- 5.4.2 Formulation Based on Independent Coordinates.- 5.4.3 Augmented Lagrangian Formulation in Canonical Form.- References.- Problems.- 6 Static Equilibrium Position and Inverse Dynamics.- 6.1 Static Equilibrium Position.- 6.1.1 Computation of Derivatives of Potential Energy.- Derivatives of the Potential of External Forces.- Derivatives of the Potential of External Torques.- Derivatives of the Potential Energy of Translational Springs.- Derivatives of the Potential Energy of Rotational Springs.- Derivatives of the Potential Energy of Gravitational Forces.- 6.1.2 Method of the Lagrange Multipliers.- 6.1.3 Penalty Formulation.- 6.1.4 Virtual Power Method.- Theoretical Development.- Practical Computation of Derivatives.- 6.1.5 Dynamic Relaxation.- 6.
Mechanical engineering, an engineering discipline born of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solu tions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for informa tion in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that will cover a broad range of concentrations important to mechanical engineering graduate edu cation and research. We are fortunate to have a distinguished roster of consulting editors, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the front page of the volume. The areas of concentration are applied mechanics, biomechanics, computa tional mechanics, dynamic systems and control, energetics, mechanics of material, processing, thermal science, and tribology. Professor Leckie, the consulting editor for applied mechanics, and I are pleased to present this volume of the series: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge by Professors Garcia de Jal6n and Bayo. The selection of this volume underscores again the interest of the Mechanical Engineering Series to provide our readers with topical monographs as well as graduate texts. Austin Texas Frederick F. Ling v The first author dedicates this book to the memory of Prof F. Tegerizo (t 1988), who introduced him to kinematics.