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Physics - Classical Continuum Physics | Kinematic and Dynamic Simulation of Multibody Systems - The Real-Time Challenge

Kinematic and Dynamic Simulation of Multibody Systems

The Real-Time Challenge

Garcia de Jalon, Javier, Bayo, Eduardo

Softcover reprint of the original 1st ed. 1994, XVI, 440 pp. 268 figs.

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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.

Content Level » Research

Keywords » applied mechanics - biomechanics - kinematics - mechanical engineering - mechanics - science - simulation

Related subjects » Classical Continuum Physics

Table of contents 

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.- 2.2.3.1 Rigid Body Constraints.- 2.2.3.2 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.- 5.1.3.1 Integration of a Mixed System of Differential and Algebraic Equations.- 5.1.3.2 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.- 5.3.1.1 Definition of Base Body Motion.- 5.3.1.2 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.- 6.1.1.1 Derivatives of the Potential of External Forces.- 6.1.1.2 Derivatives of the Potential of External Torques.- 6.1.1.3 Derivatives of the Potential Energy of Translational Springs.- 6.1.1.4 Derivatives of the Potential Energy of Rotational Springs.- 6.1.1.5 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.- 6.1.4.1 Theoretical Development.- 6.1.4.2 Practical Computation of Derivatives.- 6.1.5 Dynamic Relaxation.- 6.2 Inverse Dynamics.- 6.2.1 Newton’s Method.- 6.2.2 Method of the Lagrange Multipliers.- 6.2.2.1 Constraint Forces in Planar Multibody Systems.- 6.2.2.2 Constraint Forces in Three-Dimensional Multibody Systems.- 6.2.2.3 Calculation of Reaction Forces at the Joints.- 6.2.3 Method of the Lagrange Multipliers with Redundant Constraints.- 6.2.4 Penalty Formulation.- 6.2.5 Virtual Power Method.- 6.2.5.1 Calculation of Motor Forces.- 6.2.5.2 Calculation of Reactions at the Joints.- 6.2.6 Inverse Dynamics of Open Chain Systems.- References.- 7 Numerical Integration of the Equations of Motion.- 7.1 Integration of Ordinary Differential Equations.- 7.1.1 General Background.- 7.1.2 Runge-Kutta Methods.- 7.1.3 Explicit and Implicit Multistep Methods.- 7.1.4 Comparison Between the Runge-Kutta and the Multistep Methods.- 7.1.5 Newmark Method and Related Algorithms.- 7.2 Integration of Differential-Algebraic Equations.- 7.2.1 Preliminaries.- 7.2.2 Solutions by Backward Difference Formulae.- 7.2.3 Solutions by Implicit Runge-Kutta Methods.- 7.3 Considerations for Real-Time Simulation.- References.- Problems.- 8 Improved Formulations for Real-Time Dynamics.- 8.1 Survey of Improved Dynamic Formulations.- 8.1.1 Formulations O(N3): Composite Inertia.- 8.1.2 Formulations O(N): Articulated Inertia.- 8.1.3 Extension to Branched and Closed-Chain Configurations.- 8.2 Velocity Transformations for Open-Chain Systems.- 8.2.1 Dependent and Independent Coordinates.- 8.2.2 Dependent and Independent Velocities: Matrix R.- 8.2.3 Equations of Motion.- 8.2.4 Position Problem.- 8.2.5 Velocity and Acceleration Problems.- 8.2.5.1 Formulation A.- 8.2.5.2 Formulation B.- 8.2.6 Element-by-Element Computation of Matrix R.- 8.2.7 Computation of Mass Matrices Mb.- 8.2.8 Computation of the Matrix Product RTMR.- 8.2.9 Computation of the Matrix Product RTISc.- 8.2.10 Computation of the Term RT(Q—C).- 8.3 Velocity Transformations for Closed-Chain Systems.- 8.4 Examples Solved by Velocity Transformations.- 8.4.1 Open-Chain Example: Human Body.- 8.4.2 Closed-Chain Example: Heavy Truck.- 8.4.3 Numerical Results.- 8.5 Special Implementations Using Dependent Natural Coordinates.- 8.5.1 Differential Equations of Motion in the Natural Coordinates.- 8.5.2 Integration Procedure.- 8.5.3 Numerical Considerations.- References.- 9 Linearized Dynamic Analysis.- 9.1 Linearization of the Differential Equations of Motion.- 9.1.1 Independent Coordinates.- 9.1.2 Dependent Coordinates.- 9.1.3 Canonical Equations.- 9.2 Numerical Computation of Derivatives.- 9.3 Numerical Evaluation of the Dynamic Response.- References.- 10 Special Topics.- 10.1 Coulomb Friction.- 10.1.1 Review of the Coulomb Friction Hypothesis.- 10.1.2 Coulomb Friction in Multibody Systems: Sliding Condition.- 10.1.3 Coulomb Friction in Multibody Systems: Suction Condition.- 10.2 Impacts and Collisions.- 10.2.1 Known Impact Forces.- 10.2.2 Impacts Between Bodies.- 10.3 Backlash.- 10.3.1 Planar Revolute Joint.- 10.3.2 Planar Prismatic Joint.- 10.4 Kinematic Synthesis.- 10.5 Sensitivity Analysis and Optimization.- 10.6 Singular Positions.- References.- 11 Forward Dynamics of Flexible Multibody Systems.- 11.1 An Overview.- 11.2 The Classical Moving Frame Approach.- 11.2.1 Kinematics of a Flexible Body.- 11.2.2 Derivation of the Kinetic Energy.- 11.2.3 Derivation of the Elastic Potential Energy.- 11.2.4 Potential of External Forces.- 11.2.5 Constraint Equations.- 11.2.6 Governing Equations of Motion.- 11.2.7 Numerical Example.- 11.3 Global Method Based on Large Rotation Theory.- 11.3.1 Kinematics of the Beam.- 11.3.2 A Non-Linear Beam Finite Element Formulation.- 11.3.3 Derivation of the Kinetic Energy.- 11.3.4 Derivation of the Elastic Potential Energy.- 11.3.5 Constraint Equations.- 11.3.6 Governing Equations of Motion.- 11.3.7 Numerical Examples.- References.- 12 Inverse Dynamics of Flexible Multibodies.- 12.1 Inverse Dynamics Equations for Planar Motion.- 12.1.1 Inverse Dynamics Equations of an Individual Link.- 12.1.2 Solution of the Inverse Dynamics for an Individual Link.- 12.1.2.1 The Time Invariant Case.- 12.1.2.2 The Time Varying Case.- 12.2 Recursive Inverse Dynamics for Open-Chain Configurations.- 12.2.1 The Planar Open-Chain Case.- 12.2.2 The Spatial Open-Chain Case.- 12.3 Non-Recursive Inverse Dynamics.- 12.3.1 A Planar Open-Chain Example.- 12.3.2 A Planar Closed-Chain Example.- References.

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