You are previewing Computational Dynamics, 3rd Edition.
O'Reilly logo
Computational Dynamics, 3rd Edition

Book Description

Computational Dynamics, 3rd edition, thoroughly revised and updated, provides logical coverage of both theory and numerical computation techniques for practical applications.

The author introduces students to this advanced topic covering the concepts, definitions and techniques used in multi-body system dynamics including essential coverage of kinematics and dynamics of motion in three dimensions. He uses analytical tools including Lagrangian and Hamiltonian methods as well as Newton-Euler Equations.

An educational version of multibody computer code is now included in this new edition www.wiley.com/go/shabana that can be used for instruction and demonstration of the theories and formulations presented in the book, and a new chapter is included to explain the use of this code in solving practical engineering problems.

Most books treat the subject of dynamics from an analytical point of view, focusing on the techniques for analyzing the problems presented. This book is exceptional in that it covers the practical computational methods used to solve "real-world" problems. This makes it of particular interest not only for senior/ graduate courses in mechanical and aerospace engineering, but also to professional engineers.

  • Modern and focused treatment of the mathematical techniques, physical theories and application of rigid body mechanics that emphasizes the fundamentals of the subject, stresses the importance of computational methods and offers a wide variety of examples.

  • Each chapter features simple examples that show the main ideas and procedures, as well as straightforward problem sets that facilitate learning and help readers build problem-solving skills

Table of Contents

  1. Copyright
  2. PREFACE
    1. CONTENTS
    2. UNITS AND NOTATION
    3. ACKNOWLEDGMENTS
  3. 1. INTRODUCTION
    1. 1.1. COMPUTATIONAL DYNAMICS
    2. 1.2. MOTION AND CONSTRAINTS
      1. 1.2.1. Unconstrained Motion
      2. 1.2.2. Mechanical Joints
    3. 1.3. DEGREES OF FREEDOM
    4. 1.4. KINEMATIC ANALYSIS
    5. 1.5. FORCE ANALYSIS
    6. 1.6. DYNAMIC EQUATIONS AND THEIR DIFFERENT FORMS
    7. 1.7. FORWARD AND INVERSE DYNAMICS
    8. 1.8. PLANAR AND SPATIAL DYNAMICS
    9. 1.9. COMPUTER AND NUMERICAL METHODS
    10. 1.10. ORGANIZATION, SCOPE, AND NOTATIONS OF THE BOOK
  4. 2. LINEAR ALGEBRA
    1. 2.1. MATRICES
    2. 2.2. MATRIX OPERATIONS
      1. 2.2.1. Matrix Addition
      2. 2.2.2. Matrix Multiplication
      3. 2.2.3. Matrix Partitioning
      4. 2.2.4. Determinant
      5. 2.2.5. Inverse of a Matrix
      6. 2.2.6. Orthogonal Matrices
    3. 2.3. VECTORS
      1. 2.3.1. Differentiation
      2. 2.3.2. Linear Independence
    4. 2.4. THREE-DIMENSIONAL VECTORS
      1. 2.4.1. Cross Product
      2. 2.4.2. Skew-Symmetric Matrix Representation
      3. 2.4.3. Cartesian Coordinate System
      4. 2.4.4. Conditions of Parallelism
    5. 2.5. SOLUTION OF ALGEBRAIC EQUATIONS
      1. 2.5.1. Gaussian Elimination
      2. 2.5.2. Gauss-Jordan Method
      3. 2.5.3. Pivoting and Scaling
    6. 2.6. TRIANGULAR FACTORIZATION
      1. 2.6.1. Cholesky's Method
      2. 2.6.2. Numerical Solution
    7. 2.7. QR DECOMPOSITION
      1. 2.7.1. Gram-Schmidt Orthogonalization
      2. 2.7.2. Q and R Matrices
      3. 2.7.3. Householder Transformation
      4. 2.7.4. Important Identities for the QR Factors
    8. 2.8. SINGULAR VALUE DECOMPOSITION
      1. 2.8.1. Eigenvalue Problem
      2. 2.8.2. Singular Value Decomposition
      3. 2.8.3. Important Results from the SVD
    9. 2.9. PROBLEMS
  5. 3. KINEMATICS
    1. 3.1. KINEMATICS OF RIGID BODIES
      1. 3.1.1. Coordinate Transformation
      2. 3.1.2. Position Equations
    2. 3.2. VELOCITY EQUATIONS
      1. 3.2.1. Angular Velocity Vector
    3. 3.3. ACCELERATION EQUATIONS
    4. 3.4. KINEMATICS OF A POINT MOVING ON A RIGID BODY
      1. 3.4.1. Coriolis Acceleration
    5. 3.5. CONSTRAINED KINEMATICS
      1. 3.5.1. Planar Kinematics
      2. 3.5.2. Spatial Kinematics
      3. 3.5.3. Mobility Criteria
    6. 3.6. CLASSICAL KINEMATIC APPROACH
      1. 3.6.1. Singular Configurations
      2. 3.6.2. Mechanism Kinematics
      3. 3.6.3. Coriolis Acceleration
    7. 3.7. COMPUTATIONAL KINEMATIC APPROACH
      1. 3.7.1. Absolute Coordinates
      2. 3.7.2. Computational Approach
    8. 3.8. FORMULATION OF THE DRIVING CONSTRAINTS
    9. 3.9. FORMULATION OF THE JOINT CONSTRAINTS
      1. 3.9.1. Ground Constraints
      2. 3.9.2. Revolute Joint
      3. 3.9.3. Prismatic Joint
      4. 3.9.4. Cams
      5. 3.9.5. Gears
    10. 3.10. COMPUTATIONAL METHODS IN KINEMATICS
      1. 3.10.1. Kinematically Driven Systems
      2. 3.10.2. Position Analysis
      3. 3.10.3. Velocity Analysis
      4. 3.10.4. Acceleration Analysis
    11. 3.11. COMPUTER IMPLEMENTATION
      1. 3.11.1. Choice of the Coordinates
      2. 3.11.2. Standard Constraint Library
      3. 3.11.3. Computer Algorithm
    12. 3.12. KINEMATIC MODELING AND ANALYSIS
      1. 3.12.1. Prescribed Rotation of the Crankshaft
      2. 3.12.2. Prescribed Motion of the Slider Block
    13. 3.13. CONCLUDING REMARKS
    14. 3.14. PROBLEMS
  6. 4. FORMS OF THE DYNAMIC EQUATIONS
    1. 4.1. D'ALEMBERT'S PRINCIPLE
    2. 4.2. D'ALEMBERT'S PRINCIPLE AND NEWTON-EULER EQUATIONS
      1. 4.2.1. Newton Equations
      2. 4.2.2. Euler Equation
      3. 4.2.3. Remarks
    3. 4.3. CONSTRAINED DYNAMICS
    4. 4.4. AUGMENTED FORMULATION
    5. 4.5. LAGRANGE MULTIPLIERS
    6. 4.6. ELIMINATION OF THE DEPENDENT ACCELERATIONS
      1. 4.6.1. Generalization
    7. 4.7. EMBEDDING TECHNIQUE
      1. 4.7.1. Illustrative Example
    8. 4.8. AMALGAMATED FORMULATION
    9. 4.9. OPEN-CHAIN SYSTEMS
      1. 4.9.1. Equilibrium of the Separate Bodies
      2. 4.9.2. Equilibrium of the Subsystems
    10. 4.10. CLOSED-CHAIN SYSTEMS
      1. 4.10.1. Equilibrium of the Separate Bodies
      2. 4.10.2. Equilibrium of the Subsystems
    11. 4.11. CONCLUDING REMARKS
    12. 4.12. PROBLEMS
  7. 5. VIRTUAL WORK AND LAGRANGIAN DYNAMICS
    1. 5.1. VIRTUAL DISPLACEMENTS
    2. 5.2. KINEMATIC CONSTRAINTS AND COORDINATE PARTITIONING
      1. 5.2.1. Constraint Jacobian Matrix
      2. 5.2.2. Absolute Coordinates
      3. 5.2.3. Nonholonomic Constraints
    3. 5.3. VIRTUAL WORK
      1. 5.3.1. Generalized Forces
      2. 5.3.2. Generalization
      3. 5.3.3. Coordinate Transformation
      4. 5.3.4. Conservative Forces
    4. 5.4. EXAMPLES OF FORCE ELEMENTS
      1. 5.4.1. Gravity
      2. 5.4.2. Spring-Damper-Actuator Element
      3. 5.4.3. Rotational Spring-Damper Element
      4. 5.4.4. Coulomb Friction
    5. 5.5. WORKLESS CONSTRAINTS
    6. 5.6. PRINCIPLE OF VIRTUAL WORK IN STATICS
      1. 5.6.1. Equipollent Systems of Forces
      2. 5.6.2. Principle of Virtual Work
      3. 5.6.3. Constraint Forces
      4. 5.6.4. Equilibrium Equations
      5. 5.6.5. Illustrative Example
    7. 5.7. PRINCIPLE OF VIRTUAL WORK IN DYNAMICS
      1. 5.7.1. Connectivity Conditions
      2. 5.7.2. Dynamic Equations
      3. 5.7.3. Illustrative Example
    8. 5.8. LAGRANGE'S EQUATION
    9. 5.9. GIBBS-APPEL EQUATION
    10. 5.10. HAMILTONIAN FORMULATION
      1. 5.10.1. Canonical Equations
      2. 5.10.2. Conservation Theorem
    11. 5.11. RELATIONSHIP BETWEEN VIRTUAL WORK AND GAUSSIAN ELIMINATION
    12. 5.12. PROBLEMS
  8. 6. CONSTRAINED DYNAMICS
    1. 6.1. GENERALIZED INERTIA
      1. 6.1.1. Equipollent Systems
      2. 6.1.2. Parallel Axis Theorem
    2. 6.2. MASS MATRIX AND CENTRIFUGAL FORCES
      1. 6.2.1. Centrifugal Inertia Forces
      2. 6.2.2. Centroidal Body Coordinate System
    3. 6.3. EQUATIONS OF MOTION
      1. 6.3.1. Newton-Euler Equations
    4. 6.4. SYSTEM OF RIGID BODIES
    5. 6.5. ELIMINATION OF THE CONSTRAINT FORCES
      1. 6.5.1. Coordinate Partitioning
      2. 6.5.2. Embedding Technique
      3. 6.5.3. Identification of the System Degrees of Freedom
    6. 6.6. LAGRANGE MULTIPLIERS
      1. 6.6.1. Equipollent Systems of Forces
      2. 6.6.2. Lagrange Multipliers
      3. 6.6.3. Multiple Joints
    7. 6.7. CONSTRAINED DYNAMIC EQUATIONS
      1. 6.7.1. Theoretical Proof
      2. 6.7.2. Augmented Formulation
    8. 6.8. JOINT REACTION FORCES
      1. 6.8.1. Virtual Work
    9. 6.9. ELIMINATION OF LAGRANGE MULTIPLIERS
      1. 6.9.1. QR Decomposition
      2. 6.9.2. Singular Value Decomposition
    10. 6.10. STATE SPACE REPRESENTATION
    11. 6.11. NUMERICAL INTEGRATION
      1. 6.11.1. Euler's Method
      2. 6.11.2. Higher-Order Numerical Integration Methods
      3. 6.11.3. Runge-Kutta Methods
      4. 6.11.4. Multistep Methods
      5. 6.11.5. Adams Methods
    12. 6.12. ALGORITHM AND SPARSE MATRIX IMPLEMENTATION
    13. 6.13. DIFFERENTIAL AND ALGEBRAIC EQUATIONS
      1. 6.13.1. Constraint Stabilization Methods
    14. 6.14. INVERSE DYNAMICS
    15. 6.15. STATIC ANALYSIS
    16. 6.16. PROBLEMS
  9. 7. SPATIAL DYNAMICS
    1. 7.1. GENERAL DISPLACEMENT
    2. 7.2. FINITE ROTATIONS
      1. 7.2.1. Orthogonality of the Transformation Matrix
      2. 7.2.2. Simple Rotations
      3. 7.2.3. Successive Rotations
    3. 7.3. EULER ANGLES
      1. 7.3.1. Relationship between Euler Angles and Direction Cosines
    4. 7.4. VELOCITY AND ACCELERATION
      1. 7.4.1. Velocity
      2. 7.4.2. An Alternative Representation
      3. 7.4.3. Relative Angular Velocities
      4. 7.4.4. Acceleration
    5. 7.5. GENERALIZED COORDINATES
      1. 7.5.1. Another Representation
      2. 7.5.2. Remarks
    6. 7.6. GENERALIZED INERTIA FORCES
      1. 7.6.1. Mass Matrix
      2. 7.6.2. Parallel Axis Theorem
      3. 7.6.3. Principal Moments of Inertia
      4. 7.6.4. Centrifugal Forces
    7. 7.7. GENERALIZED APPLIED FORCES
      1. 7.7.1. Force Vector
      2. 7.7.2. System of Forces and Moments
      3. 7.7.3. Spring-Damper-Actuator Element
      4. 7.7.4. Rotational Spring-Damper-Actuator Element
    8. 7.8. DYNAMIC EQUATIONS OF MOTION
      1. 7.8.1. Centroidal Coordinate System
    9. 7.9. CONSTRAINED DYNAMICS
      1. 7.9.1. Kinematic Equations
      2. 7.9.2. Constrained Dynamic Equations
    10. 7.10. FORMULATION OF THE JOINT CONSTRAINTS
      1. 7.10.1. Spherical Joint
      2. 7.10.2. Cylindrical Joint
      3. 7.10.3. Revolute Joint
      4. 7.10.4. Prismatic Joint
      5. 7.10.5. Universal Joint
      6. 7.10.6. Rigid Joint
      7. 7.10.7. Remarks
    11. 7.11. NEWTON-EULER EQUATIONS
    12. 7.12. D'ALEMBERT'S PRINCIPLE
      1. 7.12.1. Newton Equations
      2. 7.12.2. Euler Equations
    13. 7.13. LINEAR AND ANGULAR MOMENTUM
    14. 7.14. RECURSIVE METHODS
      1. 7.14.1. Recursive Kinematic Equations
      2. 7.14.2. Dynamic Equations
      3. 7.14.3. An Alternative Matrix Approach
    15. 7.15. PROBLEMS
  10. 8. SPECIAL TOPICS IN DYNAMICS
    1. 8.1. GYROSCOPES AND EULER ANGLES
      1. 8.1.1. Ignorable Coordinates
      2. 8.1.2. Precession at a Steady Rate
    2. 8.2. RODRIGUEZ FORMULA
      1. 8.2.1. Angular Velocity
      2. 8.2.2. Application of Rodriguez Formula
    3. 8.3. EULER PARAMETERS
    4. 8.4. RODRIGUEZ PARAMETERS
    5. 8.5. QUATERNIONS
      1. 8.5.1. Quaternion Algebra
      2. 8.5.2. Three-Dimensional Rotations
    6. 8.6. RIGID BODY CONTACT
      1. 8.6.1. Parameterization of the Contact Surfaces
      2. 8.6.2. Contact Constraints
      3. 8.6.3. Multibody System Formulation
    7. 8.7. STABILITY AND EIGENVALUE ANALYSIS
      1. 8.7.1. System Natural Frequencies
      2. 8.7.2. State Space Formulation
      3. 8.7.3. Lightly Damped Modes
    8. 8.8. PROBLEMS
  11. 9. MULTIBODY SYSTEM COMPUTER CODES
    1. 9.1. INTRODUCTION TO SAMS/2000
      1. 9.1.1. Planar and Spatial Systems
      2. 9.1.2. Type of Analysis
      3. 9.1.3. Joint Constraints
      4. 9.1.4. Force Elements
      5. 9.1.5. User Constraints and Forces
      6. 9.1.6. Solution Procedures
      7. 9.1.7. Flexible Body Modeling
      8. 9.1.8. PRESAMS Preprocessor
      9. 9.1.9. Impact Dynamics
      10. 9.1.10. User-Differential Equations
    2. 9.2. CODE STRUCTURE
      1. 9.2.1. Mass Module (MASMOD)
      2. 9.2.2. Constraint Module (CONMOD)
      3. 9.2.3. Force Module (FRCMOD)
      4. 9.2.4. Numerical Module (NUMMOD)
    3. 9.3. SYSTEM IDENTIFICATION AND DATA STRUCTURE
      1. 9.3.1. System Identification
      2. 9.3.2. Data Structure
    4. 9.4. INSTALLING THE CODE AND THEORETICAL BACKGROUND
      1. 9.4.1. Theoretical Foundation
    5. 9.5. SAMS/2000 SETUP
      1. 9.5.1. SAMSWORD.CFG
      2. 9.5.2. SAMSFORT.CFG
    6. 9.6. USE OF THE CODE
      1. 9.6.1. Problem Description
      2. 9.6.2. Building the Model
    7. 9.7. BODY DATA
      1. 9.7.1. Inertia Tab
      2. 9.7.2. Coordinates Tab
      3. 9.7.3. Velocities Tab
      4. 9.7.4. Forces Tab
      5. 9.7.5. Graphics Tab
      6. 9.7.6. Statics Tab
    8. 9.8. CONSTRAINT DATA
    9. 9.9. PERFORMING SIMULATIONS
    10. 9.10. BATCH JOBS
    11. 9.11. GRAPHICS CONTROL
      1. 9.11.1. Appearance Tab
      2. 9.11.2. Model Tab
      3. 9.11.3. Camera and View Tab
    12. 9.12. ANIMATION CAPABILITIES
    13. 9.13. GENERAL USE OF THE INPUT DATA PANELS
      1. 9.13.1. Individual Element Panels and Tabulated Data Panels
    14. 9.14. SPATIAL ANALYSIS
    15. 9.15. SPECIAL MODULES AND FEATURES OF THE CODE
      1. 9.15.1. Rail Module
      2. 9.15.2. Eigenvalue Analysis
      3. 9.15.3. Spline Function Representation
      4. 9.15.4. Integration Methods
      5. 9.15.5. Change of the Degrees of Freedom
      6. 9.15.6. Solution Procedures
      7. 9.15.7. Subsystem Models
      8. 9.15.8. User Subroutines
      9. 9.15.9. Flexible Body Modeling
  12. REFERENCES