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Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems

Book Description

This book is an integrated approach to kinematic and dynamic analysis. The matrix techniques presented are general and fully applicable to two- or three-dimensional systems. They lend themselves to programming and digital computation and can act as the basis of a usable tool for designers. Techniques have broad applicability to the design analysis of all multibody mechanical systems. The more powerful and more flexible the approach, and the less specialisation and reprogramming required for each application, the better. The matrix methods presented have been developed using these ideas as primary goals. Matrix methods can be applied by hand to such problems as the slider-crank mechanism, but this is not the intent of this text, and often the rigor required for such an attempt becomes quite burdensome in comparison with other techniques. The matrix methods have been extensively tested, both in the classroom and in the world of engineering industry.

Table of Contents

  1. Coverpage
  2. Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems
  3. Title page
  4. Copyright page
  5. Dedication
  6. Dedication
  7. Contents
  8. Preface
  9. About the Authors
  10. 1 Concepts and Definitions
    1. 1.1 Mechanical Design: Synthesis versus Analysis
    2. 1.2 Multibody Systems and Mechanisms
    3. 1.3 Planar, Spherical, and Spatial Mechanisms
    4. 1.4 Mechanical Body
    5. 1.5 Mechanical Chain and Kinematic Inversion
    6. 1.6 Joints and Joint Elements
    7. 1.7 The Six Lower-Pairs
    8. 1.8 Higher-Pairs and Kinematic Equivalence
    9. 1.9 Restraints versus Constraints
    10. References
  11. 2 Topology and Kinematic Architecture
    1. 2.1 Introduction
    2. 2.2 The Incidence Matrix
    3. 2.3 Connectedness and Assemblies
    4. 2.4 Kinematic Loops
    5. 2.5 Kinematic Paths
    6. References
    7. Problems
  12. 3 Transformation Matrices in Kinematics
    1. 3.1 Introduction
    2. 3.2 Homogeneous Coordinates of a Point
    3. 3.3 Line Coordinates and Plücker Vectors
    4. 3.4 Three-dimensional Orientation
    5. 3.5 Transformation of Coordinates
    6. 3.6 Positions, Postures, and Displacements
    7. 3.7 Euler's and Chasles’ Theorems
    8. 3.8 Euler-Rodrigues Parameters
    9. 3.9 Displacement of Lines
    10. 3.10 Quaternions
    11. References
    12. Problems
  13. 4 Modeling Mechanisms and Multibody Systems with Transformation Matrices
    1. 4.1 Introduction
    2. 4.2 Body Coordinate Systems
    3. 4.3 Joint and Auxiliary Coordinate Systems
    4. 4.4 Specifying Data for a Coordinate System
    5. 4.5 Modeling Dimensional Characteristics of a Body
    6. 4.6 Modeling Joint Characteristics
      1. 4.6.1 Helical Joint
      2. 4.6.2 Revolute Joint
      3. 4.6.3 Prismatic Joint
      4. 4.6.4 Cylindric Joint
      5. 4.6.5 Spheric Joint
      6. 4.6.6 Flat Joint
      7. 4.6.7 Rigid Joint
      8. 4.6.8 Open Joint
      9. 4.6.9 Parallel-Axis Gear Joint
      10. 4.6.10 Involute Rack-and-Pinion Joint
      11. 4.6.11 Straight-Tooth Bevel-Gear Joint
      12. 4.6.12 Point on a Planar-Curve Joint
      13. 4.6.13 Line Tangent to a Planar-Curve Joint
    7. Problems
  14. 5 Posture Analysis by Kinematic Equations
    1. 5.1 Introduction
    2. 5.2 Consecutive Transformations
    3. 5.3 Denavit-Hartenberg Transformations
    4. 5.4 Absolute Position
    5. 5.5 The Loop-closure Equation (Kinematic Equation for Position Analysis)
    6. 5.6 Closed-form Solution of Kinematic Equations for Joint-variable Positions
    7. 5.7 General Styles for Closed-Form Solutions of Kinematic Equations
    8. References
    9. Problems
  15. 6 Differential Kinematics and Numeric Solution of Posture Equations
    1. 6.1 Introduction
    2. 6.2 Differential Kinematics of a Helical Joint
    3. 6.3 Derivative Operator Matrices
      1. 6.3.1 Helical Joint
      2. 6.3.2 Revolute Joint
      3. 6.3.3 Prismatic Joint
      4. 6.3.4 Cylindric Joint
      5. 6.3.5 Spheric Joint
      6. 6.3.6 Flat Joint
      7. 6.3.7 Rigid Joint
      8. 6.3.8 Open Joint
      9. 6.3.9 Parallel-axis Gear Joint
      10. 6.3.10 Involute Rack-and-Pinion Joint
      11. 6.3.11 Straight-tooth Bevel-gear Joint
      12. 6.3.12 Point on a Planar-Curve Joint
      13. 6.3.13 Line Tangent to a Planar-Curve Joint
    4. 6.4 Screw Axes and Ball Vectors for Differential Displacements
    5. 6.5 Numeric Solution of Kinematic Posture Equations
      1. 6.5.1 Solution for a Nearby Posture
      2. 6.5.2 Avoiding Convergence to a False Solution
      3. 6.5.3 Numeric Solution of the Loop-closure Equation
    6. 6.6 Identification of Generalized Coordinates
    7. 6.7 Scaling Internal Length Units
    8. 6.8 Quality Index
    9. 6.9 Convergence and Robustness
    10. References
    11. Problems
  16. 7 Velocity Analysis
    1. 7.1 Introduction
    2. 7.2 Definition of Velocity
    3. 7.3 First Geometric Derivatives of Joint Variables
    4. 7.4 Velocities of Joint Variables
    5. 7.5 First Geometric Derivatives of Body Postures
    6. 7.6 Velocities of Bodies
    7. 7.7 First Geometric Derivatives of Point Positions
    8. 7.8 Velocities of Points
    9. Reference
    10. Problems
  17. 8 Acceleration Analysis
    1. 8.1 Definition of Acceleration
    2. 8.2 Derivatives of the Qh Operator Matrices
      1. 8.2.1 Helical (Screw) Joint
      2. 8.2.2 Revolute Joint
      3. 8.2.3 Prismatic Joint
      4. 8.2.4 Cylindric Joint
      5. 8.2.5 Spheric Joint
      6. 8.2.6 Flat Joint
      7. 8.2.7 Rigid Joint
      8. 8.2.8 Open Joint
      9. 8.2.9 Parallel-Axis Gear Joint
      10. 8.2.10 Involute Rack-and-Pinion Joint
      11. 8.2.11 Straight-Tooth Bevel-Gear Joint
      12. 8.2.12 Point on a Planar-Curve Joint
      13. 8.2.13 Line Tangent to a Planar-Curve Joint
    3. 8.3 Derivatives of the Dh Operator Matrices
    4. 8.4 Second Geometric Derivatives of Joint Variables
    5. 8.5 Accelerations of Joint Variables
    6. 8.6 Second Geometric Derivatives of Body Postures
    7. 8.7 Second Geometric Derivatives of Point Positions
    8. 8.8 Accelerations of Bodies
    9. 8.9 Accelerations of Points
    10. Reference
    11. Problems
  18. 9 Modeling Dynamic Aspects of Mechanisms and Multibody Systems
    1. 9.1 Introduction
    2. 9.2 Modeling Kinetic Energy
    3. 9.3 The Inertia Matrix
    4. 9.4 Systems of Units
    5. 9.5 Modeling Gravitational Effects
    6. 9.6 Modeling Joint Stiffness
    7. 9.7 Modeling Joint Damping
    8. 9.8 Modeling Point-to-Point Springs
    9. 9.9 Modeling Point-to-Point Dampers
    10. 9.10 Modeling External Forces and Torques Applied with Joint Variables
    11. 9.11 Modeling External Forces and Torques Applied to Bodies
    12. References
    13. Problems
  19. 10 Dynamic Equations of Motion
    1. 10.1 Introduction
    2. 10.2 Lagrange's Equation
    3. 10.3 Generalized Momentum
    4. 10.4 D’Alembert Inertia Forces
    5. 10.5 Generalized Restoring Forces
    6. 10.6 Generalized Applied Forces
    7. 10.7 Complete Equations of Motion
    8. References
    9. Problems
  20. 11 Linearized Equations of Motion
    1. 11.1 Introduction
    2. 11.2 Linearization Assumptions
    3. 11.3 Linearization
    4. 11.4 Linearized Equations of Motion
    5. 11.5 Dynamic Equations with Specified Input Motions
    6. Problems
  21. 12 Equilibrium Posture Analysis
    1. 12.1 Introduction
    2. 12.2 Seeking a Nearby Posture of Equilibrium
    3. 12.3 Seeking Equilibrium with Some Generalized Coordinates Specified
    4. 12.4 Large Increments of the Generalized Coordinates
    5. 12.5 Stable versus Unstable Equilibrium
    6. 12.6 Postures of Neutral Equilibrium
    7. Reference
    8. Problem
  22. 13 Frequency Response of Mechanisms and Multibody Systems
    1. 13.1 Introduction
    2. 13.2 Homogeneous First-order Equations of Motion
    3. 13.3 Modal Coordinates
    4. 13.4 Laplace Transformed Equations of Motion
    5. 13.5 Frequency Response
    6. References
    7. Problems
  23. 14 Time Response of Mechanisms and Multibody Systems
    1. 14.1 Inverse Laplace Transform
    2. 14.2 Cauchy's Residue Theorem
    3. 14.3 Systems with Repeated Eigenvalues
    4. 14.4 Time Integration Algorithm
    5. 14.5 Adaptive Time-step Control
    6. References
    7. Problem
  24. 15 Collision Detection
    1. 15.1 Introduction
    2. 15.2 Vertex-Face Contact
    3. 15.3 Edge-Edge Contact
    4. 15.4 Finding the Time Increment until Contact
    5. References
  25. 16 Impact Analysis
    1. 16.1 Applied Impulsive Loads
    2. 16.2 Location and Type of Contact
    3. 16.3 Simple Impact Model
    4. 16.4 Impact Model with Tangential Impulse
    5. 16.5 Impact Model with Normal Torsional Impulse
    6. 16.6 Impact Model with Moment Impulse
    7. 16.7 Integrated Model of Impact
    8. 16.8 Impact Analysis with SGCs
    9. References
    10. Problem
  26. 17 Constraint Force Analysis
    1. 17.1 Introduction
    2. 17.2 Fictitious Displacements
    3. 17.3 Fictitious Derivatives
    4. 17.4 Lagrange Equation for Constraint Force
    5. References
    6. Problems
  27. Index