Multibody Dynamics

This page provides links to a set of notes and exercises related to multibody dynamics – kinematical and dynamical analyses of constrained mechanical systems consisting of many interconnected rigid bodies. Emphasis is on formulations that are conducive to being developed into computer algorithms.

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Example Multibody System

Notes

Converting Vector Operations to Matrix Operations

Representation of Vector Operations as Matrix Operations

Coordinate Transformations

  1. Coordinate Transformation Matrices
  2. Orientation Angles of a Rigid Body in Three Dimensions
  3. Orientation of a Rigid Body using Euler Parameters
  4. Time Derivative of the (Coordinate) Transformation Matrices
  5. Conversion of Direction Cosines to Euler Parameters
  6. Conversion of Direction Cosines to 1-2-3 Body-Fixed Angle Sequence

Angular Velocity

  1. Angular Velocity Components and Orientation Angles
  2. Angular Velocity Components and Euler Parameters
  3. Maple: Euler Parameters and Angular Velocity Components

Generalized Coordinates

Generalized Coordinates, Quasi-Coordinates, and Generalized Speeds

Angular Velocity and Partial Angular Velocity

  1. Angular Velocity & Partial Angular Velocity Using Absolute Coordinates
  2. Coordinate Transformation Matrices Using Relative Coordinates
  3. Angular Velocity & Partial Angular Velocity Using Relative Coordinates

Angular Acceleration

  1. Angular Acceleration Using Absolute Coordinates
  2. Angular Acceleration Using Relative Coordinates

Velocities and Partial Velocities

  1. Velocities and Partial Velocities Using Absolute Coordinates
  2. Velocities and Partial Velocities Using Relative Coordinates

Accelerations

  1. Accelerations Using Absolute Coordinates
  2. Accelerations Using Relative Coordinates

Body-Connection Array

  1. Body-Connection Array

Connecting Joints

  1. Connecting Joints – Part I
  2. Connecting Joints – Part II

Inertia Matrices and Second-Order Dyadics

  1. Matrices and Second-Order Dyadics
  2. Moments and Products of Inertia and the Inertia Matrix
  3. Principal Moments of Inertia and Principal Directions

Constraint Types

  1. Constraints for Multibody Systems

Lagrange’s Equations
(see more content in 3D Dynamics eBook)

  1. Lagrange’s Equations for MDOF Systems with Constraints
  2. Constraint Relaxation Method: Meaning of Lagrange Multipliers
  3. Four Simulink Models of a Simple Pendulum

D’Alembert’s Principle
(see more content in 3D Dynamics eBook)

  1. d’Alembert’s Principle for MDOF Systems
  2. Examples using d’Alembert’s Principle

More on Generalized Speeds, Partial Angular Velocities, and Partial Velocities

  1. Generalized Speeds, Partial Angular Velocities, and Partial Velocities

Kane’s Equations
(see more content in 3D Dynamics eBook)

  1. Kane’s Equations for MDOF Systems
  2. Examples using Kane’s Equations

Equations of Motion for Unconstrained
Multibody Systems

  1. Equations of Motion Using a Mix of Absolute and Relative Coordinates
  2. A Second Example Using a Mix of Absolute and Relative Coordinates
  3. Time Derivative of Relative Transformation Matrices
  4. Equations of Motion using only Relative Coordinates with Euler Parameters
  5. Equations of Motion using only Relative Coordinates with Orientation Angles
  6. Equations of Motion of Systems with Joint Constraints

Exercises

  1. Exercises #1     (Exercises #1 Answers)
  2. Exercises #2     (Exercises #2 Answers)
  3. Exercises #2.1     (Exercises #2.1 Answers)
  4. Exercises #3     (Exercises #3 Answers)
  5. Exercises #4     (Exercises #4 Answers)
  6. Exercises #5     (Exercises #5 Answers)
  7. Exercises #6     (Exercises #6 Answers)
  8. Exercises #7     (Exercises #7 Answers)
  9. Exercises #8     
    (Exercises #8 Model Results) (MATLAB/Simulink/SimMechanics®)
  10. Exercises #9     (Exercises #9 Answers)