Learn Multibody Dynamics

Last Updated: Sep 11, 2025

Version: 0.4.dev0+73f630c

This online book aims to teach multibody dynamics using interactive code woven into the text. It generally follows the organization and methods presented in [Kane1985] and can be thought of a retelling of many topics in the book. Each page can be downloaded as a Python script or Jupyter Notebook. The book is also available in PDF format. This book is used as a companion resource for TU Delft’s Multibody Dynamics course taught by Jason K. Moore but it is designed to stand alone from the course.

Table of Contents

• Introduction
  • What You Will Learn
  • Prerequisites
  • Purpose
  • Choice of dynamics formalism
  • Choice of programming language
  • History
  • Acknowledgements
  • Tools Behind the Book
• License
• Install the Software
  • 1) Miniconda
  • 2) Create and Activate an Environment
  • 3) Install Packages
  • 4) Open Jupyter Notebook
  • Software Versions
• Jupyter and Python
  • Learning Objectives
  • Introduction
  • The Jupyter Notebook
  • Python
  • Learn More
• SymPy
  • Learning Objectives
  • Introduction
  • Import and Setup
  • Symbols
  • Undefined Functions
  • Symbolic Expressions
  • Printing
  • Differentiating
  • Evaluating Symbolic Expressions
  • Matrices
  • Solving Linear Systems
  • Simplification
  • Learn more
• Orientation of Reference Frames
  • Learning Objectives
  • Reference Frames
  • Unit Vectors
  • Simple Orientations
  • Direction Cosine Matrices
  • Successive Orientations
  • SymPy Mechanics
  • Euler Angles
  • Alternatives for Representing Orientation
  • Learn more
• Vectors
  • Learning Objectives
  • What is a vector?
  • Vector Functions
  • Addition
  • Scaling
  • Dot Product
  • Cross Product
  • Vectors Expressed in Multiple Reference Frames
  • Relative Position Among Points
• Vector Differentiation
  • Learning Objectives
  • Partial Derivatives
  • Product Rule
  • Second Derivatives
  • Vector Functions of Time
• Angular Kinematics
  • Learning Objectives
  • Introduction
  • Angular Velocity
  • Angular Velocity of Simple Orientations
  • Body Fixed Orientations
  • Time Derivatives of Vectors
  • Addition of Angular Velocity
  • Angular Acceleration
  • Addition of Angular Acceleration
• Translational Kinematics
  • Learning Objectives
  • Introduction
  • Translational Velocity
  • Velocity Two Point Theorem
  • Velocity One Point Theorem
  • Translational Acceleration
  • Acceleration Two Point Theorem
  • Acceleration One Point Theorem
• Holonomic Constraints
  • Learning Objectives
  • Four-Bar Linkage
  • Solving Holonomic Constraints
  • General Holonomic Constraints
  • Generalized Coordinates
  • Calculating Additional Kinematic Quantities
• Nonholonomic Constraints
  • Learning Objectives
  • Motion Constraints
  • Chaplygin Sleigh
  • Rolling Without Slip
  • Kinematical Differential Equations
  • Choosing Generalized Speeds
  • Snakeboard
  • Degrees of Freedom
• Mass Distribution
  • Learning Objectives
  • Particles and Rigid Bodies
  • Mass
  • Mass Center
  • Distribution of Mass
  • Inertia Matrix
  • Dyadics
  • Properties of Dyadics
  • Inertia Dyadic
  • Parallel Axis Theorem
  • Principal Axes and Moments of Inertia
  • Angular Momentum
• Force, Moment, and Torque
  • Learning Objectives
  • Force
  • Bound and Free Vectors
  • Moment
  • Couple
  • Equivalence & Replacement
  • Specifying Forces and Torques
  • Equal & Opposite
  • Contributing and Noncontributing Forces
  • Gravity
  • Springs & Dampers
  • Friction
  • Aerodynamic Drag
  • Collision
• Generalized Forces
  • Learning Objectives
  • Introduction
  • Partial Velocities
  • Nonholonomic Partial Velocities
  • Generalized Active Forces
  • Generalized Active Forces on a Rigid Body
  • Nonholonomic Generalized Active Forces
  • Generalized Inertia Forces
  • Nonholonomic Generalized Inertia Forces
• Unconstrained Equations of Motion
  • Learning Objectives
  • Dynamical Differential Equations
  • Body Fixed Newton-Euler Equations
  • Equations of Motion
  • Example of Kane’s Equations
  • Implicit and Explicit Form
• Simulation and Visualization
  • Learning Objectives
  • Numerical Integration
  • Numerical Evaluation
  • Simulation
  • Plotting Simulation Trajectories
  • Integration with SciPy
  • Animation with Matplotlib
• Three Dimensional Visualization
  • Learning Objectives
  • pythreejs
  • Creating a Scene
  • Transformation Matrices
  • Geometry and Mesh Definitions
  • Scene Setup
  • Animation Setup
  • Animated Interactive 3D Visualization
• Equations of Motion with Nonholonomic Constraints
  • Learning Objectives
  • Introduction
  • Snakeboard Equations of Motion
  • Simulate the Snakeboard
  • Animate the Snakeboard
  • Calculating Dependent Speeds
• Equations of Motion with Holonomic Constraints
  • Learning Objectives
  • Introduction
  • Four-bar Linkage Equations of Motion
  • Simulate without constraint enforcement
  • Animate the Motion
  • Correct Dependent Coordinates
  • Simulate Using a DAE Solver
• Exposing Noncontributing Forces
  • Learning Objectives
  • Introduction
  • Double Pendulum Example
  • Apply Newton’s Second Law Directly
  • Auxiliary Generalized Speeds
  • Auxiliary Generalized Active Forces
  • Auxiliary Generalized Inertia Forces
  • Augmented Dynamical Differential Equations
  • Compare Newton and Kane Results
• Energy and Power
  • Learning Objectives
  • Introduction
  • Kinetic Energy
  • Potential Energy
  • Total Energy
  • Energetics of Jumping
  • Simulation Setup
  • Conservative Simulation
  • Conservative Simulation with Ground Spring
  • Nonconservative Simulation
  • Simulation with Passive Knee Torques
  • Simulation with Active Knee Torques
• Equations of Motion with the Lagrange Method
  • Learning Objectives
  • Introduction
  • Inertial Forces from Kinetic Energy
  • Conservative Forces
  • The Lagrange Method
  • Generalized Momenta
  • Constrained Equations of Motion
  • Lagrange’s vs Kane’s
• Unconstrained Equations of Motion with the TMT Method
  • Learning Objectives
  • Example Formulation
  • Create the TMT Components
  • Formulate the reduced equations of motion
  • Evaluate the equations of motion
• Notation
  • General
  • Orientation of Reference Frames
  • Vectors and Vector Differentiation
  • Angular and Translational Kinematics
  • Constraints
  • Mass Distribution
  • Force, Moment, and Torque
  • Generalized Forces
  • Unconstrained Equations of Motion
  • Equations of Motion with Nonholonomic Constraints
  • Equations of Motion with Holonomic Constraints
  • Energy and Power
  • Lagrange’s method
  • Figure Sign Conventions
• References

Prior Versions

• Version 0.3: Final version after the 2024 course.

• Version 0.2: Final version after the 2023 course.

• Version 0.1: Final version after the 2022 course.

Lecture Videos

2023 Lecture Video Playlist

There are companion lecture videos that align with many of the chapters. The 2023 YouTube playlist is shown below:

2022 Lecture Video Playlist

There are companion lecture videos that align with many of the chapters. The 2022 YouTube playlist is shown below: