This page provides links to a set of notes 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.

**Notes**

**Converting Vector Operations to Matrix Operations**

**Coordinate Transformations**

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

**Angular Velocity**

- Angular Velocity and Orientation Angles
- Euler Parameters and Angular Velocity Components
- Maple: Euler Parameters and Angular Velocity Components

**Generalized Coordinates**

Generalized Coordinates, Quasi-Coordinates, and Generalized Speeds

**Angular Velocity and Partial Angular Velocity**

- Angular Velocity & Partial Angular Velocity Using Absolute Coordinates
- Coordinate Transformation Matrices Using Relative Coordinates
- Angular Velocity & Partial Angular Velocity Using Relative Coordinates

**Angular Acceleration**

**Velocities and Partial Velocities**

- Velocities and Partial Velocities Using Absolute Coordinates
- Velocities and Partial Velocities Using Relative Coordinates

**Accelerations**

**Body-Connection Array**

**Connecting Joints**

**Inertia Matrices and Second-Order Dyadics**

- Matrices and Second-Order Dyadics
- Moments and Products of Inertia and the Inertia Matrix
- Principal Moments of Inertia and Principal Directions

**Constraint Types**

**Lagrange’s Equations**

(see more content in 3D Dynamics eBook)

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

**D’Alembert’s Principle**

(see more content in 3D Dynamics eBook)

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

**Kane’s Equations**

(see more content in 3D Dynamics eBook)

**Equations of Motion for Unconstrained**

Multibody Systems

Multibody Systems

- Equations of Motion Using a Mix of Absolute and Relative Coordinates
- A Second Example Using a Mix of Absolute and Relative Coordinates
- Time Derivative of Relative Transformation Matrices
- Equations of Motion Using Relative Coordinates Only

**Exercises**

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