This page contains links to set of **notes**, **example problems**, and **exercises** associated with a second course in rigid body dynamics. The notes cover topics relevant to two-dimensional and three-dimensional motion. A more complete presentation of three-dimensional, rigid body dynamics is provided in the 3D Dynamics eBook.

A set of exercises with answers are provided at the bottom of this page. Two of the exercises recommend practice with MATLAB/Simulink/SimMechanics®.

Trademark: MATLAB, Simulink, and SimMechanics (now called Simscape Multibody) are all registered tradmarks of The Mathworks, Inc. The Mathworks does not warrant the accuracy of the examples provided below.

**References:**

- L. Meirovitch, Methods of Analytical Dynamics, McGraw-Hill, 1970.
- T.R. Kane, P.W. Likins, and D.A. Levinson, Spacecraft Dynamics, McGraw-Hill, 1983
- T.R. Kane and D.A. Levinson, Dynamics: Theory and Application, McGraw-Hill, 1985
- R.L. Huston, Multibody Dynamics, Butterworth-Heinemann, 1990
- H. Baruh, Analytical Dynamics, McGraw-Hill, 1999
- H. Josephs and R.L. Huston, Dynamics of Mechanical Systems, CRC Press, 2002
- R.C. Hibbeler, Engineering Mechanics: Dynamics, 13th Ed., Pearson Prentice Hall, 2013
- J.L. Meriam and L.G. Craig, Engineering Mechanics: Dynamics, 3rd Ed, 1992
- F.P. Beer and E.R. Johnston, Jr. Vector Mechanics for Engineers: Dynamics, 4th Ed, 1984

**Angular Motion and Angular Velocity**

**Derivatives in Different Reference Frames**

**Kinematics of Points Fixed on Rigid Bodies**

- Relative Kinematics of Two Points Fixed on a Rigid Body
- Example System I
- Mechanical Configuration of a Robot

**Kinematics of Points Moving on Rigid Bodies**

**Motion Constraints**

- Systems with Closed Kinematic Chains
- Rolling Constraints – Point Contact
- Thrust Bearing Example
- Rolling Constraints – Line Contact
- Differential Gear Set (T.R. Kane)

**Orientation Angles and Angular Velocity**

- Orientation Angles for Rigid Bodies
- Angular Velocity and Orientation Angles
- Orientation of a Rigid Body Using Euler Parameters

**Angular Momentum and Kinetic Energy of a Rigid Body**

- Moments and Products of Inertia and the Inertia Matrix
- Angular Momentum of a Rigid Body about its Mass Center
- Angular Momentum of a Rigid Body about an Arbitrary Point
- Kinetic Energy of a Rigid Body
- Simple Crank Shaft – Angular Momentum & Kinetic Energy
- Misaligned Disk on Shaft – Angular Momentum & Kinetic Energy
- Example System II
- Example System II – Angular Momentum & Kinetic Energy

**Newton/Euler Equations of Motion**

- Newton/Euler Equations of Motion
- Bearing Loads on a Simple Crank Shaft
- Bearing Loads on a Misaligned Disk
- Newton/Euler Equations of Motion: Example System II

**Degrees of Freedom and Partial Velocities**

- Degrees of Freedom of Mechanical Systems
- Partial Velocities and Partial Angular Velocities
- Partial Velocities and the Slider Crank Mechanism

**Generalized Forces and the Principle of Virtual Work**

**Lagrange’s Differential Equations of Motion**

(Independent Generalized Coordinates)

(Independent Generalized Coordinates)

- Introduction to Lagrangian Dynamics
- Lagrange’s Equations for Multi-Degree-of-Freedom Systems
- Lagrange’s Equations Examples (2D)
- Lagrange’s Equations: Example System II

**Linearization of Equations of Motion, Natural Frequencies and Mode Shapes**

**Lagrange’s Equation of Motion**

(Dependent Generalize Coordinates)

(Dependent Generalize Coordinates)

- Configuration Constraints for Mechanical Systems
- Lagrange’s Equations for Multi-Degree-of-Freedom Systems with Dependent Generalized Coordinates
- Constraint Relaxation Method: Meaning of Lagrange Multipliers
- Example – Equations of Motion of a Slider Crank Mechanism

## Example Problems

- Example #1 – Angular Velocity and Angular Acceleration
- Example #2a – Calculation of Velocity using Direct Differentiation

Example #2b – Calculation of Velocity using the Derivative Rule - Example #3 – Calculation of Acceleration using the Derivative Rule
- Example #4 – Calculation of Velocity using the Two-Point Formula
- Example #5 – Calculation of Acceleration using the Two-Point Formula
- Example #6 – Calculation of Velocity and Acceleration using the Formulae for a Point Moving on a Body
- Example #7 – Calculation of Velocity using the Formula for a Point Moving on a Body
- Example #8 – Calculation of Acceleration using Formula for a Point Moving on a Body
- Example #9 – Calculation of Velocity using the Formula for a Point Moving on a Body
- Example #10 – Calculation of Acceleration using the Formula for a Point Moving on a Body
- Example #11 – Angular Momentum & Kinetic Energy of a Simple Crank Shaft
- Example #12 – Angular Momentum & Kinetic Energy of a Misaligned Disk
- Example #13 – Angular Momentum & Kinetic Energy of a Two DOF System
- Example #14 – Bearing Loads on a Simple Crank Shaft
- Example #15 – Bearing Loads on a Misaligned Disk
- Example #16 – Newton Equations of Motion for a Two DOF System
- Example #17 – Partial Velocities for a Slider-Crank Mechanism
- Example #18 – Principle of Virtual Work (Simple Mechanism)
- Example #19 – Principle of Virtual Work (Simple Mechanism)
- Example #20 – Lagrange’s Equations (1 DOF system)
- Example #21 – Lagrange’s Equations (2 DOF system)
- Example #22 – Lagrange’s Equations (2 DOF system)
- Example #23 – Lagrange’s Equations (3D, 2 DOF system)
- Example #24 – Linearization of Equations of Motion
- Example #25 – Linearization of Equations of Motion
- Example #26 – Natural Frequencies and Mode Shapes

**Exercises**

- Exercises #1 (Exercises #1 Answers)
- Exercises #2A
- Exercises #2B (Exercises #2 Combined Answers)
- Exercises #3 (Exercises #3 Answers)
- Exercises #4 (MATLAB/Simulink/SimMechanics®)
- Exercises #5 (Exercises #5 Answers)
- Exercises #6 (Exercises #6 Answers)
- Exercises #7 (Exercises #7 Answers)
- Exercises #8a (Exercises #8a Answers)
- Exercises #8b (Exercises #8b Answers)
- Exercises #9 (MATLAB/Simulink/SimMechanics®)
- Exercises #10 (Exercises #10 Answers)
- Exercises #11 (Exercises #11 Answers)