Intermediate Dynamics

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 trademarks of The MathWorks, Inc. The MathWorks does not warrant the accuracy of the examples provided below.

ME555Homework01Figure03
Three Degree-of-Freedom Rotating Disk

References:

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

Angular Motion and Angular Velocity

  1. Simple Angular Motion
  2. Summation Rule for Angular Velocities

Derivatives in Different Reference Frames

The Derivative Rule

Kinematics of Points Fixed on Rigid Bodies

  1. Relative Kinematics of Two Points Fixed on a Rigid Body
  2. Example System I
  3. Mechanical Configuration of a Robot

Kinematics of Points Moving on Rigid Bodies

  1. Kinematics of a Point Moving on a Rigid Body
  2. Example: Slider on a Rotating Bar

Motion Constraints

  1. Systems with Closed Kinematic Chains
  2. Rolling Constraints – Point Contact
  3. Thrust Bearing Example
  4. Rolling Constraints – Line Contact
  5. Differential Gear Set (T.R. Kane)

Orientation Angles and Angular Velocity

  1. Orientation Angles for Rigid Bodies
  2. Angular Velocity and Orientation Angles
  3. Orientation of a Rigid Body Using Euler Parameters

Angular Momentum and Kinetic Energy of a Rigid Body

  1. Moments and Products of Inertia and the Inertia Matrix
  2. Angular Momentum of a Rigid Body about its Mass Center
  3. Angular Momentum of a Rigid Body about an Arbitrary Point
  4. Kinetic Energy of a Rigid Body
  5. Simple Crank Shaft – Angular Momentum & Kinetic Energy
  6. Misaligned Disk on Shaft – Angular Momentum & Kinetic Energy
  7. Example System II
  8. Example System II – Angular Momentum & Kinetic Energy

Newton/Euler Equations of Motion

  1. Newton/Euler Equations of Motion
  2. Bearing Loads on a Simple Crank Shaft
  3. Bearing Loads on a Misaligned Disk
  4. Newton/Euler Equations of Motion: Example System II

Degrees of Freedom and Partial Velocities

  1. Degrees of Freedom of Mechanical Systems
  2. Partial Velocities and Partial Angular Velocities
  3. Partial Velocities and the Slider Crank Mechanism

Generalized Forces and the Principle of Virtual Work

  1. Generalized Forces
  2. Principle of Virtual Work
  3. Principle of Virtual Work – Example

Lagrange’s Differential Equations of Motion
(Independent Generalized Coordinates)

  1. Introduction to Lagrangian Dynamics
  2. Lagrange’s Equations for Multi-Degree-of-Freedom Systems
  3. Lagrange’s Equations Examples (Two Dimensions)
  4. Lagrange’s Equations: Example System II

Linearization of Equations of Motion, Natural Frequencies and Mode Shapes

  1. Linearization of Functions of One or Many Variables
  2. Natural Frequencies and Mode Shapes

Lagrange’s Equation of Motion
(Dependent Generalized Coordinates)

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

Example Problems

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

Exercises

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