1. The grades are ready. Send me e-mail if you want to know your grade.

Many (but not all) of the the following topics will be covered not in the same order and not to the same level of detail:

1. Quick review of Newtonian dynamics
2. Relative motion, generalized coordinates and constraints
3. Principle of virtual work and D'Alembert's principle
4. Hamliton's principles
5. Lagrange's equations
6. Hamilton-Jacobi equations Not covered
7. Kinematics and dynamics of rigid bodies
8. Stability of dynamic systems
9. Gibbs-Appell equations and Kane's equations
10. Canonical transformations Not covered
11. Computational aspects of dynamics to solve real problems
12. Dynamics of lightly flexible bodies with some highlights of variational calculus
13. Chaotic dynamics (as an appetizer for ENM 601) Not covered
14. Special theory of relativity (just a glimpse!) Not covered

The course can be divided into five parts.

1. Beginning with a quick review of Newtonian dynamics, we will move onto analytical dynamics. Here, we focus on
   energy methods and integrals of the equations of motion: the description of dynamical systems with simplified models in which the order of differential equations is reduced by exploiting conservation laws.
2. Rigid body kinematics: the description of the motion of systems of rigid bodies.
3. Rigid body kinetics: the study of the forces that cause motion and the relationship between the forces and the motion. This relationship is generally described by equations of motion. The main focus will be on analytical dynamics, a set of principles that allow us to write the equations of motion using analytical methods (as opposed to graphical or numerical methods).

Some advanced topics on analytical dynamics (extension of part 1) will be discussed at this point.
4. Dynamics of elastic continua: the study of the dynamics of deformable bodies in which distributed models of compliance, inertia, and damping are developed as partial differential equations using the tools of variational calculus.
5. Miscellaneous topics: dynamics being the subject that has been thought out by many of the greatest minds of humankind, there are many interesting topics that we can talk about in this "advanced" dynamics class. Chaotic dynamics is one such topic.

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Material covered in previous years should serve you as a guide even though some minor variations exist in this year's coverage, mostly in the ordering of the topics.

• MEAM 535: Spring 1997 Schedule
• MEAM 535: Spring 1998 Schedule
• MEAM 535: Fall 1998 schedule

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1. Students are expected to have studied kinematics and kinetics in a sophomore level course (MEAM 211 at Penn) and must be familiar with Newton's laws and their application to particles and systems of particles in two and three dimensions.
2. We will assume that everybody is familiar with matrices and determinants, and has had a basic course in ordinary differential equations and multi-variable calculus (MEAM 240 and 241 at Penn).
3. Students must also know how to manipulate, multiply and differentiate vectors.
4. Familiarity with numerical and symbolic manipulation software such as Matlab and Maple is desired.

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1. All homeworks are due on Tuesday by 6 PM unless noted otherwise either here on the web or in the class. Extension of the deadline will be given for valid reasons only. Extension must be requested before the deadline.
2. All homeworks will be graded and counted towards the final grade for the class.
3. Late homeworks will be penalized if extension is not granted before the homework is due.
4. Undergraduates will be exempted from solving a few problems in a few homeworks, and also in all the three exams. If an undergraduate chooses to do those problems anyway, he/she will be given extra credit.
5. Extra credit will be given to all students who show initiative, enthusiasm and go that extra mile by doing more than what is asked formally in homeworks and the project. Extra points will be given in everything that is graded if your work deserves it. So, it pays to work hard and learn more.

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• Required Text
  [HB] Analytical Dynamics by Haim Baruh, WCB/McGraw-Hill, Boston, ISBN 0-07-365977-0. Meet the author on the web. He has an errata there!
  This is a modern exposition of analytical dynamics for first-level graduate stduents and senior undergraduate students. This book presents the material in almost the same order as what we would like to do in this course this year. Many examples and exercise problems are included. History of dynamics and brief biographies of famous dynamicisits in an appendix is a highlight of this book.
• Strongly recommended text
  [DG1] Classical Dynamics by Donald T Greenwood, Dover. It costs only $10 since it is in Dover. But it!
  It is a very good book for self-study and presents the analytical dynamics clearly and in depth. Includes such topics as Poisson and Lagrange brackets and introduces relativistic dynamics. It includes many examples and well-formulated end-of-the-chapter problems.

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• Introduction to Dynamics by Bruce H. Karnopp.
  A refresher on Newtonian Dynamics. The book (particularly the problems) is very entertaning as is the author if you were to take his class at the University of Michigan.
• Principles of Dynamics by Donald T. Greenwood, Second Edition, Prentice Hall, ISBN 0-13-709981-9.
  This book, as opposed to Greenwood's other book recommended above, emphasizes more elementary concepts. Neverthelss it is a great book with many solved examples and good problems at the end of the chapters.

• A Treatise on the Analytical Dynamics of Particles & Rigid Bodies" by E. T. Whittaker.
  This is a classic treatise on analytical mechanics. The book is so analytically oriented that there are no figures in this book!
• A Treatise on Analytical Dynamics by L. A. Pars.
  An old classic which is held in high regard by many like the Whittaker's book.
• Classical Mechanics H. Goldstein, Addison-Wesley, Reading, 1989.
  A classic textbook that covers many of the topics we would cover in this course. Somewhat newer than other books listed in this category.
• Variatinal Principles of Mechanics by C. Lanczos, Dover.
  A very good book on analytical dynamics even though the title doesn't indicate that. A good presentation of variational principles. This book is also famous for its insighful remarks on the history of dynamics.
• Principles of Mechanics by J. L. Synge and B. A. Griffith
  It is a good book, but has very little on analytical mechanics.
• Analytical Dynamics of Discrete Systems R. M. Rosenberg, Plenum Press, New York, 1977. ISBN 0-306-31014-7.
  A very useful book for the material covered in this course.
• Rational Mechanics by C. W. Kilmister and J. E. Reeve
  Essential mathematics and concepts for dynamics are discussed well in this book.

• Tensor Calculus and Analytical Dynamics" by John G. Papastavridis, CRC Press, ISBN 0-8493-8514-8.
  This is a recent book writiten for the theoretically oriented and tensor-savvy readers. It has three or four figures in the entire book. The same author is writing another book to be released in 2000 which he refers to as (or so I hear) 2000 AD where AD stands for Analytical Dynamics.
• Dynamics: Theory and Applications, T. R. Kane and D. A. Levinson, McGraw Hill, New York, 1985. ISBN 0-07-037846-0.
  It is an excellent book with a very systematic notation and a clear presentation of concepts. For those interested in learning about how the equations of motion of complex multi-body systems can be easily generated using computers, this is a great book. Problems given at the end of the book on each chapter are carefully designed to emphasize the main concepts. The only drawback of this book is that its notation is so unique that (i) you have to read it carefully from the first line to the last (and between the lines too) and (ii) it is not easy to switch to another book after reading this.
• Methods of Analytical Dynamics by L. M. Meirovitch. McGraw-Hill, Classic Textbook Re-Issue
  A very good textbook that is very systematic in its presentation and covers a wide range of topics.
• Computatioonal Dynamics by A. A. Shabana, Wiley-InterScience, ISBN 0-471-30551-0
  A modern book that focuses on computational aspects of multi-body dynamic systems.
• Applied Dynamics: With Applications to Multi-Body and Mechatronic Systems by F. K. Moon, Wiley-Interscience; ISBN: 0471138282.
  An intermediate level textbook which introduces the concepts of multibody dynamics, mechatronic dynamics, and nonlinear dynamics as well as traditional dynamics and examines underlying principles in the context of present-day computational methods.
• Analytical Dynamics: A New Approach by F. E. Udwadia and R. E. Kalaba
  It is an interesting book because it begins from the lesser known Gauss's principle of mechanics. It has good background material on the necessary mathematics.
• The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach M. Lesser, World Scientific Series on Nonlinear Science, 1995. ISBN 981-02-2209-2.

I haven't read this one as I don't have it. The book is out of print. Check amazon.com if you are curious.

• Nonlinear Dynamics and Chaos by S. Strogatz, Addison-Wesley.
  This book is used for ENM 601.
• Chaotic Dynamics by G. L. Baker and J. P. Gollub, Cambridge University Press.

• A Brief History of Time by Stephen Hawking.
  If you haven't read this great book already, do so in the Fall break or Thanksgiving break. It was a best-seller for a good reason. A great theoretical physicist of our time has come down to our level to explain profound concepts lucidly.
• Relativity Simply Exlained by Martin Gardner, Dover Pub, ISBN: 0486293157
  A well-written book that explains deep concepts of special and general relativity in an easy-to-read manner with many insightful figures. Strongly recommended for those interested in getting an intuitive understanding of how Einstein revolutionized mechanics.
• Concepts of Mass in Classical and Modern Physics by Max Jammer.
  If you don't want to take the concept of mass for granted, read this great book.
• Concepts of Force by Max Jammer, Dover.
  If you don't want to take the concept of force for granted, read this great book.
• Concepts of Space by Max Jammer, Dover.
  If you don't want to take the concept of space for granted, read this great book.Don't miss the Foreword written by Einstein!

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1. A print-out of the content of the web page as it was on 9/9/1999.
2. A handout on vector analysis. Seven single-sided pages. 9/14/1999.
3. A handout on the derivation of Kane's definition of angular velocity vector. Six single-sided pages. 9/26/1999
4. Recitation problem sheet for 10/14/99. One single-sided page. 10/12/99.
5. Analytical dynamics cheat-sheet -- in-class fun quiz (10/19/99)
6. A two-page handout on rigid bidy representations
7. Summary of rigid body dynamics-- one double sided sheet (11/16/99)
8. Stability -- five double-sided sheets (11/30/99)
9. Notes on variational calculus and vibrations of continuous systems (11/30/99)
10. The anecdote on the Brachistochrone problem -- one double-sided sheet (12/2/99)

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Homeworks and their solutions

1. Homework 1 (Assigned: 9/9/1999 and Due: 9/14/1999) (Solution is in Towne library reserve).
2. Homework 2 (Assigned: 9/14/1999 and Due: 9/23/1999) (Solution is in Towne library reserve).
3. Homework 3 (Assigned: 9/24/1999 and Due: 9/28/1999) (Solution is in Towne library reserve).
4. Homework 4 (Assigned: 9/28/1999 and Due: 10/5/1999) (Solution is in Towne library reserve).
5. Homework 5 (Assigned: 10/5/1999 and Due: 10/12/1999) (Solution is in Towne library reserve).
6. Homework 6 (Assigned: 10/12/1999 and Due: 10/19/1999) (Solution is in Towne library reserve).
7. Homework 7 (Assigned: 10/22/1999 and Due: 10/28/1999) (Solution is in Towne library reserve).
8. Homework 8 (Assigned: 11/11/1999 and Due: 11/18/1999) (Solution is in Towne library reserve)
9. Homework 9 (Assigned: 11/30/1999 and Due: 12/7/1999)

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The project in this course will require you to model a real system of sufficient complexity, formulate the equations of motion, and simulate the dynamics. Some programming will be required for this. Matlab is highly recommended unless you are more comfortable with something else.

You all have chosen a problem for the project (I am sure almost all of you did; if not, do it immediately).

A project report is due on Dec. 9th, 1999. The report should be typed up and it should include:

• Adequate description of your real system
• Model of your system: generalized coordinates, constraints, forces, etc.
• Governing equations of your system
• Simulation of your system in Matlab (or something else). Attach Matlab code, important figures, solution data, etc.
• Conclusions (what you learnt by studying this system)

Here is the list of projects undertaken this semester.

MEMS gyroscope	Stapler
Umbrella arm	Vibrating chip detector
Car with a trailor	Mouse-trap car
Leg of a SONY dog	Crank-piston mechanism
Swing-arm lamp	Stability of an underwater vehicle
Bicycle	Tank
Dynamics of whiplash injury	Unicycle
Four-bar mechanism	Boomerang's trajectory

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1. Books on reserve and their charged/not-charged status can be found right here on the web.
   Go to Franklin catalog and click on Course Reserve under Search Type and choose "MEAM535" or "Ananthasuresh".
2. Homework #1 solution
3. Homework #2 solution
4. Homework #3 solution
5. Recitation class 1 (10/5/99) notes
6. Homework #4 solution
7. Recitation class 2 (10/7/99) notes
8. Recitation class 3 notes (10/14/99)
9. Homework #5 solution
10. Recitation class 4 notes
11. Homework #6 solution
12. Recitation class 5 notes
13. Homework #7 solution
14. Recitation class 6 notes
15. Mid-term I solution
16. Homework #8 solution

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1. Summary of week 1
2. Summary of week 2
3. Summary of week 3
4. Summart of week 4
5. Summart of week 5
6. Summart of week 6
7. Summart of week 7
8. Summart of week 8
9. Summart of week 9
10. Summart of week 10
11. Summart of week 11
12. Summart of week 12

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Created: September 3, 1999
Last Updated: December 28, 1999.