ME6443 Variational Methods in Engineering

Fall 2007

	Professor Al Ferri
	Office: GTL Rm 205 107 Phone: 03 87 20 39 27 Email Address: al.ferri@me.gatech.edu Office Hours: Mondays and Wednesdays, 11am-11:30am Class Meeting Time/Place: Tuesdays and Thursdays, 9:30am-11am, Red Classroom (216) Textbook: Calculus of Variations, R. Weinstock, Dover Publications, 1974 The best way to reach Prof. Ferri is via email.

Course Information

Class Syllabus

Prerequisite: Graduate Standing

Course Catalog Description: 

Calculus of Variations, Hamilton's principle and Lagrange's equations, Sturn-Liouville problems, approximation techniques.

Announcements/New Items:

• Announcement 1: I will be out of town next week (Sept 3 through Sept 7).
• Announcement 2: Homework Solution 1 and Homework Assignment 2 are available in the table below.
• Announcement 3: Homework Solution 2 and Homework Assignment 3 are available in the table below.
• Announcement 4: Homework Solution 3 and Homework Assignment 4 are available in the table below.
• Announcement 5: Test 1 will be Thursday, Oct 4 from 9:00-10:00am. It will be closed-book, closed-notes, and closed-calculators; You will be allowed to bring ONE 8.5inch-11inch or A4 sheet of paper with you, on which you can write whatever formulae or other information you think is important. The formula sheet MUST BE HANDWRITTEN.
• Announcement 6: Solution to Test 1
• Announcement 7: Grade Distribution for Test 1
• Announcement 8: Homework Solution 5 and Homework Assignment 6 are available in the table below.
• Announcement 9: Homework Solution 6 and Homework Assignment 7 are available in the table below.
• Announcement 10: Homework Solution 7 is available in the table below. Note that the solutions use slightly different notation than that used in the problem statement.
• Announcement 11: Test 2 will be on Tuesday, Nov 13, from 9:45am-11am. It will cover all of the material that we covered since the last test, up to and including the vibrating string. Specific topics include: 1. Differential and finite constraints, 2. Optimal control, 3. Fermat's principle for wave propagation, 4. The "delta-notation", 5. Lagrange's equations and Hamilton's Principle, 6. Treatment of holonomic and nonholonomic constraints, 7. Two-dimensional problems in the Calculus of Variations, 8. The vibrating string, 9. Geometric and Natural BC's, 10. Eigenvalues and eigenfunctions, 11. Orthoganality/normalization of eigenfunctions, 12. Total solution in the form of an infinite summation of eigenfunctions. The test will be closed-book and closed notes.You will be allowed 2 handwritten sheets of paper (8.5inch-11inch or A4 size), which may be 2-sided. If you prefer, you can use 4 single-sided, handwritten sheets. Bring your calculators.
• Announcement 12: Homework assignement 8 (click here) is available in the table below. The assignement asks you to modify a Matlab program which is available at this link.
• Announcement 13: Solution to Test 2
• Announcement 14: Grade Distribution for Test 2
• Announcement 15: Here is a write-up on Weak Formulations.
• Announcement 16: Homework Solution 8 and Homework Assignment 9 are available in the table below.
• Announcement 17: Handout on the Method of Weighted Residuals.
• Announcement 18: Powerpoint presentation on Finite Element Method
• Announcement 19: Powerpoint presentation (updated) on Ritz Method for Dynamics/Vibrations Problems
• Announcement 20: Example on the Ritz Method for Linear Vibration.
• Announcement 21: The Course Instructor Online Survey (CIOS) website is now open for Fall semester courses. Please log in and let me know how you liked the course.
• Announcement 22: Problem 2 on HW9 has been corrected.
• Announcement 23: Homework Solution 9 is available in the table below.
• Announcement 24: The exam will be from 2pm-5pm on Wednesday, Dec. 12. The exam will be closed-book, closed notes, and closed calculator (bring one). You are allowed to bring three 8.5x11inch sheet of paper (two-sided) on which you can handwrite anything you want. It is important that this be handwritten- NO PHOTOCOPIES!. The exam will focus on the material since the second test. This includes: Two-dimensional calculus of variations, vibrating strings/rods/beams, Hamilton's principle for continuous systems, geometric or essential BC's and natural BC's, separation of variables, eigenvalues/natural frequencies, eigenfuncations/modeshapes, Ritz approximations, weak formulations, and the method of weighted residuals. The three finals (Fall 2002, Fall 2004, and Fall 2005) will give you a good idea of the scope.
• Announcement 25:
• Announcement 26:

Assignments

Over the course of the semester, there will be approximately 12 homework assignments worth 10 points each. I will drop your lowest grade. Working in groups is allowed but each student must turn in their own homework. Some assignments will be time-consuming and some will require computers to solve the problem and/or plot the results. The homework assignments are designed to help you to learn the material. If you do not put in sufficient effort, or if you are relying too heavily on the aid of others in your study-group, it will negatively impact your test and exam grades. Solutions to the homework will be posted to the website after the due-date is past; it is your responsibility to review these solutions in detail and to compare them with your returned homework assignments.

Links to the solutions are posted in last column when they are available.

Assignment	Due Date	Solutions
Hwk 1: Minimum/maximum of functions	August 30, 2007	HW Solution 1
Hwk 2: Introduction to Calculus of Variations	September 11, 2007	HW Solution 2
Hwk 3: Brachistochrone Problem	September 18, 2007	HW Solution 3
Hwk 4: Free Endpoint Problems	September 25, 2007	HW Solution 4
Hwk 5: Optics and Optimal Control	October 16, 2007	HW Solution 5
Hwk 6: Lagrange's Equations	October 23, 2007	HW Solution 6
Hwk 7: Lagrange's Equations with constraints	Nov 8, 2007	HW Solution 7
Hwk 8: Weak Formulations	Nov 27, 2007	HW Solution 8
Hwk 9: Method of Weighted Residuals (corrected)	Dec 6, 2007	HW Solution 9
Hwk 10
Hwk 11
Hwk 12

Grading Policy:

Test 1 TBA 25%
Test 2 TBA 25%
Final Exam TBA 35%
Homework 15%.

Academic Integrity:

Academic honesty is essential to achieve high-quality education and to maintain the value of a Georgia Tech diploma. While I encourage you to work together and to form study groups, it is important that you take responsibility for the content of all assignments. Cheating on tests and final exams will not be tolerated. When uncovered, violations will be reported to the Dean of Students immediately. A valuable resource for the Georgia Tech Student Code of Conduct and the Academic Honor Code is: http://www.honor.gatech.edu/honorcode/honorcode.html.

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Sample Tests from Previous Semesters and Quarters:

ME6443 Test 1, Fall Semester 2002

ME6443 Test 1, Fall Semester 2004

ME6443 Test 1, Fall Semester 2005

ME6443 Test 2, Fall Semester 2002

ME6443 Test 2, Fall Semester 2004

ME6443 Test 2, Fall Semester 2005

ME6443 Final Exam, Fall Semester 2002

ME6443 Final Exam, Fall Semester 2004

ME6443 Final Exam, Fall Semester 2005

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Resources and References

• Netscape Navigator (obtained from the Netscape home page) or Internet Explorer (obtained from the Microsoft IE home page) to view Web pages.
• MATLAB 5.3, available in ME computer clusters