16.241 Advanced Structural Dynamics (Fall '00)

Prof. Carlos E. S. Cesnik (instructor) 33-316 x2-1518 ccesnik@mit.edu

Ms. Cathy Chase (course secretary) 33-309 x3-6339 cmchase@mit.edu

This course is a collection of special topics in structural dynamics. It builds on basic material presented in 16.221 and introduces students to the more advanced concepts and tools used in this field. At the end of this course, the students will:

• become more familiar with the important issues and philosophies associated with time-dependent problems in structures,
• become conversant in the terminology of structural dynamics, and
• achieve a working understanding of these issues applied to various structures with a particular focus on aerospace structures.

There will be three types of assignments during the term: problem sets, take-home exams, and a term project.

Problem Sets: There will be approximately six problem sets during the term. These will be handed out on a relatively regular basis and generally have a week period for completion. Late submission of problem sets is discouraged. Maximum credit on late problem sets will be degraded by 33% per class day. After 3 class days, a zero will automatically be recorded. This policy may be changed if necessary to get solutions out in a timely fashion (before a test or other event). Solutions will be handed out on average one week after problem sets are due (and hopefully when they are returned). Solutions may be obtained earlier in special cases in which case the date when no credit is recorded for the problem set will be moved up.

Exams: There will be two take-home exams during the term, one at approximately mid-term, and one during the last week of classes. Several days will be given for the completion of the exam. The exact handout and due dates will be announced at least two weeks prior to the exam.

Project: The term project consists of studying the technique of Volterra Series (impulse response) to create reduced order models for transient solutions of nonlinear problems. The specific structural problem will be chosen by the students. A written description of the problem, along with direct time simulations of it, must be submitted to Prof. Cesnik by October 5th. Detailed description of the method and its implementation, the problem of choice, and numerical simulations will be handed by the student in a final written report by December 5th. Also, each student will summarize the findings in an oral presentation to the class (time length to be determined) in December 7th. In this way, the entire class can benefit from the term projects of all students.

Grading: The final grade will be calculated approximately as follows:

Attendance, participation, general evaluation ±5%

The course will be graded on an absolute scale using the letter grades as defined in the MIT Faculty Rules and Regulations:

A - Exceptionally good performance, demonstrating a superior understanding of the subject matter, a foundation of extensive knowledge and a skillful use of concepts and/or materials.

B - Good performance, demonstrating capacity to use the appropriate concepts, a good understanding of the subject matter, and an ability to handle the problems and materials encountered in the subject.

C - Adequate performance, demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation for moving on to more advanced work in the field.

D - Minimally acceptable performance, demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems, but also demonstrating deficiencies serious enough to make it inadvisable to proceed further in the field without additional work.

F - Unsatisfactory performance.

Plusses and minuses will be used throughout the term.

A Note on Submission of Work: The manner in which you present your work can be just as important (and in some cases more so) than the final answer. Be sure to delineate each step along the way. Show a clear and logical approach to your solution. That makes your problem sets a better reference to you and easier for us to give you partial credit (if so deserving). That is also the way a good engineer works, so it is an excellent habit to acquire. Remember, in this course the solution may be an essay or a report, not just a number. So, students are encouraged to fully explain their approach and implementation of a problem solution. The mere manipulation of equations is not the only important skill, but the ability to properly approach the problem and apply the knowledge gained to date.

The fundamental topics to be covered during the term can be summarized as:

I. Introduction: continuous structures; discrete point methods; modal methods; substructuring; Rayleigh-Ritz; finite element; condensation.

II. Dynamics Response and Transient Stresses: general equations for dynamic response and application to aircraft; transient stresses; force summation and mode displacement methods; aircraft landing problem; aircraft gust problem; more accurate analysis; integration schemes; building in earthquake; Timoshenko beam.

III. Dynamic Instability of Structures: instability from nonconservative forces, static, dynamic; cantilever beam with follower force; frequency coalescence, mode shapes, effect of damping; flow through cantilever pipe; introduction to parametric excitation; Mathieu equation; general harmonic balance for parametric solution.

IV. Instabilities of Rotating Structures: rotating beam with disk; effect of static unbalance and gravity; whirl motions; shaft with many disks; ground resonance of windmills; multiblade coordinate; stability.

V. Effects of Nonlinearities: introduction to nonlinearities; beam vibrations; Duffing equation; harmonic, superharmonics, subharmonics; nonlinear panel flutter; other nonlinear problems.

A post factum syllabus will be handed out at the end of the term.

The fundamental principle of academic integrity is that you must fairly represent the authorship of the intellectual content of the work you submit for credit. In the context of this course, this means that if you consult with others in the process of completing homework, you must acknowledge their contribution in a way that reflects their true ownership of the ideas and methods you borrowed.

Discussion among students to understand the home problems or to prepare for quizzes is encouraged. Some exercises will be deliberate team exercises, in which one cooperative piece of work will be handed in. Even in cases where individual answers are expected, collaboration on homework is allowed so long as all references (both literature and people) used are named at the end of the assignment. Word-by-word copies of someone else's solution or parts of a solution handed in for credit will be considered cheating unless there is a reference to the source for any part of the work which was copied verbatim. Failure to cite another student's contribution to your homework solution will be considered cheating. Official Institute policy regarding academic honesty can be found in the current Bulletin under "Academic Procedures and Institute Regulations". Cases of academic dishonesty are a severe breach of the student's and engineer's code and will be treated appropriately.

Study Group Guidelines: Study groups are considered an educationally beneficial activity. However, at the end of each problem on which you collaborated with another student, you must cite the students and the interaction. The purpose of this is to acknowledge their contribution to your work. Some examples follow:

1. You discuss concepts, approaches, and methods that could be applied to a home problem before either of you start your written solution. This process is encouraged. You are not required to make a written acknowledgment of this type of interaction.

2. After working a problem independently, you compare answers with another student, which confirms your solution. You should acknowledge that the other student's solution was used to check your own. No credit will be lost if the solutions are correct and the acknowledgment is made.

3. After working a problem independently, you compare answers with another student, which alerts you to an error in your own work. You should state at the end of the problem that you corrected your error on the basis of checking answers with the other student. No credit will be lost if the solution is correct and the acknowledgment is made.

4. You and another student work through a problem together exchanging ideas as the solution progresses. You both should state at the end of the problem that you worked together jointly. No credit will be lost if the solutions are correct and the acknowledgment is made.

5. You copy all or part of a solution from a reference such as a textbook or a bible. You should cite the reference. No credit will be lost assuming the solution is correct and pertinent to the problem statement. You receive credit because you showed the judgment to look in the literature for solutions to the problems. However, doing this without understanding the solution is unwise!

6. You copy verbatim all or part of a solution from another student. This process is strongly discouraged. You will lose credit for verbatim copying from another student when you have not made any intellectual contribution to the work you are both submitting for credit.

7. Verbatim copying of any material, which you submit for credit without reference to the source, is considered to be academically dishonest. At the least, you will receive no credit for the problem set or test in question.

Main References:

• Bisplinghoff, R., Ashley, H., and Halfman, R. L., Aeroelasticity, Dover, 1955. (TL570.B622)
• Bisplinghoff, R. and Ashley, H., Principles of Aeroelasticity, Dover, 1962. (TL570.B623)
• Bolotin, V. V., Dynamic Stability of Elastic Systems, Holden-Day, 1964. (QA931.B693 c.6)
• Craig, R. R., Structural Dynamics: An Introduction to Computer Methods, Wiley, 1981. (TA654.C72)
• Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, 1986. (QA935.M53)
• Meirovitch, L., Analytical Methods in Vibrations, MacMillan, 1967.
• Washizu, K., Variational Methods in Elasticity and Plasticity, Pergamon, 2nd ed., 1974. (QA931.W319)

Others:

• Atluri, S. N. (editor), Computational Nonlinear Mechanics in Aerospace Engineering, Progress in Astronautics and Aeronautics, Vol. 146, 1992. (TL790.P964 v.146)
• AGARD Lecture Series 191, Non Linear Dynamics and Chaos, 1993. (TL507.N867 no. 191)