Department
of Mechanical and Materials Engineering
This is Dr. Levy’s EML 3222 System Dynamics Summer 2007 page
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My office is in EC3462, and email address is levyez@fiu.edu
my tel no. is 305-348-3643, department fax no. is 305-348-1932
Office hours are M, W and R 1300-1430
Please start reading Chapters 1 and 2
lecture 1: page 1, page2, example, page 3
lecture 2: page 4, page 5, page 6, page 7, page 8
HW # 1 is due 5/14 Monday: Problems 1.1, 1.4, 2.2, 2.4
lecture 3: page 9, page10, page11, examples, page12
lecture 4: page 12, page 13, page 14,
lecture 5: page 15, page 16, page 17, the following information deals with electrical elements: capacitors, inductors and resistors. They are like electrical “masses”, “springs” and “dampers”. Please read and understand. The material will not be covered in depth in class but you will be responsible for it. I will skim over the material to cover the main elements, transformers and transducers. Just note the similar way we can describe electrical elements.
The information on electrical elements begins at the end of page 17, page 18, page 19, page 20,
Lesson 6: examples with electrical elements, examples with electrical elements 2, page 21, page 22, page 23, page 24, page 25,
HW1 solution: problem 1, problem 2a 2b, problem 3a 3b, problem 4
HW #2 is due 5/25 Friday: Problems 2.11, 12, 13, 14, 15
lecture 7: page 25, page 26, page 27, page 28, page 29
lecture 8: page 30, page 31, page 32, page 33, page 34, example solution. I will not be here on Wednesday 5/23. Here is the tape for Wednesday class for you to view from the comfort of your home/office.
WE WILL HAVE
lecture 9 page 35, page 35a,, page 35b (note problem 2-23 here is like problem 6.1b in your book), page 36, page 37,
HW #3: Do 6.1b in your book to replace the system by an equivalent rotational system (i.e., find Jeq) and also replace the system by an equivalent linear mechanical system (i.e., find m eq). Also do problem 2.17 in your book. Finally, please do problem 2.33 on the following pages page HW3prob3, page HW3prob3a Also read chapter 3 and chapter 4. HW due on Friday June 1
Monday June 4 is the first quiz-more information next
week.
lecture 10: page 37, page 38, page 39, page 40, page 41, page 42, page 43, page 44
Because of a prior commitment, there will be no class on Friday June 1. However, I will be videotaping the June 1 class. The videotape should be available for viewing over the weekend as I will be putting it and the material on the website and in the library annex. I have your second homework graded and you can pick it up from my office from 11-1 and 3-5pm on 5/31.
The first exam is Monday June 4 and it will be in
EC1104 just for Monday. Insure that ALL cellphones are off and that all bathroom
requirements are completed prior to the start of the exam. No one will be allowed out of the classroom
until the end of the exam and the exam has been turned in.
The exam will consist of two parts.
The first part
is closed book. You should know similarities and differences
between systems. You should know the
elemental and constitutive equations for the different type elements. You should know across variables, through
variables, integrated across variables, integrated through variables for each
and every system. You should know how
systems and system elements can be defined, e.g., static, dynamic, stochastic,
and what kind of governing equations can be generated by different types of
elements involved in your system. You
should know how nonlinear elements can be linearized.
The second part
only allows the handout called appendix 2 modeling of system elements
that contains formulas for dampers, springs, inductances etc. You will be asked to do some basic
computations and solve problems of the type given to you for homework.
I should be in on Thursday
5/31 from 11-1 and from 3-5, and, Friday 6/1 from 1-3 for any questions you
might have; or, you can contact me via email. Homeworks due on Friday
June 1 need to be turned in to my office by 1pm on Friday, or can be emailed or
faxed to me by 1pm. Fax is 305-348-1932.
Lecture 11: Here is the tape for Friday class, page 44, page 44solution, page 45-46 examples, page 45-46solns, page 47, page 48, page 49, page 50,
HW2 solution: problem 1, problem 2a, 2b, problem 3, problem 4, problem 5
Friday 6/1
update: Please note change in classroom where the exam will be…it has been
changed BACK to our classroom EC1115
HW3 has been turned in and no other submissions are now accepted. Here are the solutions 1, 2, 3
Lecture 12: exam 1
Lecture 13: The following pages are review from the tape: page 47, page 48, page 49,page 50, we begin new material from here: page 51, page 52, page 53,
Lecture 14: page 53, page 54, page 55,
HERE IS HOMEWORK 4.
PLEASE DO PROBLEMS 4.2, 4.3, 4.6d and 4.11d – FOR EACH OF THE PROBLEMS
DRAW THE SYSTEM GRAPH AND DERIVE THE CONTINUITY AND COMPATIBILITY EQUATIONS FOR
EACH. HOMEWORK IS DUE ON FRIDAY 6/15
Lecture 15: page 55, page 56 . These are the solution to page 54 top and bottom problems. Please note that we will work on solving the governing equations in class. We also worked on problem 4.4 from your book. The problem can be obtained from your fellow classmates if you were unable to attend. A handout was given which will be discussed in Wednesday 6/13 lecture. When you come to class you will be given a copy.
Lecture 16: Please bring the handout given last class with
you. page 57, page 58, page 59. Here we will discuss state variables and how
to get the equations
Lecture 17: page 58, page 59, page 60, page 61, page 62,
HERE ARE SOME MORE EXAMPLES YOU CAN USE TO DERIVE THE
ELEMENTAL, NODE, PATH and STATE EQUATIONS ( 1
AND 2 )
THE SECOND REGULAR EXAM WILL BE ON JULY 2 and will cover
the materials from exam 1 through the development of the state equations and
the solution of the equations.
You are given the following assignment (HW 5) due 6/22: develop the state equations for three problems -- the two problems we did (given on page 54, 55 and 56 of the notes on the website) and problem 6.15 in your book.
The week of 6/17-6/22 we will be going over the analytical method of solution of the state equations. The website contains the material to be covered for that lesson (see pages 61-63). Please download and bring the sheets with you to class.
Lecture 18: page 61, page 62, page 63 ,
page 64 Here we will discuss solutions of the state
variables equations analytically.
For those who need a review of matrices, here it is.
I AM INCLUDING HERE TWO PDF FILES THAT MIGHT HELP YOU: ONE IS
ON LINEAR GRAPHS AND ONE IS ON STATE EQUATIONS. THESE DOCUMENTS APPEAR TO BE THE PRECURSOR
DOCUMENTS THAT LED TO THE BOOK SYSTEM
DYNAMICS, AN INTRODUCTION, BY LOWELL AND WORMLEY.
Also look at the STATE EQUATIONS FILE which comes from your book to find more examples for deriving state equations. Please note that the figure numbers used in the file can be found at the end of the file.
Do the following problems for HW6 : Solve these 
and 
and 
These are due on Wednesday 6/27
HERE are HOMEWORK 4
solutions. PROBLEMS 4.2, 4.3, 4.6d and 4.11d
Lecture 19: We
will be solving a simple heat transfer problem given in problems 4.14. and 5.20
in your books. We will also do some
review of obtaining the state equations for the following problem 2 and here is the solution for that problem with
deletion of two state variables.
Lecture 20: we solved problem 4.9 in the handout (Shearer
and Kulakowski) given and we did the following matrix problem,
Lecture 21: we will start numerical methods given in page 65, page 66, page 67 and a handout to be given and explained in class.
Here are the solutions to the problems of page 54, 55 and
56 (HW5): Problem 1, Problem 2. The following is the solution for
problem 6.15 (a), (b). Once solutions are posted, no further
homework submissions will be accepted.
As far as exam 2 is concerned:
It will be on July 2—
1- YOU WILL BE ALLOWED TO BRING IN 7 8.5
X 11 SHEETS OF EQUATIONS BUT NO SOLUTIONS OF ANY KIND. BRING A WORKING CALCULATOR...YOU WILL NOT BE
ALLOWED TO BORROW ONE.
2- The materials that will be covered on the exam include: LINEAR GRAPHS, SYSTEM GRAPHS, DERIVING COMPATIBILITY,
CONTINUITY, ELEMENTAL EQUATIONS, STATE VARIABLE EQUATIONS. SO GIVEN A SYSTEM, YOU SHOULD BE ABLE TO
DERIVE THE SYSTEM GRAPH, THE CONTINUITY, COMPATIBILITY, ELEMENTAL EQUATIONS AND
THE STATE VARIABLE EQUATIONS.
3- YOU SHOULD ALSO BE ABLE TO SOLVE ANALYTICALLY (SUCH AS THE
HOMEWORK 6 ASSIGNMENT) MATRIX DIFFERENTIAL EQUATIONS OF THE FORM 
.
4- PLEASE MAKE SURE YOUR EQUATION SHEETS INCLUDE THE MATERIAL
ON TRANSFORMERS AND TRANSDUCERS AS WELL.
5- NUMERICAL METHODS WILL NOT BE COVERED ON THIS EXAM.
By Wednesday, you
are to form a team of 4: you will be asked to choose a process that
you can model, with my OK. You will then
be asked to define the linear graph and get all equations as well as state
equations and you will be asked to solve the state equations for some
reasonable set of data.
Lecture 22: page 67, a handout will be given in class on the Runge-Kutta method. The handout was from Applied Numerical Analysis by Gerald and Wheatley, 7th Edition, pages 340-342. I have copies in my office. If you missed class, stop by my office to get your copy. Also, Please read the first two sections of chapter 11 of your book on numerical methods. We looked at several examples and how they could be used in the different numerical methods discussed, namely: Euler, Modified Euler, Runge-Kutta Second Order method, Runge-Kutta Fourth Order Method.
Here is the solution to the problems for HW6 – Problem 1 and 2, Problem 3. No more homeworks will be accepted once the solutions are revealed.
Lecture 23: This problem was discussed in class. Also here is the solution . Friday 6 July we will also start discussion of the behavior of first order systems (such as a damper and spring, or mass and damper) and second order systems (such as mass-spring-damper). Read Section 8.3, and Chapter 9.
Lecture 24: exam 2.
Lecture 25: Friday 6 July we will also start discussion of the behavior of first order systems (such as a damper and spring, or mass and damper) and second order systems (such as mass-spring-damper). Read Section 8.3, and Chapter 9. Here is some reading material from the authors of your book that covers the topics we will touch on (First and Second Order response)
Project Information: You will be expected to produce a project
report by the 18th of July in a form similar to your lab
reports. The report will include:
An introduction detailing what you
are planning to model, who your team members are and what they have contributed
to the project;
Modeling section in which you give
the modeling of your project and the matrix form of those equations;
A results section in which you
detail the numerical equations you will be using to solve the problem, the parameters
you will be using in your equations and where they have been obtained, the
graphs of the state variables as a function of time, the outputs you want to
find as a function of time;
and finally a Discussion of your
results. Also discuss the step-size you use and its effect on the accuracy of
the results
You will need to vary at least 3
parameters to see the effect of parameter variation on your solutions. Your discussion section should be in depth.
Here are the solutions to problem 1 of both tests. Here is one of the solutions to problem 2 and here is the other solution to problem 2.
Lecture 26-27: In lesson
26 we covered the solution to the homogeneous part of the equation: 
. which we rewrote as 
Here are the pages dealing with what was discussed on July 9 and what we will discuss on July 11. page 1, page 2, page 3, page 4. Note that you have some materials covered in class (on overshoot, time constants and first order equations for example), that are not found in these sheets. Also, compare your notes to these pages to insure that the same notation is used and that the variables are defined in the same manner.
Please do the following problems due on Monday July 16: Do problems 8.9d and problem 8.13 all parts. Here the function us(t) represents the Heaviside step function (=1 for t>0 and 0 for t<0). Also do problem 9.11. For part (a) only do part (i).
Lecture 28: We finish the problem we started on Wednesday 7/11. Here it is for you to look over and here is the solution graph. Note that the solution is tending towards Vm =0.2 m/s which is the steady state solution of the problem due to the step increase to Vm=0.2 m/s that occurs at t=0+
Lecture 29: There will be no lecture on July 18. However, your homeworks have been graded and here is the solution to some of the problems. You can pick up your homework when you turn in your projects. Projects are due by 1pm tomorrow in my office EC3462.
Final exam is at 1140 am on Friday 20 July in EC 1115. Don’t forget you will be allowed to bring in 10 sheets of 8.5 x 11 inch paper with whatever information you wish on it. Materials from the last exam to the end will be emphasized but the exam is comprehensive. Make sure you have a working calculator and that you apply time management during the examination.