Department of Mechanical and Materials
Engineering
This is Dr. Levy’s EML3222 System Dynamics Summer 2018 page
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Here is the (8/23/18)
updated syllabus
for the course.
My office is in EC3442, and email address is levyez@fiu.edu
My tel. no. is 305-348-3643. My fax no. is the department fax no. is 348-1932
My Office
hours: M and W 230-400pm
TA
information: there is no TA at the present.
Photocopies
of the 3 material selections relating to vibrations will be available from the
department office starting May 7. Please
make up your groups of four and one of you come to get the materials. Cost is $15 per set. They
will be used in the vibrations section of the course, which will occur later on
in the semester. HWs will be assigned
from this source later in the semester as well.
Here is Lecture 1 video and here are the pages related to the first lecture: page 1, page2, example, page 3, page 4, page 5
Here is Lecture 2 video and the pages related to the video: page 6, page 7, and page 8. The materials we cover in class on 5/7 goes to the 1 hour mark on the video tape; the rest on 5/9.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Here is the material of the 5/9 classes: Lecture 3 video: At the 45 minute mark of the video we begin the topic of system dynamics by defining the through and across variables, the elemental and constitutive equations, the ideal and pure element. page 9, page10, page11, examples, page12,
HW # 1 is due 5/18
Friday: Problems 1.1, 1.4, 2.2, 2.4 from
your Rowell and Wormley books
Here is the rest of the material for 5/9 that goes with Lecture 4 video: The video covers the rotational mechanical system elements page12, page 13, page 14, page 15, page 16, page 17,
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.
Here is the 5/11 material that will go with Lecture 5 video: The information on electrical elements begins at the end of page 17 and continues on page 18, page 19, page 20,
The information on electrical elements: capacitors, inductors and resistors are like electrical “masses”, “springs” and “dampers”. Please read and understand.
Start reading chapter 6 on
transducers in one energy domain (transformers) and transducers in multi energy
domains (transducers)
Lesson 6 continues the rest of
the 5/11 materials discussed in Lecture 5: examples
with electrical elements, examples
with electrical elements 2, page 21,
Here are the pages that go with Lecture 6 on Fluid systems: page 22, page 23, page 24, page 25, page 26, page 27,
Here is the material that will go with Lecture 7 video that covers more on the fluid systems and also begins thermal systems as well: page 26, page 27, page 28, page 29, page 30,
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Here is the material that will go with Lecture 8 video on thermal systems: page 30, page 31, page 32, page 33, page 34, example solution
Please read chapters 3 and 4 of Rowell and Wormley book on system graphs. We will start talking about those during the next lecture.
Here are the Problems 1.1, 1.4, 2.2, 2.4 and the solutions: problem 1, problem 2a 2b, problem 3a 3b, problem 4
Here is the material that goes with Lecture 9 video: page 36, page 37, page 38, page 39, page 40, page 41, page 42.
Here is the material that goes with Lecture 10 video: page 42, page 43, page 44, page 44solution, page 45-46 examples, page 45-46solns. Here is the material that goes with Lecture 10partb video: page 47, page 48, page 49,
Here is the material that goes with Lecture 11 video: page 49, page 50, and page 51. Here is the material that goes with Lecture 11 part b video: page 52, page 53, and page 54. The last portion deals with an example that involves a transformer and how to represent it in a system graph.
Please do problems 4.1, 4.2, 4.5
and 4.11 in your system dynamics books. Their solutions will be revealed before
the first examination.
Here is the material that goes with Lecture 12 video: page 54, These are the solution to page 54 top and bottom problems page 55, page 56. We also cover page 57, page 58,
We now begin talking about state equations.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
I AM INCLUDING HERE TWO PDF FILES THAT MIGHT HELP YOU: ONE IS
ON LINEAR
GRAPHS AND ONE IS ON STATE
EQUATIONS. THESE DOCUMENTS ARE THE
PRECURSOR DOCUMENTS THAT LED TO THE BOOK SYSTEM
DYNAMICS, AN INTRODUCTION, BY ROWELL AND WORMLEY.
You can find more examples for deriving state equations in these files. Please note that the figure numbers used in the file can be found at the end of the file.
You are also suggested to
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.
We now begin talking about state equation solutions. This solution methodology depends on understanding matrices. For those who need a review of matrices, here it is.
Here is the material that goes with Lecture 13 video: discussed state variables and how to get the state variable equations and how to solve state variable equations- page 57, page 58, page 59. Note that there is no link to page 60 for this semester.
As a review of system
graph and state equation derivations we look at this problem.
We also show the state equation solution which is given in the top half
of the page.
Here is the material that goes with Lecture
14 video: We discuss the following
system including getting the system graph. We also show the state
equation solution which is given in the bottom
half of the page. We have given a
handout for the system graph for a fluid system connected to a
piston/spring. We will derive the state
equations. Also, prob. 7.29 on that
sheet will be discussed and one of the state equations will be derived but you
are to derive the remaining state equations.
The first exam date is announced as 5/25 in which you will be given a system and you will be asked
to derive the elemental equations (including transformer/transducer equations
if any), the node equations (continuity), the path equations (loop,
compatibility), and also some if not all the state equations. Five Formula sheets allowed will be discussed. The exam will be 65 minutes
with the remaining time being devoted to a lecture. Lastly,
you should remember how to linearize nonlinear equations.
HERE are the solutions
to PROBLEMS 4.2, 4.3, 4.6d and 4.11d
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Here is the material that goes with Lecture
14-15 video: Halfway through the video we start discussing the numerical solution of
the state variable differential equations given in page 61,
page 62,
and page
63. This material will NOT be on the
first exam, but will be part of the second exam.
We continue our discussion of numerical solutions of the state equations in Lecture 16. The rest of the numerical solution of sets of first order differential equations will be discussed. A handout will be given in class on the Runge-Kutta method. The handout is from Applied Numerical Methods by James, Wilford and Smith, pages 339-345 and 351-353. We will look 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
We continue our discussion of numerical solutions of the state equations in Lecture 17. We will work on several problems (4th order and 2nd order RK methods).
The first exam date is announced as 5/25 in which you will be given a system and you will be asked to
derive the elemental equations (including transformer/transducer equations if
any), the node equations (continuity), the path equations (loop,
compatibility), and also some if not all the state equations. Five Formula sheets are allowed as
discussed. The exam will be 65 minutes with the
remaining time being devoted to a lecture. Lastly, you should remember how to linearize nonlinear
equations (see section 3.5 of Rowell book or my notes for first
lecture), know similarities and differences between systems, and basic
definitions. The examination will cover
Chapters 1-6 of the Rowell and Wormley book BUT NOT the NORMAL TREE topic.
The second exam is scheduled for June 4 and it will cover materials
related to the numerical methods part of the course, namely lectures
14-17. More about what materials you can
bring in to the exam later.
Please start reading the first and second section-Chapter 1 and 2 materials from the materials obtained from the office manager (EC3475). I will be assigning examples out of those materials. So it would be to your benefit to come pick up the copies of the materials for yourself.
Out of the photocopies of the do the
following: Problems 1.7 to 1.10, 1.13,
1.16, 1.19, 1.22, 1.26, 1.27, 1.29, 1.31, 1.32, and 1.36. Solutions will be posted later.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Please make sure that you have the following information about second order systems, namely:
where g is the acceleration of gravity and Dst is the static displacement of the system; that
is, weight of the system = k*Dst.
Please use the Greek capital letter D for the static displacement instead of the
normal d, as the letter d is to
be used for the logarithmic decrement, and they should not be confused.
And for systems with damping included: =2ςωn and =ς and Ccrit=2√(mk) =2mωn =2k/ωn
Here is Lecture 18 video: equation of motion for linear mechanical, linear rotational, inverted pendulum
Here is Lecture 19 video: equivalent springs both linear and rotational, springs in parallel and series
Here is Lecture 20 video: equivalent springs and masses using equivalent potential and kinetic energies
Now for underdamped systems: the logarithmic
decrement, δ, equals
where ζ is the damping ratio.
Also we show that δ =
(1/N) * ln(xo/xN) = (1/N) * ln(xi/xi+N)
where N = the number of cycles between the
first and last measurement. x here is
the displacement and the subscript “o” means the first measurement value. The formula also applies between any N
cycles, meaning starting from cycle i and going to cycle i+N.
We will discuss the forced vibration of systems
and cover the topic of resonance.
Exam 2 will be on June 4 and will cover the numerical solution and analytical solution of state equations. You will be allowed pens, pencils, straight edge, erasers and your calculator. You will be allowed two (CORRECTED) 8.5 x 11 formula sheets.
Please ensure your formula sheets include the formulas for the Euler, Modified Euler, Runge Kutta 2nd order, Runge Kutta 2nd order Midpoint, 4th order Runge Kutta and how to handle numerically the solution of more than one state variable equation. Any solutions found on your formula sheets will have the formula sheet confiscated.
A REMINDER: I have given you a chance to redo exam 1 in its entirety and to turn it in with the original exam. The retest needs to be turned in by Noon, June 1—NO EXCEPTIONS! So, please turn in the retest at the start of the class.
Please start reading the first and second section-Chapter 1 and 2 materials from the materials obtained from the office manager (EC3475). I have assigned examples out of those materials. So it would be to your benefit to come pick up the copies of the materials for yourself.
June 4 we will have the numerical methods exam and you will be allowed 2 (TWO) 8.5 x 11 formula sheets. We will also be having a class as well on the mechanical vibrations of undamped single degree of freedom systems. The materials for this class is in Chapter 1 and Chapter 2 of the notes obtained from the office manager.
The retest and the second
exam were returned on Wednesday June 6.
Because the average of the class was close to 75, no retest of the
second examination will be given. On
Friday June 6, we begin discussion of the damped vibration systems. Please read Chapter 3 in your handouts
obtained from the secretary. Please note
that we have the third exam tentatively scheduled for June 11 and it will cover
Chapters 1 and 2 of the Handout.
Here is Lecture 21 video: we look at damped systems, derive equations and talk about overdamped, critically damped and underdamped systems
Please make sure that you have the following information about second order systems, namely:
where g is the
acceleration of gravity and Dst is the
static displacement of the system; that is, weight of the system = k*Dst.
Please use the Greek capital letter D for the
static displacement instead of the normal d, as the
letter d is to be used for the logarithmic decrement,
and they should not be confused.
And for systems with damping included: =2ςωn and =ς and Ccrit=2√mk =2mωn =2k/ωn
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Now for underdamped systems ONLY: the logarithmic
decrement, δ, equals
where ζ is the damping ratio. Also we show that δ =
(1/N) * ln(xo/xN)
= (1/N) * ln(xi/xi+N)
where N = the
number of cycles between the first and last measurement. x here is the
displacement and the subscript “o” means the first measurement value. The formula also applies between any N
cycles, meaning starting from cycle i and going to
cycle i+N.
Here is Lecture 22 video: we look at underdamped systems, derived logarithmic decrement and talk about two problems.
Look at Problems 2.1-2.4, 2.6, 2.7, 2.17 to 2.19, 2.28, 2.38, 2.45, 2.52, 2.60, 2.80, 2.82, 2.83, and 2.97 in the handouts obtained from the department.
Monday 11 June is the third of
the inclass exams that covers the vibration section
of the course. This will include free vibration of
undamped system. Free vibration of
damped systems (underdamped, overdamped, critically damped). You are allowed 3 (NOTE CHANGE) 8.5x11 inch
pages of formula sheets only.
On 13 June we will discuss forced vibration of single degree of freedom systems found in Chapter 3 of the notes obtained from the department office manager in room EC3475. You will also be given handouts related to this.
Here are solutions to some of the problems… problems 1-7 and 1-8, 1-9 and 1-10
Here are solutions for problems 1-13a and 1-13b, 1-29, 1-30, 1-35 and 1-35b
Final exam is comprehensive
(vibrations-free and forced vibrations of undamped and damped systems, as well
as system graphs-node equations, elemental equations, loop equations and state
variable equations, and numerical and analytical solutions of systems) and will be given on June 15 between
1200 and 215pm. Location is our
classroom. More on what you can
bring in will be announced on June 11. Please keep all your formula
sheets as you will need them for the final—you will be allowed 9 formula pages.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Here are problems similar to 2-4, 2-6 and 2-7 . Problem 2-7 requires moving the springs to the location of the mass.
Here are problems 2-17, 2-18, 2-19. In problem 2-17 both springs see the same displacement. For 2-18 use the equivalent k for extension of a wire.
Here are problems like 2-28, 2-38
Here is the rest of 2-38, and problems 2-68, 2-79
Here is problem 2-45
Here is problem 2-60
Here, the second problem is like problem 2.80. Note, overshoot represents the maximum displacement above the x=0 line, i.e., when the velocity=0.
Here are problems the first two are like 2.83, 2.97
The following problems deal with forced vibrations from Chapter 3 of the handouts obtained from the
department office manager. Here are
problems
3.1, 3.2, 3.8, 3.10. Also try problems 3.25, 3.26—No solutions will
be given for these…
The retest and the second
exam were returned on Wednesday June 6.
Because the average of the class was close to 75, no retest of the
second examination will be given. On Friday
June 6, we begin discussion of the damped vibration systems. This material is discussed in Lecture 21. The third
exam is tentatively scheduled for Monday June 11 and will cover Chapters 1 and
2 of the Handout obtained from the secretary. Please read Chapter 3 in your
handouts obtained from the secretary.
The
third exam is tentatively scheduled for Monday June 11 and will cover Chapters
1 (undamped vibration of SDOF systems) and Chapter 2 (damped vibration of SDOF
systems) of the Handout obtained from the secretary. You are allowed 3 (NOTE CHANGE) 8.5x11 inch pages of formula sheets only.
PLEASE PICK UP ALL YOUR PREVIOUS EXAMS FROM MY OFFICE TO HELP YOU PREPARE FOR THE EXAM
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Friday’s class (6/8) covered damped systems with friction and with viscous
damping. The following handouts were
given that related the subject of friction and with viscous damping friction document, viscous
document-1, viscous
document-2, viscous
document-3, viscous
document-4. Also we gave out the
following documents and did several problems: sheet
1-problem statement and solutions sheets
1
Please make sure that you have the following information about second order systems, namely:
where g is the
acceleration of gravity and Dst is the
static displacement of the system; that is, weight of the system = k*Dst.
Please use the Greek capital letter D for the static
displacement instead of the normal d, as the
letter d is to be used for the logarithmic decrement,
and they should not be confused.
And for systems with damping included: =2ςωn and =ς and Ccrit=2√mk =2mωn =2k/ωn
Now for underdamped systems ONLY: the logarithmic
decrement, δ, equals
where ζ is the damping ratio. Also we show that δ =
(1/N) * ln(xo/xN)
= (1/N) * ln(xi/xi+N)
where N = the
number of cycles between the first and last measurement. x here is the displacement
and the subscript
“o” means the first measurement value.
The formula also applies between any N cycles, meaning starting from
cycle i and going to cycle i+N.
Please
note that the materials handed out Friday including the solutions are highlighted
in blue above. Also the link to the last
two pages have been fixed. The solution
sheets also have some important formulas that might be useful to you.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Here
is a video related to quick
review of damped SDOF systems, introduction to forced vibration, beats,
undamped forced motion. The relevant
parts to forced vibration part of the course is the undamped forced motion
discussion
Here
is a video related to rotating
unbalance. The material starts at
the 26 minute mark.
Here are problems 3.1, 3.2, 3.8, 3.10
Try problems 3.25,
3.26—No solutions will be given for these
Here are problems 3.29 and 3.33part1, 3.33part2 and 3.32
Here are problems 3.54, 3.58
Here are solutions to problem 3.53. Problems 3.25 and 3.26 solutions can be viewed privately.
èFinal
exam will be on June 15; time: 12:00-130pm.
EC 1112 is the exam room.
Final exam is comprehensive
(vibrations-free and forced vibrations of undamped and damped systems, as well
as system graphs-node equations, elemental equations, loop equations and state
variable equations, and numerical solutions of systems)
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, except for NO solutions
from the book or from problems we have worked out or have been assigned.
PLEASE PICK UP ALL YOUR PREVIOUS EXAMS FROM MY OFFICE TO HELP YOU PREPARE FOR THE EXAM
Please
make sure to be seated so that we can start immediately at 1200. I would like to complete and clear the room
by 130pm.
You
will be given 3 problems: read all the problems. 1 you will be required to do, and you must
choose 1 of the remaining two problems to do.
The choice problems are marked "choose this problem or problem
xx".
Good
luck tomorrow and use good test taking strategy that we talked about during
Wednesday's review.
If you wish, we can have a review
session at 2pm, Thursday for 90 minutes. I will ask about this on Wednesday’s class.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.
Attached are the notes from the Wednesday
class with one problem
worked out. Please note that we also
reviewed some of the other problems from handouts given in class on Monday or
mailed to you if you are doing the course remotely due to work (forced
vibration problems of Chapter 4 of Vierck as well as
the solutions of many of those problems).
Please remember the exam is comprehensive and will cover system graphs, numerical methods and vibrations (undamped, damped, forced damping). Please make sure that your formula sheets reflect those topics. Good luck!
With respect to the sheet that you are allowed to bring
into the exam in which you have your numerical methods formulas:
Please make sure you have the information for all the
numerical methods we discussed either in class and/or in the
numerical methods handout given in class, namely
1)
Euler
2)
Modified Euler
3)
Runge-Kutta
2nd order which is like the modified Euler
4)
Runge-Kutta
2nd order known as the midpoint method
5)
Runge-Kutta
3rd order
6)
Runge-Kutta
4th order
and how to use these
methods if you have more than one state variable equation, as well.
Attached are the problems worked out in the review session and the system graph worked out in the review session. The system graph in the review session did not have a fluid capacitance in the piston cylinder whereas this one does. The problems that were completed were those on rotating unbalance, an overdamped, critically damped and underdamped system. The last 3 were from a paper that was handed out in class last week or was mailed to you.
This
material and all the linked materials provided, except where stated
specifically, are copyrighted © Cesar Levy 2018 and is provided to the students
of this course only. Use by any other
individual without written consent of the author is forbidden.