Department of Mechanical and Materials Engineering

This is Dr. Levy’s EML3222 System Dynamics Spring 2019 page

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Here is the (01/07/2019) updated syllabus for the course. Please note that syllabus is updated

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 305-348-1932

My Office hours: M 10am-12pm and W 3pm-430pm or by appointment

TA information: there is no TA. If you have any questions please come to see me in my office.

It is suggested that you form a group of four students as your study group.

This material and all the linked materials provided, except where stated specifically, are copyrighted © Cesar Levy 2019 and is provided to the students of this course only. Use by any other individual without written consent of the author is forbidden.

Photocopies of the 3 material selections relating to vibrations will be available from the department office starting January 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.

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: Problems 1.1, 1.4, 2.2, 2.4 from your Rowell and Wormley books. It will be due Wednesday Jan 16, 2019 at the start of class (11am).

Here is the material 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,

Here is the 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 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. Please read Ch 2 and 3 of Rowell and Wormley. The information on electrical elements: capacitors, inductors and resistors are like electrical “masses”, “springs” and “dampers”. Please read and understand.

Also, start reading chapter 6 on transducers in one energy domain (transformers) and transducers in multi energy domains (transducers)

Lesson 6 continues the materials discussed in Lecture 5: examples with electrical elements, examples with electrical elements 2, page 21,

HW # 1 is: Problems 1.1, 1.4, 2.2, 2.4 from your Rowell and Wormley books. It will be due Wednesday Jan 16, 2019 at the start of class (11am).

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,

Here is the material that will go with Lecture 8 video: 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

HW was assigned. It is due Monday Feb 4 at the beginning of class: Problem 3.8 from page 29. Please do what the problem asks.

ALSO, b) for the same problem determine the fluid resistance of the penstock and determine the energy lost due to fluid resistance for the time period you found from the original problem statement. Hint: Assume that the power that you find when the penstock acts as the fluid resistor is constant.

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,

Please start reading Chapter 4 of your Rowell and Wormley book. This chapter will deal with line graph representations, compatibility and continuity equations, system graphs and how to determine the correct number of equations for unknowns for each system.

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. These problems are due on Wednesday, Feb 20.

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.

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. 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.

As a review of system graph and state equation derivations we look at this problem. We will release the state equation solution at a later date.

First exam is scheduled for Monday, March 4 at 12pm. Please be in your seats by 1155am so that the exams can be handed out. The exam will be 55 minutes long. The exam will test your knowledge of determining the system graph, the elemental, nodal and loop equations given a real system. Also, you will be required to identify the state variables and sources, and to derive the state variable equations for all or some of the state variables. You will be allowed 3 8.5x11 inch formula sheets, but those sheets cannot contain solved problems. You may add your own notes that help you to solve problems of the type we have discussed in class. All systems (linear mechanical, rotational mechanical, electrical, fluid and thermal) are testable.

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. Also, prob. 7.27 on that sheet was discussed and one of the state equations was derived but you should derive the remaining state equations to get used to deriving the equations yourself.

HERE are the solutions to PROBLEMS 4.2, 4.3, 4.6d and 4.11d

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. 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 2^nd order midpoint 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 (4^th order and 2^nd order RK methods). The video covers numerical solutions up to the 42 minute mark and does a system graph problem for the rest of the time.

Exam 2 will be on Wednesday, March 20 and will cover the numerical solution of state equations. You will be allowed pens, pencils, straight edge, erasers and your calculator. You will be allowed pens, pencils, straight edge, erasers and your calculator. You will be allowed three 8.5 x 11 formula sheets. Please ensure your formula sheets include the formulas for the Euler, Modified Euler, Runge Kutta 2^nd order, Runge Kutta Midpoint, 4^th order Runge Kutta and how to handle numerically the solution of more than one state variable equation. The exam will be one hour long. Any solutions found on your formula sheets will have the formula sheet confiscated.

On March 18, upon our return from spring break, we begin the vibrations portion of the course. Please make sure you have gotten the handouts from the department manager (EC3475) related to this part of the course.

ANNOUNCEMENT—

Ladies and gentlemen:

Because of the low class average on exam 1, I will allow you to redo the examination IN ITS ENTIRETY and turn it in by Thursday, March 21 at 1pm. You may drop off the re-test at my office, EC3442, or scan the exam (must be a readable scan) and email it to me (levyez@fiu.edu) to arrive by 1pm my time. You must do your own work. The pledge you signed on the original exam is still in effect. This is the only exam for which you will be given a re-test. Your retest grade and your original grade will be averaged. If you had a good score and do not plan to submit a re-test, please inform me of such.

Test 2 is this Wednesday. It will only cover numerical methods. You are allowed 3 8.5 x 11 sheets of formulas. No solutions allowed. Make sure that your sheets include: Euler, Modified Euler, Runge-Kutta 2nd order, 2nd order midpoint, 4th order for one state variable equation. You should have all these methods for two or more state variable equations, as well.

Please pick up the vibrations packet at the department office EC3475. Cost for the packet is 15 dollars. We will have a class after the examination.

Exam 2 on numerical methods is lecture 21

Here is Lecture 19 video: We begin the vibrations part of the course by deriving the equation of motion for linear mechanical, linear rotational, inverted pendulum (56 min. mark)

Here is Lecture 20 video: we define equivalent springs both linear and rotational, springs in parallel and series that can be used to solve vibrations problems.

Please make sure that you have the following information about second order systems, namely:

where g is the acceleration of gravity and D_stis the static displacement of the system; that is, weight of the system = k*D_st.

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: =2ςω_n and =ς and Ccrit=2√mk =2mω_n =2k/ω_n

ANNOUNCEMENT—

Due to an incident that occurred during the 2^nd examination, a new restroom policy is being instituted. Please take care of your restroom needs before any examination. No restroom breaks will be allowed for the remaining two examinations. Please be mindful of this policy before bringing in any food or drink into any future examination.

Please pick up the vibrations packet at the department office EC3475. Cost for the packet is 15 dollars.

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 for your group. Out of the photocopies of the first two selections 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 in a week’s time.

Here is Lecture 21 video: equivalent springs and masses using equivalent potential and kinetic energies

Please review the videos for Lessons 19-21 as we will be doing some of those problems next week. Also, the first announcement of the 3^rd examination is that it will be given on April 15.

Lesson 22: we continued speaking about equivalent springs and masses.

Here is Lecture 23 video: we look at damped systems, derive equations and talk about overdamped, critically damped and underdamped systems

Lesson 24: we continue to talk about damped systems.

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

And for systems with damping included we will need the following parameters:

=2ςω_n and =ς and Ccrit=2√mk =2mω_n =2k/ω_n

Here is Lecture 25 video: we look at underdamped systems, derived logarithmic decrement and talk about two problems.

Now for underdamped systems: the logarithmic decrement, δ, equals where ζ is the damping ratio. Also we show that δ = (1/N) * ln(x_o/x_N) = (1/N) * ln(x_i/x_i+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, where i is any number.

Your 3^rd exam is announced for April 15. It will cover solution of first order equations (Chapter 8 in the System Dynamics book: time constants, general solution of the equation) as well as undamped vibrations, equivalent masses and springs, damped vibrations with friction and damped viscous vibrations. You are allowed 3 8.5 x 11 formula sheets.

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.

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…

On April 17 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 is Lecture 26 video: we look at forced vibration and derive the displacement function for a harmonic force F(t)=P sinw_ft. We discussed the effect of frequency ratio r=w_f/w_n and damping ratio ζ.

I am adding one more videotape in which we discuss how to find the displacement due to forced vibrations Lecture 8a (just review the first 45 minutes of the tape). Please see it before the exam. The book from which the problems are taken are Vierck, Vibration Analysis.

Here is Lecture 27 video: we cover two problems, one forced undamped and one forced damped vibration problem at the beginning of the videotape (up to the 45 minute mark).

Exam 3 was returned. Students were given the option of doing a retest that will be averaged with your actual exam3 score. Exam retest is to be scanned and sent by email by 12 noon Friday April 19, 2019. Students are still under exam conditions and must do their own work.

Also a review session is scheduled between 12pm and 2pm on Monday 22 April in room EC1107.

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 April 24, Wednesday, from 215 to 415pm. Location is EC1107.

Don’t forget you will be allowed to bring in 9 sheets of 8.5 x 11 inch paper with whatever information you wish on it, except for solutions from the book.

PLEASE PICK UP ALL YOUR PREVIOUS EXAMS FROM MY OFFICE TO HELP YOU PREPARE FOR THE EXAM

With respect to the sheet that you are allowed to bring into the exam in which you have your numerical methods formulas:

You should devote 3 sheets to 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 2^nd order which is like the modified Euler

4) Runge-Kutta 2^nd order known as the midpoint method

5) Runge-Kutta 3^rd order

6) Runge-Kutta 4^th order

and how to use these methods if you have more than one state variable equation, as well.

3 sheets should deal with the materials in vibrations and 3 sheets should cover the materials on system dynamics/system graphs/ transformers/transducers/ elemental-nodal-loop equations.

Please make sure to be seated so that we can start immediately at 215pm as there is another examination in the classroom right after our class.

Remember, you will be given 4 problems: read all the problems and follow the instructions. Good luck during the exam and use good test taking strategy that we talked about during class.

In the final class, we reviewed forced vibration to include finding Xss, Ftrans,max for the normal situation of a periodic force acting on the mass. We will NOT cover rotating unbalance and so you will not be tested on it.