ME 305, Introduction to System Dynamics

ME 305, Introduction to System Dynamics

Fall 2003

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In the following, textbook reference is: K. Ogata, SYSTEM DYNAMICS, Prentice-Hall, 3rd Ed. 1998.

9/02/03

We covered introductory material from Chapter 1, including concepts such as: dynamics, system, mathematical modeling, etc. Chapter 1 was completed. The course structure was explained so that a full road map for the entire semester was clearly given per topic/chapter. It was emphasized that this course moves very quickly through different materials and if you do not pay attention, it becomes very difficult to catch up. The most important aspect of this class is homework – as it is assigned after each class, and is due the next class. I solve all homework problems in the beginning of the class, hence you need to submit your work in the beginning of the class. It will be graded, and will form a basis for your total grade in the class. As I mentioned, not submitting any homework will result in loss of at least one full grade point. You are highly encouraged to attempt and submit the homework – even if you are uncertain if you did it correctly or not. This will ensure a decent score on the homework and will make you ready for the next lecture when we solve these problems.

Keep in mind that this course basically involves: (a) Deriving a model of the system (using e.g. Newton's laws), (b) Setting up of a mathematical equation based on the model - including the initial conditions and possible conditions of disturbances or external excitations (these equations are usually ordinary differential equations ODEs), and (c) Solving the equations, i.e. ODEs using Laplace transform techniques.

Complex numbers were reviewed.

NO HOMEWORK!

9/04/03

The concept of Laplace transformations (LT) was covered from Chapter 2. An analogy of log transform for multiplying two numbers was provided. We also covered the Final Value and Initial Value theorems. Problems B-2-2a, B-2-4, B-2-9 and several additional ones were solved. It was emphasized that the approach is based on utilizing Laplace Transform tables to solve the problems.

Assigned Homework: Read related problems, and hand-in, B- 2-1, B-2-2(b), B-2-3, B-2-10, B-2-15. (Since many of you do not have the text-book, click the links for Laplace transform tables, and the homework problems.)

9/09/03

All homework problems were solved. The inverse LT using partial fraction expansion was considered. Please make sure you pay attention to three different cases, (1) distinct real poles, (2) distinct poles with complex conjugates, and (3) multiple poles. For multiple roots, you need to consider a method where derivatives are required. We did examples for all the cases including the HW problem B-2-15.

Laplace transforms for derivatives were introduced, and problems similar to B-2-28 were solved to show how to solve ordinary differential equations. Students are encouraged to pay special attention to the inverse LT in case of B-2-28, where we had to manipulate the original terms for obtaining two types of inverse transform terms.

Assigned Homework: Read related problems, and hand-in, B- 2-19, B-2-21, B-2-26.

9/11/03

All homework problems were solved. The inverse LT using partial fraction expansion was further considered. Solution to ordinary differential equations using LT was also reviewed. Students are encouraged to pay special attention to the inverse LT in cases such as B-2-27, where we had to manipulate the original terms for obtaining two types of inverse transform terms. Problems B-2-16 through B-2-22 and B-2-26 through B-2-28 were fully solved in the class as a review for the Quiz I, which will be given at the beginning of the class next Tuesday. Please submit the homework in the beginning of the lecture.

Assigned Homework: Read related problems, and hand-in, B- 2-29, B-2-30.

9/16/03

All homework problems were solved. The inverse LT using partial fraction expansion was further explained. We solved problems similar to B-2-16 (b) and B-2-28 in the class as a further review for the Quiz I. Later, Quiz I was given.

Assigned Homework: None.

While we have not covered the use of MATLAB in this class, you are encouraged to study the text-book material on using MATLAB for solving partial fractions. MATLAB is available for your use on SUN workstations located on second floor of GITC. You do not need to fully understand the MALTAB software. However, you can get Dr. Dave's own notes on using MATLAB by clicking here .

Before you do that, you will need ADOBE Acrobat Reader software installed on your computer. If you do not have that, to load this "free" software, please click here .

For self study, consider a MATLAB problem below involving multiple identical real roots from the text-book (page 33).

The below is the printout (in green color) of an interactive session on MATLAB. The information from page 33 is entered as num and den polynomial vectors, and then the residue function is applied to obtain the solution to the problem of partial fraction expansion.

>> num = [0 1 2 3]

num =

0 1 2 3

>> den = [1 3 3 1]

den =

1 3 3 1

>> [r, p, k] = residue(num,den)

r =

    1.0000
   -0.0000
    2.0000

p =

   -1.0000
   -1.0000
   -1.0000

k =

[]

You are encouraged to try out all the solved problems from the text-book.

9/18/03

In the class, all Quiz I problems were solved. Note that the class average for this quiz was 22. We began Chapter 3, Mechanical Systems. Mechanical elements - inertia and springs were introduced. Equivalent spring (when in parallel or in series) rules were explained. Food for thought was: What happens to k of a spring cut in half?

We also did a spring-mass problem, with and without consideration of gravity. Note that the gravity force cancels out with the initial extension (or compression) in the spring. We explained basic rules about when gravity can be cancelled. Next, a number of different problems with various combinations of springs in parallel and series were done.

Assigned Homework: Read related problems, and hand-in, B-3-6, 8 and 10.

9/23/03

All homework problems were solved. Class of problems having pivoted motion were considered. A problem similar to B-3-12 was solved in the class. The best way to solve this class of problems is by taking the summation of moments at the point of rotation, i.e., the pivot point.

Another important item we learned was the ability to determine if the gravity forces will cancel out in your problem or not. This is a very useful information, as ignoring the gravity when it does cancel out greatly simplifies your solution. However, you must be sure that it cancels out. The way to know this apriori is by checking if the gravity plays a role as a restoring potential. You must include gravity into your analysis (FBDs) when gravity is either always pulling you away from equilibruim OR it is always pulling you tawards equilibruim. In all other cases (i.e. when it pulls you away on one side of equilibrium but pulls you towards it on the other side of equilibrium), gravity can be ignored in the analysis. Importance of parallel axis theorem was shown.

Note that one can determine that the springs are in parallel if all of them experience the same magnitude of motion, and they are in series when all of them experience the same magnitude of force.

No Homework.

9/25/03

A spring-mass-damper system was solved. We also considered a number of multiple degrees of freedom spring-mass-damper systems. Please pay special attention to signs of the forces and when an element involves motion at both the ends. When you get your answer, you should check the signs of each motion variable and its derivatives. They all must be same for each motion variable. In other words, if you have X, Y and Z as three independent variables and you obtain three equations of motion. Then, in each equation, all X, dX/dt, and d2X/dt2 must have the same sign. A number of problems were solved to demonstrate this fact. It was also shown how to derive these equations without even setting up free bosy diagrams.

Systems with rolling friction were considered. A more difficult version of problem B-3-22 was considered. In problems such as these, it is easier to solve them by taking summation of moments about the point of contact of rolling.

Assigned Homework: Read related problems, and hand-in, B-3-12, 13(a), 15 and 22.

9/30/03

All homework problems were solved. Energy method for deriving the equations of motion was introduced. This is a "simpler version" of the Lagrange approach that some of you may have learned in ME 314. Here, we only consider the conservative (i.e. non-dissipative) cases, and then apply the principle of energy conservation. Several example problems were solved, i.e., spring-mass, and others. Note that the sum of kinetic and potential energy is conserved.
The most fundamental equation to keep in mind here for the calculation of kinetic energy is: T = = ½ m V_cg²+ = ½ I_cg w²
Here note that the velocity is of the center of mass, and the moment of inertia is also at the center of mass. Regarding the potential energy, it comes from gravity and springs. For gravity contribution, one must have a datum for zero potential (usually the equilibrium is taken as the datum).

A problems similar to B-3-12 and B-3-22 along with classical spring-mass and pendulum problems were solved in the class. It was pointed out that the energy method is a scalar approach in contrast to Newton's method which is a vectorial approach, requiring free body diagrams.

Assigned Homework: Read related problems, and hand-in, B-3-12 using energy method.

10/02/03

Problem B-3-12 was done using energy method. A new class of problems involving spring-mass-pulley systems was introduced.

We solved simplified versions of problems SP1, through SP4, which are spring-mass-pulley system problems. In these problem, difficulty lies in determining the amount of spring extension. Two methods were illustrated, the slack method, and the instant center of rotation method. Please pay attention to these techniques. The instant center method is a systematic approach, and one can easily find the instant center of rotation by placing imaginary surfaces at various locations and recognizing where the pulley would have a rolling contact. A rolling contact is also the instantaneous center of rotation. All these were solved by both the Newton's and energy methods.

Having known two ways to solve these problems, one can solve the spring-mass-pulley problems without explicitly knowing the extension in the spring. For example, if the motion of the mass is x, then assume that the spring extension is y = cx, where c is simply a constant that is unknown. We assume that the spring extension is some linear function of x. Now, solve the problem using both Newton's and energy methods. When you compare two answers, you can solve for the unknown, c.

Note that there will be a mid-term exam one week from today. This will have problems from chapter 3, selected from four types, (1) spring-mass-damper, (2) pivoting elements such as pendulums and their variations, (3) rolling contacts such as B-3-22, and (4) spring-mass-pulley problems. You need to be able to handle both Newton's method and energy method. This exam will be an open book plus one sheet of your own notes.

Please go back to "Courses" page and look for the exam review!

Homework: Read related problems, and hand-in, SP5 using both Newton's and Energy Methods.

10/07/03

Homework problem SP5 was done using both methods. A full review of the mid-term I was conducted, and 4 challenging problems were solved, the last 3 done by both Newton's and energy methods.

Note again that the mid-term exam will be given during next class. This exam will be an open book plus one sheet of your own notes. Note that I have a sample exam posted on this web-site - so please go back to "Courses" page and look for the exam review!

Homework: Read related problems, be ready for the test.

10/09/03

The mid-term exam was given during the class.

Homework: None.

10/14/03

Graded mid-term exams were handed back. Problem 1 was solved in the class.

Material from chapter 4 on electrical systems was presented, starting from the definition of electrical elements such as resistors, capacitors, and inductors, followed by loop-law analysis for electrical circuits for derivation of models. Next, the concept of Transfer Functions (TF) was introduced. It is important to note that a TF is simply a characteristic of the system, and does not have anything to do with initial conditions or the external input. However, TF along with the external input may be analyzed to study system response. Transfer function material was reviewed, and the concept of complex impedance (for resistor, capacitor, and inductor) was introduced. Note that the use of this concept makes derivation of TF for electrical circuits very simple. Moreover, recognizing that the laws of the resistors in parallel and series work also for complex impedances, and that makes the whole analysis much simpler. A number of multiple-loop electrical circuits were analyzed.

Homework: Read related problems, and hand-in B-4-6, B-4-13, B-4-16.

10/16/03

All homework problems were solved. The concept of generality between the electrical and mechanical systems was given - including why the topic of this course- the "Systems Dynamics" is more relevant than the old days when only mechanical vibrations were taught at the undergraduate level. Using the concept of force-voltage analogy, equivalence between the electrical and mechanical systems was shown. This type of analysis gives us a broader picture of how the dynamic systems from different fields essentially behave similarly - for instance, inertia elements are associated with second order derivatives, dissipative terms are associated with first order derivatives and the restoring terms are associated with the basic variables.

Homework: Read related problems, and hand-in B-4-19, B-4-20.

10/21/03

All homework problems were solved. It was shown through these two and four more examples how one can go from a mechanical to an electrical system (and vice versa) by inspection, without having to derive equations of motion. Note that the elements appearing in series in the mechanical systems appear in parallel in the electrical systems, and conversely, the elements appearing in parallel in the mechanical systems appear in parallel in the electrical systems. This knowledge allows you to very quickly and easily convert one system from another, and also gives you a better understanding of each system.

Review for Quiz II (to be given at 10:30 am, Thursday, October 23, 2003) was done. Three types of problems are: (1) derive a transfer function based on the concept of complex impedances for linear and op-amp circuits (e.g. problems B-4-13 through 16), (2) convert a mechanical system to an electrical system using the force-voltage analogy (e.g. problem 4-19), and (3) convert an electrical system to a mechanical system using the force-voltage analogy (e.g. problem 4-20). If you are confident, you need not show the equations in doing the problems type 2 and 3.

Chapter 6 was started. It was emphasized that Chapters 6 and 7 are regarding study of the response of dynamical systems such as mechanical and electrical. Response due to the initial conditions (initial position or initial velocity) is usually called natural response, while due to the input force is called forced response. Typically, one studies a dynamical system by first analyzing the response due to the initial conditions (this is done for the thermometer system, with guidelines for design insights obtained through the analysis, and was also done for a spring-mass-damper system). One also then applies various forces, typical are, impulse, step, ramp, and sinusoidal. Study of the response due to sinusoidal forces is covered in chapter 7.

Homework: Read related problems, and be ready for the quiz.

10/23/03

Quiz II was given and introductory material of Chapter 6 was done. It was emphasized that Chapters 6 and 7 are regarding study of the response of dynamical systems such as mechanical and electrical. Response due to the initial conditions (initial position or initial velocity) is usually called natural response, while due to the input force is called forced response. Typically, one studies a dynamical system by first analyzing the response due to the initial conditions (this was done for the thermometer system, with guidelines for design insights obtained through the analysis, and was also done for a spring-mass-damper system). One also then applies various forces, typical are, impulse, step, ramp, and sinusoidal. Study of the response due to sinusoidal forces is covered in chapter 7.

Introductory material for the first order system such as a mercury bulb thermometer was done, and it was shown how "systems analysis" helps in designing this device.

10/28/03

The analysis of a spring-mass-damper system for natural response shows that there are three cases, (1) under damped, where damping ratio is less than one, and the characteristic equation yields complex conjugates as roots, and the response is decaying sinusoid, (2) critically damped, where damping ratio is equal to one, and the characteristic equation yields two identical real roots, and the response is a sum of one decaying exponential and another decaying exponential multiplied by time, and (3) over damped, where damping ratio is greater than one, and the characteristic equation yields two distinct real roots, and the response is a sum of two decaying exponentials. While the algebraic manipulations are complicated, the crux of the problem is what is stated here, and you must be able to understand and recognize these three conditions.

We solved a problem with numerical values for the mass, spring constant, and three different values for the damping constant to show all three cases.

The concept of logarithmic decrement and its practical use in determining spring constant and damping in a spring-mass-damper system was shown. The equation for the log decrement is given as:
log decrement = {ln(first peak/last peak)}/{number of cycles from first to last peak}
A problem, B-6-8 was solved. It was emphasized that in problem B-6-8, one can compute the value of damping ratio from the displacement chart given, by noting that peak to peak decrement of 25% occurs in 0.5 cycles. We also discussed in general sense, how to handle for a spring-mass-damper system, different types of forces, such as impulse, step and ramp. In all the cases, one still has to be concerened about three different possibilities - under-damped, over-damped, and critically damped.

Homework:Read related problems, and hand-in, B-6-7.

10/30/03

HW problem was solved. Transfer function approach was introduced and a problem (B-6-5) was solved. Analysis of a spring-mass-damper system under impulse force was also considered. We also discussed design problems based on the concept of logarithmic decrement.

Homework: Read related problems, and hand-in, B-6-4, B-6-10, B-6-12.

11/04/03

All homework problems were solved. A review of Chapter 6 was done. Three classes of problems from this chapter are: (1) first or second order systems with initial conditions and impulse and step inputs; (2) design problems based on he concept of logarithmic decrement; and (3) derivation of transfer functions and solving response with impulse and step functions.

Next, chapter 7 was started, and first order as well as second order systems were solved using the sinusoidal function approach.

For the next class, please do a review of the Mid-Term II (Make sure you are using MS Internet Explorer for a Sample Mid-term II and its review, click here !) The concept of the sinusoidal transfer function was presented in detail and spring-mass and spring-mass-damper problems were solved using this concept.

Homework: Read related problems, and hand-in, B-7-1.

11/06/03

Carry out review from this web-site.

Homework: Read related problems

11/11/03

Homework problem was solved. We reviewed sinusoidal transfer functions. We derived Transmissibility Ratios (TR) of force transmitted to the ground. TR for motion was also defined. Concept of vibration isolation was also introduced. Keep in mind that in order for achieving a good vibration isolation, TR should be less than one. That can be achieved by keeping the frequency ratio, b, much greater than 1.4. On the other hand, if the damping ratio is high, TR is usually well-controlled around a value of 1, but even for a large value of b it stays close to one, and cannot be made much smaller than one. Thus if the frequency ratio can be assured to be high, then damping ratio should be fairly small.

Note that the second mid-term will be next week, November 18. Next lecture will involve a detailed review for this test, so it is important that all students attend the class.

Homework: Read related problems

11/13/03

A full review for the mid-term was carried out where a number of highly relevant problems were solved in the class. The mid-term is next Tuesday.

Homework: Read related problems

11/18/03

Mid-term exam was given.

11/25/03

We reviewed sinusoidal transfer functions. We derived Transmissibility Ratios (TR) of force transmitted to the ground. TR for motion was also defined.

Problems B-7-9 and 10 were solved. It was shown that for the same mechanical system, motion (transmitted from the ground to the body/mass) and force (transmitted to the ground due to the sinusoidal force acting on the body/mass) transmissibility values are same - i.e. you get the same final expression. As you may recall, this was the case even in a simple spring-mass-damper system. We also solved problem similar to B-7-7. Dynamic vibration absorber system was also examined qualitatively.

Note that for the material from Chapter 7 - there are three types of problems that you must be able to solve- (1) Response due to sinusoidal input force - this is done through sinusoidal transfer functions and you must be able to find the pertinent transfer function, G(s), and then apply the relation, x(t) = |G(jw)| P sin(wt + f), where f is the angle of G(jw), e.g. problem B-7-4, (2) Compute motion or force Transmissibility Ratio of any mechanical system, e.g. B-7-8, 9, 10, and (3) Applied/design type problems in vibration isolation, e.g. B-7-7.

Homework: Read related problems

12/02/03

Mid-term exam grades were handed out, and the entire exam was solved in the class. The class performance was not very good, and it seems that we need to review some of this material more carefully. Important aspects of Ch 7 were again pointed out.

Homework: Read related problems

12/04/03

A full review of the make-up mid-term was conducted. This make-up exam is designed to give you an opportunity to improve your grade. Please note that it will cover chapters 6 and 7.