ME 305, Introduction to System Dynamics
Dr. Rajesh N. Dave (Go to home-page)
Spring 2004
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In the following, textbook reference is: K. Ogata, SYSTEM DYNAMICS, Prentice-Hall,
4th Ed. 2004.
1/20/04
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. The concept of Laplace transformations (LT) was covered from Chapter 2. An analogy of log transform for multiplying two numbers was provided. Problems B-2-2a, B-2-4, and several additional ones were solved. It was emphasized that the approach is based on utilizing Laplace Transform (LT) tables to solve the problems.
Assigned Homework: Read related problems, and hand-in, B- 2-1, B-2-2(b), B-2-3
1/22/04
Homework problems were solved. We covered the Final Value and Initial Value theorems, and did problem B-2-9 for final value as well as initial value. When the limits are indeterminate, we can apply L'Hopital rule, or divide by S the numerator as well as the denominator as necessary. Problem B-2-10 was done to find its final value, unlike what is asked, the initial value. That will be a homework problem.
Next, the inverse Laplace transform 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/poles, you need to consider a method where derivatives are required. We did a number of examples for all the cases including the problems B-2-13, and 14.
Assigned Homework: Read related problems, and hand-in, B- 2-10, B-2-15 , B-2-17
1/29/04
All homework problems were solved. Laplace transforms for derivatives were introduced, and five different problems similar to B-2-24 were solved to show how to solve ordinary differential equations. Students are encouraged to pay special attention to the inverse LT in case of the last version, where we had to manipulate the original terms for obtaining two types of inverse transform terms. Problems B-2-20, B-2-22, and B-2-23 were also solved.
Assigned Homework: Read related problems, and hand-in, B- 2-24, B-2-25
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.
2/03/04
All homework problems were solved. It was emphasized that the problem s such as B-2-24 are very important in this course, and when the input is a step force, the problem gets more complicated.
We began Chapter 3,
Mechanical Systems. Mechanical elements - inertia, springs and dampers
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 that in problems such as these
gravity can be cancelled, but in the next lecture, this concept will be
explained in more detail. Next, a number of different problems with various
combinations of springs in parallel and series were done. 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.
Assigned Homework: Read related problems, and hand-in, B-3-7, 8 and 10.
2/05/04
All homework problems were solved. Next, we considered a classic spring-mass-damper problem. It was shown how to handle the force due to the dashpot, which is always opposing the velocity. Several problems with multiple degrees of freedom involving several masses, springs and dampers were solved. A problem where the spring and damper are in series was also considered. In this case, the only proper way to do such a problem is to introduce a new motion variable (also placing an imaginary mass of zero in between) in between the spring and the dashpot.
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.
Class of problems having pivoted motion were considered. These included many versions of the pendulum, horizontal as well as vertical, including inverted case. 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. Please also pay special attention to computing the moment of inertia by using the parallel axis theorem.
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.
Assigned Homework: Read related problems, and hand-in, B-3-12, 13 and 14.
2/10/04
All homework problems were solved. The lecture was utilized for a detailed review for two upcoming quizzes. Quiz I is scheduled for Thursday, 2/12/04, covering Chapter 2, and Quiz II is scheduled for Tuesday, 2/17/04, covering mechanical systems from Chapter 3.
Two typical problems from Chapter two, one involving inverse Laplace transform for a case of Multiple identical roots, and one on solving an ordinary differential equation that involves complex conjugates, were considered in detail. For the mechanical systems, it was shown that one can write down the equations of motions simply by inspection. This was followed by solving problems involving combinations of spring-mass-damper systems. Please note that if there is a spring and a dashpot in series, you must assign an independent motion variable in between, and place an imaginary zero mass there. This helps setting up the free body diagrams. Problems involving pivoting elements were also considered. Quiz II will concern with two types of problems; spring-mass-damper and pivoting elements.
Assigned Homework: Study for the quizzes.
2/12/04
Quiz I was given in the class.
2/17/04
Quiz II was given in the class.
2/24/04
Quiz I and Quiz II results were handed out, and all the problems were solved. We began new material, involving spring-mass-pulley systems. Special problem set was introduced, and SP1 was solved. Two methods were taught for determining the extension in the spring - the method of instant center of rotation, and the "slack-in-the-rope" method.
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 Vcg 2
+ = ½ Icg
w 2
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 along with classical spring-mass, spring-mass-pulley 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.
For the pulley problems, it was shown that one need not explicitly determine the extension of the spring - simply assume it to be some linear scale of the mass motion x, i.e. cx, where c is a constant. When you solve a problem using both Newton's and Energy methods, you can compare the solutions to determine the value of c.
Assigned Homework: Read related problems, and hand-in, B-3-12 by Energy method and SP2 by Newton's method and Energy method.
2/26/04
All homework problems were solved. For SP2, the instant center of rotation and the slack methods were explained. Next we solved problem SP3, which is a spring-mass-pulley system problem. In such problems, difficulty lies in determining the amount of spring extension. Both the methods were illustrated again, 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 an 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 problems were solved by both the Newton's and energy methods.
It is emphasized that 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.
It was also shown that there are only three cases of the pulley and spring arrangements. In one case, pulley only changes the direction of motion, and the other two cases are such that in one case, mechanical advantage is created, and the other the "distance advantage". Recognize that most problems simply involve combinations of these three cases.
Systems with rolling friction were also considered. Three versions of spring and a rolling cylinder type problem were considered. In problems such as these, it is easier to solve them by taking summation of moments about the point of contact of rolling. Both the Newton's method and Energy method were used.
Assigned Homework: Read related problems, and hand-in SP4 by Newton's method and Energy method.
3/02/04
Homework problem was solved using both the methods. Next, SP5 was solved using both the methods, and assuming spring extension in terms of the unknown constant c. However, we also utilized the instant center method to obtain the spring extension directly. This completed the material from Chapter 3. Note that we have covered two type of problems that are not in this text-book, the ones with rolling contact, and ones with pulleys.
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 combinations, including external force, p(t), (2) pivoting elements such as pendulums and their variations such as B-3-12, (3) rolling contacts as done in the class, and (4) spring-mass-pulley problems such as all special 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.
You are encouraged to try out the sample mid-term. Please go back to "Courses" page and look for the exam review!
Next, material from chapter 6 on the electrical systems was presented, starting from the "road-map" of this chapter. Note that this chapter is useful for understanding the system approach, where we learn that many systems, such as mechanical and electrical systems behave in a similar manner and have similar governing equations. Moreover, there is a direct analogy between the mechanical and electrical systems, for instance, the springs are like the capacitors, and the dash-pots are similar to the resistors.
Formal definition of electrical elements such as resistors, capacitors, and inductors was provided, followed by loop-law analysis for electrical circuits for derivation of models. We analyzed several circuits with single and multiple loops and derived their governing equations.
Homework: Read related problems, and hand-in B-6-4.
3/04/04
Homework problem was solved. Next, we did a full review for Mid-Term I, which included fully solving a sample test, where problems 2, 3 and 4 were solved using both the Newton's and Energy methods. The sample test can be downloaded as a PDF file here!
Homework: Prepare for the Mid-term - send any good challenging problems to Prof. Dave by e-mail!
3/09/04
Mid-term I was given in the class.
3/11/04
Graded Mid-term exams were returned. Selected problems were solved in the class. The class average was high.
We continued with Chapter 6 on Electrical Systems, and solved the homework problem. Transfer function material was introduced, and the concept of complex impedance (for resistor, capacitor, and inductor) was also 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. 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. We also showed the electrical-mechanical analogy through several examples, and using force-voltage equivalence.
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, do problems from Chapter 6, B-6-9, B-6-11.
3/23/04
Homework problems were solved.
Electrical circuit material was completed.
Chapter 8 was started. It was emphasized that Chapters 8 and 9 are regarding study of the response of dynamical systems such as mechanical and electrical, and form the crux of this course. You will learn many practical things about second order linear dynamic systems. 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 as well as simple inputs such as impulse, step and ramp. Such analysis was done for the thermometer system, with guidelines for design insights obtained through the analysis. Applied forces, typically are, impulse, step, ramp, and sinusoidal. Study of the response due to sinusoidal forces is covered in chapter 9.
Homework: Read related material, do B-6-17, B-6-18.
3/25/04
Homework problems were solved. 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.
Homework: Read related material.
3/30/04
We reviewed material covered in the previous lecture. 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, {ln(first peak/last peak)}/{number of cycles from first to last peak}. Analysis of a spring-mass-damper system under impulse force was also considered. Problems, similar to the homework assignment were solved.
Homework: Read related problems and do B-8-4, B-8-7, B-8-10, B-8-11.
4/01/04
All homework problems were solved and Chapter 8 material was fully reviewed. Introduction was provided for Chapter 9.
No home work was assigned.
4/06/04
We continued introduction to Chapter 9 which was started on April 1. This chapter was contrasted with Chapter 8, where we applied external forces such as impulse, step, or ramp with or without initial conditions. There, the focus was on inverse Laplace transform, and examining transient response. Here, we need not take inverse Laplace transforms, because of the sinusoidal transfer function approach. The concept of the sinusoidal transfer function was presented and a spring-mass and spring-mass-damper problems were solved using this concept. We also solved a problem that is a variation of problem B-9-4, which is the homework problem.
Homework: Read related problems, and hand-in B-9-4.
4/08/04
Homework problem solved in detail, and it was shown how all problems of these class have a similar structure when you solve them.
Next, we began the material on sinusoidal forces due to rotating unbalanced mass, and the concept of vibration isolation. The first topic involved the Transmissibility Ratios (TR) of force transmitted to the ground. TR for motion was also defined. Both quantities were derived, and it was shown that for a given mechanical system, they are the same. Concept of vibration isolation was related to TR. 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.
A problem similar to B-9-7 was solved in the class.
Homework: Read related problems, and hand-in B-9-7.
4/13/04
Homework problem was solved and its variations were considered. The design concept for vibration isolation was discussed. As mentioned above, frequency ratio, b, should much greater than 1.4, and the damping ratio should be kept small, but non-zero.
We also solved problem B-9-9, and discussed the concept of dynamic vibration absorber.
Homework: Read related problems, and hand-in B-9-10.