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# Analysis & Design of Engr Syst ES 205

RHIT

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This 14 page Class Notes was uploaded by Napoleon Turcotte II on Monday October 19, 2015. The Class Notes belongs to ES 205 at Rose-Hulman Institute of Technology taught by Staff in Fall. Since its upload, it has received 77 views. For similar materials see /class/225124/es-205-rose-hulman-institute-of-technology in Engineering & Applied Science at Rose-Hulman Institute of Technology.

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Date Created: 10/19/15

E8205 Analysis and Design of Engineering Systems Homework Probems Lesson 09 Problem 91 In the electromechanical system shown below iA is the current passing through the motor 9L 0L and m are the load position velocity and acceleration and TL is the effective torque acting on the load this is the combined effect of the torque applied through the springlike shaft and the damping torque acting on the load OpAmp Circuit Rf Cf Load DC Motor einftl Ri eA RA LA Obtain the EOM for this system then develop models in two forms a As a set of six 1St order differential equations b As a state space matrix and output matrix of the form 8A 8A 8A A iA 1 iA 9M 6M em L 6M em A B and wL C D QM QM em a QM em 6139 6L T 6L m m m Page 1 of 2 E8205 Analysis and Design of Engineering Systems Homework Probems Problem 92 For the tape drive shown below determine the necessary equations to find the differential equation of motion EOM that relates the known input voltage V3 to the unknown output displacement X1 Assume the output shaft of the motor is rigid Hint The tension in the tape will change from one side of the idler wheel to the other Do not find the EOM but number the equations that you would use and generate a list of the unknown variables If you need a parameter that is not explicitly given in the figure below be sure you clearly define your notation Takeup capstan Head kT kc r B1 1 F known Armature controlled M DC motor Page 2 of 2 f y Output equations 0 State equations ES205 Understanding Operations and Matrices in Matlab Matlab is an excellent tool for performing a wide variety of operations on scalar vector and matrix values Scalar A single number gtgt a 5 Real scalar 5 gtgt b 89883 423411 Complexscalar 89883i4234 Vector A row or column of numbers gtgt vectoriA 2 pi 17 9 Rowvector 2 314 17 62 gtgt vectoriB 62 94 9 Columnvector 94 A row vector can also be created by use of the colon operator gtgt vectoric 57 9 5 6 7 gtgt vectoriD10 O575 9 10095 90 85 8075 Matrix A set of columns and rows of numbers 3 4 14 28 0 1 2 gtgt matrixiA 1 2 3 4 9 gtgt matrix B 14 28 O Inf NaN i 9 1nf NaN 0z391 where Inf and NaN are special terms for 00 and 00 How would you type in the matrix 11 44 77 as a single line of code 22 55 88 33 66 99 gtgt X11 33 77 22 55 88 33 66 99 or X 11 33 77 22 55 88 33 66 99 Vectors and Matrixes use brackets and to enclose lists of array elements Elements of a row are separated by either blanks 0r commas Rows are separated by either quotEnterquot or semicolons Subscripts of Arrays What if we want to extract or operate on entries within a vector or matrix We need to employ subscripts to pull out individual elements or subsets De ne G as shown below gtgt F 60 90 100 150 avectorwith4elements a matrix with 6 elements gtgt G l O 5 O 3 4 2 9 8 3 5 Examples of using subscripts to pull out speci c elements or subsets of F or G gtgt F SubsetA FM gtgt FisubsetA Fl3 gtgt GisubsetA Gl2 gtgt GisubsetB Gl223 Mathematical Operators in Matlab In addition to the usually math operators A Matlab has a number of mathematical operations that are no what you think of as standard mathematical operations These are used to be work efficiently with matrices To be able to use Matlab to its full potential you need to understand how the different operators work Operators in order of importance parenthesis Highest 39 colon operator transpose operators A 0 exponentiation operators multiplication and diVision addition subtraction and negation lt gt lt gt N logical comparators N logical NOT amp logical AND l logical OR Lowest Addition Scalar Matrix Addition gtgt B512345 gt15 25 35 45 55 gt 6 7 s 9 10 Matrix Matrix Addition gtgtB 1 2 3 8 2 0 gt 18 2 2 30 gt 9 0 3 gtgt C 1 2 3 5 10 2 gterror Matrix dimensions must match Subtraction Scalar Matrix Subtraction gtgtD 5 3 2 6 gt 5 3 5 2 5 6 gt 2 3 1 Matrix Matrix Subtraction gtgtE 123 10 0 4 gt1 10 2 0 3 4 gt 9 2 1 Multiplication Scalar Matrix Multiplication gtgtF 5 1 2 3 gt 51 52 53 gt 5 10 15 Full Matrix Matrixl Matrix2 Multiplication requirement columns of matrix 1 rows of matrix 2 1x3 3x1 1x1 4 G12 44 1 3 gt 1 2 4 1 gt142143gt14 3 3x1 1x3 3x3 4 4 8 16 H 4 1 31 2 4 gt 1 12 4 gt 1 2 4 3 3 6 12 Element bv Element Matrix Matrix requirement matrices must be same size I x 3 I x 3 I x 3 J 1 2 4 8 2 0 gt 18 2 2 40 gt 8 4 0 2x2 2x2 2x2 1 2 5 0 5 0 K12345 02 6gt 3 4 2 6 6 24 Division Matrix Scalar or ScalarMatrix Divison gtgtL 1 2 3 5 gt 15 25 35gt 2000 4000 6000 gtgtM4 4 10 8 gt 44 410 4 8 gt100 250 20 Full Matrix atrixMatrix or MatrixMatrix Division requirement for light division of columns must be equal requirement for left division of rows must be equal A123 B135 C9 3 5 2 1 0 2 6 4 gtgt BA gt 1 3 511 2 3 gt 02500 02500 06250 3 5 2 0 2 6 gtgt AC gt 1 2 3 9 gt 245000 3 5 2 1 175000 0 2 6 4 65000 Element bv Element Matrix Matrix Matrix Matrix Division requirement matrixes must match in size gtgt 123 4 3 0 gt 14 23 30 gt 02500 06667 Inf gtgt1 4 2 s 3 0 gt1s 43 20 gt 0125 075 000 0 1 3 9 5 1 09 15 31 Inf 500 033 Exponentiation Matrix A Scalar Exponentiation requirement matrix must be square 11 x n gtgt1234A2 gt 12 1 2 gt 7 10 3 4 3 4 15 22 Matrix A Scalar Exponentiation gtgt1234A2 gt 12 A2 gt1A2 2A2 gt 1 4 3 4 3A2 4A2 8 16 Scalar A Matrix Exponentiation gtgt 2A1 2 3 4 gt 104827 141519 212278 317106 Matrix A Matrix gElement by Element Exponentiation requirement matrix sizes must match gt3456A1234gt3A1 4A2 3 16 5A3 6A4 125 1296 Solution to 39 quot quot Given a set of 3 equations 6x1y3z10 6 1 3 x 3x 8y 5z20 gt 1 0 3 8 5 y 20 1x2y7z30 1 2 7 z 30 A X B Coef cient Unknown Constant Matrix Matrix Matrix To solve for X A 6 1 3 B 10 3 8 5 20 1 2 7 30 Method 1 Use inverse Matrix gtgt invAB gtgtXinvAB 01575 00034 00651 10 00890 01541 00719 20 00479 00445 01541 30 04452 gt x 00342 gt y 42123 gt z Method 2 Use left division gtgt AB gtgt XAB 6 1 3 10 3 8 5 20 1 2 7 30 04452 gt x 00342 gt y 42123 gt z Example You have performed an experiment where you have found the undamped natural frequency for a system and want to see how different damping coefficients affect the system response For instance you have an undamped natural frequency of a 25 rads and are interested in the response when 4 01 03 05 and 07 Knowing these characteristics equations apply to the system Percent Overshoot Rise Time Settling Time 71 12 are 7r tan 7 4 OS IOOeVI z T 9V T r152 071W S2 4 0 Write a short mflle in Matlab to calculate each of these for the four values of 4 mflle Code mydemo m a short Matlab demo wn25 zeta01 03 05 07 sqzetasqrt 1 zeta quot2 PO100exp pi zeta sqzeta Trise pi atan2 sqzeta zeta wnsqzeta Tsettling24 zetawn Results gtgt mydemo wn 25000 zeta 01000 03000 05000 07000 sqzeta 7 09950 09539 08660 07141 PO 729248 372326 163034 45988 Trise 06718 07864 09674 13141 Tsettling2 160000 53333 32000 22857 Policies and Procedures E8205 Analysis and Design of Engineering Systems Objectives This course is designed to help students integrate the material learned in E8202 E8203 ES204 and MA222 and to deepen students understanding of engineering systems modeling simulation analysis and design To meet this objective a certain level of expertise in both mathematics and conservation and accounting principles is required It is assumed that you have taken the prerequisite courses and have acquired better than a super cial level of pro ciency in applying the principles and methods taught in these courses If you are having trouble in these areas consult with your instructor to determine an appropriate course of action but do not expect class time to be devoted to this material Exams and quizzes Exams are comprehensive closedbook and closednotes Laptop computers are prohibited on exams bring your calculators You may bring a onesided 85xll helpsheet to the exam You are allowed one helpsheet for exam 1 two helpsheets for exam 2 and three helpsheets for the nal exam Helpsheets must be signed and turned in with each exam Handwritten typed or reduced photocopies of reference information are permitted Helpsheets may not contain problem solutions Students violating these requirements will receive a zero for the exam Tenminute unscheduled quizzes readiness assessment tests are given sometimes daily and sometimes weekly covering homework and lecture material No makeup quizzes are given Quizzes missed due to excused absences will not affect your grade Makeup work Makeup of required coursework is considered only for those students who consult with their instructor prior to an absence or who in the event of unforeseeable circumstances contact their instructor as soon as practicable For example if you are participating in an Instituterelated activity or if you have a court appearance or a scheduled hospital visit let your instructor know before the absence that you ll be gone and for how long If you get sick on the day of an exam call in sick If you have a family emergency call your instructor as soon as it is convenient Failure to follow these guidelines results in a grade of zero for the work missed Final course grade Your nal course grade is based on the following weighted average Quizzesamp Homework 15 Laboratory 19 Exams 218 36 Note You must pass the exams to pass the class Final exam 30 100 A weighted average score of 90 is guaranteed to be an A 80 a B 70 a C and 60 a D Below 60 is failing E8205 Policies and Procedures Page 1 of 5 Labs The laboratory experiences support the course learning objectives in modeling simulation analysis and design of engineering systems Laboratory time is used also to support the course objectives in teaming writing oral presentations and computing 80me lab periods are largely computational others are experimental and some involve demonstrations The experiments both handson and demonstration produce data that students use in model validation and parameter identi cation generally related to lecture topics being covered at the time It is not uncommon for students to have had lab experiences in other courses in which the mode of collect the data and get out was both ef cient and rational The E8205 lab experience is neither conceived nor implemented with this mode in mind Your instructors designed the lab as a time and place to provoke thought train critical thinking and engage one another in active learning The three hours of lab per week are put to full and active use Please come to lab with this active mode in mind Bring your laptops to every lab period unless noted otherwise by your instructor Of ce hours While your instructor is typically available most hours that he or she is not in class lab or Institute or departmental meetings it happens occasionally that a student and instructor have con icting schedules and cannot nd a convenient time to meet In response to this problem and to provide our students with ample opportunity for studentteacher interaction all E8205 students are welcome to seek assistance from all E8205 instructors Homework Introduction A large part of this course is concerned with problem solving and applying conservation principles to a variety of situations For this reason mastery of the homework is essential for the successful completion of this course Treat homework as an opportunity to practice and learn new skills not just an exercise in acquiring a grade 0 Homework generally is assigned daily and collected as sets on a regular basis as indicated in the schedule Solutions for most of these assignments will be available in the library AFTER they are turned in o Homework is collected on the due date indicated in the course schedule The time of day that homework is collected is determined by your instructor 0 Points are deducted if the homework format is not followed 0 If you use software Matlab 8imulink Maple in a homework solution include the software inpu and outpu elements to support your solution A penalty is assessed on each problem that references software but that does not include the relevant documentation 0 Late homework is not accepted without prior approval from the instructor A problem solution is a form of technical communication and must be clear and easy to understand For this reason you are required to follow the same homework structure used in E8201 A detailed description of the homework format can be found in Appendix A of the E8201 notes This section summarizes the relevant requirements E8205 Policies and Procedures Page 2 of 5 Requirements Use standard size 85 X 11 inch engineering paper Write on one side only Start each problem on a new page Emphasize your nal answer by double underlining it or by boxing it in Include units If you use a computer program in your solution cut and paste a copy of the relevant section of the computer session in sequence in the problem solution Present the nal answer on the engineering paper Staple the pages of each homework set together in order and include a cover page as described below Structure Every problem should have the following sections 2 3 4 5 Known Find Given Analysis Comments A photocopy of the basic problem statement can be cut out and taped to your solution For additional discussion see Appendix A of the E8201 notes Comments on analysis E8205 Policies and Procedures Be sure to identify the system or systems you will be analyzing For many problems this process involves drawing a free body diagram a kinetic diagram and other appropriate schematic drawings Problems may require more than one system Be liberal in your use of sketches to make your solution clear and easy to understand Clearly label coordinates and variables such as displacement velocity current voltage volumetric flow rates heat ow rate and points of static equilibrium As you proceed through your solution process insert descriptive comments to explain your procedure Identify assumptions as you proceed through the solution Be sure to state in words what principle you are using or if you re using a kinematic or constraint relationship Include a table of U nknownsEquations if appropriate Number the independent equations to help determine when you have enough equations to solve for the unknowns Does the number of unknowns equal the number of equations Once the number of unknowns equals the number of equations you should be able to solve for the unknowns If you have more unknowns than equations identify an additional system if there is one or identify additional equations often these come from kinematics or constraints Solution of equations and final answer Solve the resulting equations manually or using a computer algebra system whichever is easiest for the particular problem being solved and clearly indicate the answer If the solution of the equations fails to give a reasonable answer there is probably a problem in the equations Check your answer Your solution should be checked to make sure it is reasonable Ask yourself quotDoes the answer make sensequot If not check your analysis then check if your model is correctly implemented in the software package you are using if any Checking like engineering practice has to be meticulous Page 3 of 5 Keys to success 0 Do the homework daily This requires discipline You are encouraged to attempt these problems on your own first before discussing them with your peers The single most important piece of advice from students from prior years is Start your home problems on the day they are assigned 0 Get help from your professor If he or she isn t in when you drop by leave some sort of communication send an email leave a note on the door or call and leave a message on voicemail Visit another E8205 instructor The time to learn the material is now not the night before an exam If you get stuck on a home problem go to the next problem Get help from your professor early and often o The process of obtaining a solution is as important as obtaining a correct answer In engineering practice an answer without supporting documentation demonstrating a process consistent with established practice is meaningless o If your answer is incorrect go back over your solution looking for errors in sign in units or in the application of a principle 0 Pay attention to making your solution a clear technical communication 0 Get help from your classmates but do not copy There is no penalty for working as a group unless you let everyone else do the workithe penalty is that you leam nothing Working with other students in this class is encouraged The only condition placed on working with others is that you acknowledge their help This is one purpose of the honor statement Working in a group on the homework can be an excellent way to learn the material It is important to note that every student will be individually accountable for the material via the exams and inclass quizzes Do not divide up the problems but help each other when you get stuck Cover sheet The last page of this handout is the cover sheet to be used on every homework assignment Include one of the two honor statements shown as appropriate if it is required by your instructor The cover sheet may be typed E8205 Policies and Procedures Page 4 of 5 IM Studious CM 9999 Analysis and Design of Engineering Systems ESZOSO HW 1 March 6 2008 11 12 13 I pledge my honor that I did not copy any of this homework but I did receive help from the names of your helpers go here your signature OR I pledge my honor that I did not copy any of this homework and this work is entirely my own your signature

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