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Everyday Examples in Engineering

Engineering Design

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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Table of Contents: Engineering Design: Organizing Brainstorming Results

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Focus on Engineering Design: Cellular Telephone System Design

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Street Crossing Problem

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Dimensioning

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Pattern Development

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Curve Principles: Spaghetti

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Creating Suitable Criteria for Option Evaluation: Apples & Oranges

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Wanted: Everyday Examples in Engineering

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ENGAGE is an Extension Services project funded by the National Science Foundation. The overarching goal of ENGAGE is to increase the capacity of engineering schools to retain undergraduate students by facilitating the implementation of three research-based strategies to improve student day-to-day classroom and educational experience. ENGAGE strategies include: • Integrating into coursework, everyday examples in engineering (E3s) • Improving student spatial visualization skills • Improving and increasing faculty-student interaction For more information about ENGAGE contact: Susan Metz, Project Director [email protected] For more information on E3s or to submit E3s contact: Pat Campbell, Ph.D. [email protected] Jennifer Weisman, Ph.D. [email protected] Go to EngageEngineering.org for lesson plans for the following courses: Calculus and Differential Equations, Chemistry, Circuits, Control Systems, Dynamics, Elasticity and Plasticity, Engineering Design, Engineering Graphics, Fluids, Introduction to Engineering, Manufacturing, Material Failure, Mechanics, Physics, Properties of Materials, Statics, Stress and Strain and Thermodynamics.

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Using Everyday Examples in Engineering (E3) Engineering Design: Organizing Brainstorming Results Laura Bottomley North Carolina State University Engage Ask students how many of them have ever shopped in a grocery store. This is, of course, a laughable question, and they may laugh. Ask them to think about a time that they went into a grocery store to quickly pick up some specific items for a meal and had to run through the entire store to get only a few required items. Explore Ask students to brainstorm a list of categories of items found in a grocery store. They should do this in groups of three, either self-selected or assigned. You might ask some groups to share some of the categories that they have listed. Try to ensure that the groups have as complete a list as possible. Now ask students to make a simple sketch of a grocery store layout, either from memory or from exploring the Internet. If possible, students could visit a grocery store as homework and make a more detailed sketch. (The advisability of this depends on the location of your campus and the accessibility of a fairly standard grocery store.) Explain Explain that the engineering design process frequently involves an idea-generation step, sometimes accomplished through brainstorming, which can result in many diverse ideas, almost like the list of diverse items found in a grocery store. In fact, designing the layout of a grocery store is an industrial engineering problem. Engineers frequently have a need for schemes to organize quantities of diverse information. One tool for doing so is called a concept map. A concept map allows information to be grouped and placed in a hierarchy. In addition, creating a concept map for a set of information (like concepts taught in an undergraduate class) can help you monitor your understanding of the material and prepare for tests. Concept maps for a given situation are not unique, and iteration is frequently required to improve on an initial attempt. One way to approach a concept map is to identify first the question that is being addressed. In the case of our example, the question is how to organize a grocery store. The next step is to define broad groups or categories of information or related items.

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These are called concepts. If brainstorming has been done using Post-it Notes©, the broad concepts can be identified and placed on a wall or board. If the concept map is to be used as a part of the brainstorming process, the group can consider the question and first make a list of general concepts surrounding the question. Then sub-concepts are added as linkages from the broader concepts. Relationships between the concepts and the central question and/or between concepts and sub-concepts may be added as verbs written on the linkages. An example concept map is given below.

Figure 1: Example affinity diagram for design of improved digital camera1 Elaborate Discuss examples of engineering design that would benefit from the application of a concept map. Stress that every design situation may not be enhanced by using this tool. Consider asking students to generate ideas of where it would be useful. (Examples to consider include: designing a new model car, a digital camera, or a new candy, improving a cell phone or a shoe, etc.) Evaluate 1-Ask students to return to their lists of grocery store items. Ask them, in their groups, to draw a concept map that would lead to the design of the layout of a grocery store.

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2-Have them sketch a potential layout from their concept map. Ask them to consider layouts that vary from the standard, and then to analyze their results in comparison with a standard layout in terms of a. potential shopping efficiencies, b. cost (e.g. how many different cold storage units are required? Are they located away from the door which might affect their operation? ), c. overall aesthetic appeal, d. other criteria that you or they identify. 3-Ask students to draw a network or graph linking the items on the following grocery list, and to calculate the length of the graph or network for their layout and for a traditional layout, using imaginary units. Grocery list: • Tortillas •

Salsa

•

Cheese

•

Ground beef

•

Taco seasoning

•

Lettuce

•

Tomatoes

Extend If students seem interested, this activity can be extended by talking about graph theory or networking. A simple network routing algorithm could be explained (like the Bellman-Ford algorithm for evaluating ‘”shortest path” networking, and the algorithm could be used to calculate shortest routes through a grocery store.

© 2010 Laura Bottoley. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes.

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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Using Everyday Examples in Engineering (E3) Focus on Engineering Design: Cellular Telephone System Design Laura Bottomley North Carolina State University This lab is divided into two main parts that will focus on the engineering design process. Engage Take your own cell phone out and hold it in front of the class. Ask the students who have cell phones to take them out, and then ask them who has coverage in the classroom and who doesn’t. Ask them if they have ever had the experience of being on a cell conversation and losing signal. Then introduce the lab as follows: “The objectives for this lab are important for all engineering disciplines; everyone will need a knowledge of design whether he or she ultimately works as a designer, a researcher, in sales, or even as a patent attorney or physician. We have chosen a specific project to complete in this lab which is interdisciplinary enough to have direct application to a variety of disciplines (Can you identify which ones?) and indirect application to every engineering discipline. During this lab, we intend to: • •

Introduce you to the engineering design process in a hands-on manner Apply computer tools (such as Excel, the WWW, word processing) to engineering design”

Explain If you haven’t yet introduced the design process, here is an introduction: The Design Process Design is a structured problem solving activity. The engineering design process distinguishes engineering from other professions. Design is the engineer’s opportunity to use knowledge gained through the educational process to solve practical problems of interest or benefit to society. Make no mistake. This is where the most difficult problem in engineering education lies: teaching you to connect the abstract mathematical and scientific concepts that you have learned to the real world problems you must solve. The knowledge that is used includes ethical considerations, social awareness and political concerns in addition to science and math. 6

The steps usually outlined for the engineering design process include problem definition, evaluation, solution and communication stages. These stages can be further broken down as follows: • •

•

•

Problem definition: • Identify need • Outline problem Problem evaluation • Research • Identification of constraints • Selection of criteria Problem solution • Identification of multiple alternative solution techniques • Analysis and evaluation of different alternatives (may be a subset of available alternatives) • Decision Communication • Technical specification of solution • Written and/or oral communication of solution

For the communication phase of the design process, the engineer must understand the audience and purpose of each type of communication. A Specific Problem: Designing a Cellular Telephone System PART A You are a designer for a promising new cellular telephone company, Century Telecom that has just won a slice of the frequency spectrum in the Cary, NC area. You are going to provide cellular service to the Town of Cary and adjacent suburban areas. Because your company is new, and still rather small, this will be the only area that you serve for now. However, you expect to grow in the near future! A map of your projected service area is on the next page. The inner line represents the city limits, while the outer line represents the limits of your coverage area. The area of coverage is about 329 square miles. The suburban area is described by drawing a line around the city limits at a distance of about 4.5 miles. You are in charge of designing the telephone system for this application from the ground up, so to speak. By the way, there is a complication to the problem. The city council has established zoning such that structures over 30 feet are not allowed inside the city limits. This is unfortunate as the towers which you will use to support your transmit/receive antennas are much taller than that. What will you do?

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Map of Projected Service Area:

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THE SOLUTION In order to solve the problem as stated above, you will need to know a few things about HOW to design cellular telephone systems. Cellular telephony is so named because a desired coverage area is divided up into “cells” which are usually served by a central transmit/receive antenna. A map of the area that the system will serve is analyzed to locate where broadcast antennas should be placed. These antennas can broadcast to only a limited area with sufficient power to be reliable. In “real life” the cells (i.e. the area it is assumed that the antenna can broadcast to) are usually amorphous in shape, but for analysis we usually assume regularly shaped, uniformly sized cells. The shape used is that of a hexagon. A circle more accurately models the broadcast area of an omni-directional antenna, but non-overlapping circles cannot be placed in such a way as to completely cover a given area. So, in the initial design phase for a cellular system, nonoverlapping hexagons are used to cover the entire service area. Although in the previous paragraph we talked about a central antenna, there are other ways to broadcast to a cell. So another design decision that must be made at this point is how to “illuminate” the cell, or what type of antenna placement to use. Two techniques are center illumination and side illumination. In the center illumination case, one omni-directional antenna structure is placed in the center of each cell. The power levels are set so that signals sent from the antenna cover only the cell to which it is assigned. (Of course this is impossible, but we assume it to be the case and then compensate for error later.) In a side illuminated cell, three directional antenna structures are placed at three equally spaced vertices of the hexagonal cell. Each antenna is designed to broadcast over a wedge, and the three structures together cover the hexagonal cell.

The side illumination case allows for an overall reduction in antenna sites (including towers and land usage), because a tower can hold three antennas broadcasting into three adjacent cells, but requires more actual antennas. And of course each tower costs money, each antenna costs money and each piece of land on which an antenna is placed costs money. To solve your problem, you should first place the cells for your system and figure out how you will serve the downtown area (which, by the way, will have the most customers requiring service). You must place the hexagons in such a way that the entire service area is covered. The radius of a typical cell is about 5 km. Start by finding out the minimum number of cells to cover the area and then see how close you can get to that minimum by building an actual model.

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For your solution to this part of the problem, you should submit: 1-a discussion of how you will place antennas to address the political difficulties, 2-a discussion of how you decide how many cells you will need (including a calculation of the area of a cell), and 3-a map of cell locations with antenna locations indicated. PART B Now that you have the geometry of your problem figured out, you need to complete the design of the system by assigning radio frequencies to the antennas in each cell. The Federal Communications Commission has assigned you a chunk of the frequency spectrum, 33 megahertz. When cellular systems are designed, a decision is made about whether each call can use the whole allotted spectrum (e.g. the 33 MHz in this case). There are engineering tradeoffs involved. If two cells next to each other use the same frequencies, interference in conversations can occur. (Consider, for example, two people having a face-to-face conversation standing next to two other people engaged in the same activity. They are using the same part of the frequency spectrum—the audible part—and interference may occur because they can hear each other talking. The situation is much better if, say, they are standing on opposite sides of the room.) For this reason, the spectrum is usually broken up into “chunks”, and rules established about how far two cells must be apart in order to use the same frequency “chunks.” You must decide how to partition the spectrum among your cells and how frequently to reuse frequencies. The more you reuse frequencies, the more customers you can serve ($), but the greater the interference you incur (translating to worse service). The Solution Each cellular call involves at least two antennas, one at the base station (i.e. on the towers that you just placed) and one on the mobile (i.e. the telephone handset or car). They cannot use the same frequency for the same reason that you cannot easily understand someone speaking to you when you are actively talking to them. (The human voice occupies the frequencies from about 40 Hertz to about 3100 Hertz.) Indeed, both parts of the conversation have to be broadcast at frequencies much higher that those of “baseband” voice--for some reasons which are obvious and some that are not so obvious. So, the uplink (car to tower) and downlink (tower to car) are assigned to different frequency bands. This is done for every customer served in the cell. A single channel requires 25 kilohertz (times two for uplink and downlink, which makes a duplex channel). (Notice that this is more than 3100-40=3060 Hz, which is what you would expect voice to require. Some coding, which adds information to improve the quality of service, is obviously going on.) Furthermore, cells which sit next to each other in your geometry cannot use the same sets of frequencies, because mobiles on the edge of the coverage area would not be able to distinguish their call from a call in the next cell. A rule must be established as to how far away one cell must be from another cell which uses the same frequencies to serve its customers. This is called frequency reuse assignment. How you define your frequency reuse will determine the quality of service your customers will experience and how many customers you will be able to serve--a typical design tradeoff! 10

Cells are grouped together in N size groups. The entire allocated spectrum is then available to each group. This is called N cell reuse. Any value of N is possible that can be derived from this equation: Let i and j be non-negative integers and i≤j. Then N=i2 + ij + j2 For example, take i=1 and j=1, then N=3. For i=1 and j=2, N=7. For i=2, j=2, N=12. For i=1, j=3, N=13. Why do you think we use this math to determine possible values for N? Try picking a value for N that does not fit this criteria and using it in the following analysis too. Once you have N chosen, you then divide your total allotted frequencies into N groups. The groups of frequencies are then assigned to cells. Assignments are done in such a way as to maximize the distance between cells using the same group of frequencies. For our hexagonal geometry, it is done like this: from a cell which uses one set of frequencies, to get to the next cell which uses the same set, go out one wall of the cell and move i cells. Then make a 60° turn counter-clockwise and move j cells. A reuse pattern for N=3 is shown below. N=3 implies three different frequency chunks, to we call them A, B, and C.

Cells labeled with the same letter use the same set of frequencies. The value of N chosen translates directly to the number of customers able to be served (system capacity, C). C=N⋅M⋅k, where M is the number of clusters of cells in a system and k is the number of duplex channels available per cell. Note that M and k must be integers. 11

The logical conclusion of this reasoning is that we should chose N=1 and reuse the entire allotted frequency for each cell. But we already know why not to do that: the tradeoff is greater interference between conversations in other cells. This is called co-channel interference. A measure of how much interference affects a conversation is the signal to noise ratio (SNR) experienced. (You already understand signal to noise ratio in the context of writing on a chalk board. Writing on a clean board—large signal, small noise—is easier to read than writing on a dirty board—same signal, greater noise.) The SNR is obtained by dividing the desired signal power by power for all interfering signals. An acceptable signal to noise ratio is a minimum of 15 dB2. If we consider only the first layer of co-channel cells, the signal to noise ratio experienced by a user in a cell is given by: S/I = (√3N)n where n is between 2 and 4 and represents the severity of the signal environment (are there a lot of highly reflective objects around like buildings or absorbing objects like trees, etc.?) We will assume n=3 for this problem. To choose your value of N, create a spreadsheet with columns for N, S/I (in dB), and system capacity, C. Choose N to maximize capacity while staying within the signal to noise ratio requirements. After you choose N, go back to your map from part A and label each cell that uses the same frequencies with a letter like the example for N=3 above. Because your map deals with a “real-life” situation, you might not have the same number of A’s, B’s, C’s, etc. The solution to part B, then, should be summarized by enumerating the following: 1-frequency reuse factor and assignment of which cells will use the same frequencies (indicated on the cell assignment map you made for part A), 2-the expected worst case signal to noise ratio, 3-the total system capacity. Include a copy of your spreadsheet. Now that you have finished the assignment, go back over the problem and identify which pieces apply to the various engineering disciplines. Record the results in your notebook. Elaborate In practice, systems may not combine types of illumination. Other design considerations, like ability to acquire leases for land for towers, existence of interfering structures like buildings or trees, distribution of potential customers, and licensing will all contribute to a design solution. We have removed most of these factors in order to come up with a first pass design solution. Evaluate Cellular Telephony Lab Solution For the first part, they need to calculate the area of a hexagon with a radius of 5 km. The easiest way to do this is to recognize that a single hexagon is made of 6 equilateral triangles with sides 2

The decibel is a logarithmic unit which is derived by taking ten times the base 10 logarithm of a ratio of powers. So, in this case, the signal to noise ratio in dB would be calculated by taking 10 log(S/I).

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of length 5 km. The area comes out to be 64.95 km2 and 25.37 mi2. Then if you divide 329 mi2 by 25.37 mi2 you get a minimum number of about 13 hexagons. Then they cut apart the hexagons and place them on the map so that there are no antennas inside the city limits. Some of the hexes have to be side illuminated (three alternating vertices must be outside the city limits) and some of them can be center illuminated (the center must be outside the city limits). Then you will find you need somewhere between 15 and 22 hexagons. Each solution could be different—keep placing hexagons until all the kidney bean shaped area is covered. © 2010 Laura Bottomley. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes.

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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Using Everyday Examples in Engineering (E3) Engineering Design: Call for Contract Performance Plans (CPP): Street Crossing Problem on Stadium & Northwestern Robin Adams and Senay Purzer Purdue University Robin Adams and Senay Purzer would like to acknowledge the following individuals associated with this work: Cynthia J. Atman (University of Washington); Mark Henderson (Arizona State University); Sean Brophy (Purdue University); and PK Imbrie (Purdue University).

College campuses are often overcrowded with pedestrians crossing streets, since walking is a popular form of transportation for college students. One busy intersection on the Purdue campus is at Stadium and Northwestern outside Armstrong Hall (see Figure 1). This is a high traffic area (cars, buses, bicyclists, pedestrians, etc.) due to its central location to campus, neighborhood dining establishments (e.g., Café Royale, McDonald’s, Jimmy John’s, etc.), and a gas station with a convenience store. Often, people will not follow traffic rules when crossing the street (i.e., “jaywalking”). Sometimes, drivers will cut across Stadium to enter the BP gas station (see Figure 1). While the traffic signals limit certain activity (e.g., turning right on a red light from Stadium to Northwestern), accidents have happened and people have been seriously injured. NORTHWESTER N BP Gas Station Café Royale

Crosswalks

STADIU M

Armstrong Hall

Figure 1. A Sketch of the Intersection

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The President of the University has asked your team (among others) to design a cost effective method to cross Northwestern which would reduce the possibility of accidents at this intersection. Your work should contain a detailed description of your design and include any relevant information the President might need to understand and evaluate your design in relation to other designs she might receive (such as an estimate of the costs and benefits associated with your design). TEAM ASSIGNMENT For this assignment you will prepare a contract performance plan (CPP) poster, demonstrating that your team understands the problem and knows how to go about solving it. This assignment is due on [DATE]. SPECIFIC EXPECTATIONS 1. Use PowerPoint to create an electronic poster with dimensions of 36”’h x 44”w. The file to be submitted should be called StreetProject_Spr09_Team_xx.ppt (xx is the team number) 2. In your poster include the following information. a. Statement of the problem, constraints & criteria (back up with data & evidence when you identify constraints or criteria that are not explicitly stated in the problem). b. Your solution plan and objectives. c. Description/presentation of a systematic approach your team used for rich brainstorming. d. A systematic comparison of top three alternative solutions and selection of one of the solutions as your proposed solution (back up your predictions & decisions with data & evidence.) e. Description of your team members, strengths relating to this project, and contributions to this assignment. f. Include any relevant information the President might need to understand and evaluate your design in relation to other designs. Any one of you can be asked to present your team’s CPP. Make sure you allocate time for team presentation rehearsal in your project plan. You will have 5-10 minutes to give an oral presentation along with your poster. More specific presentation time will be announced on or before the presentation day. Your presentation will be graded based on the: 1) clarity of information & presentation; 2) quality of team organization; 3) quality of design process and solution plan; 4) how well you supported your predictions and decisions with research, data, and evidence; and alignment between the problem statement and selected solution. © 2010 Robin Adams and Senay Purzer. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes.

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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Using Everyday Examples in Engineering (E3) Dimensioning Sheryl Sorby Michigan Technological University Concept: Dimensions are placed in the most distinctive view Distribute a Ken doll, clothed in a swimsuit, to each student group. Alternatively, you could use something like a GI Joe, but the distinction between dimensions is not as great as it is with a Ken doll. 1. Have the students sketch Ken from the side and from the front views. Make sure the views project orthographically from one view to the next. 2. Tell the students that Ken is an idealized man whose chest is 42” and whose waist is 24”. Have them place these dimensions on their sketches. 3. Discuss which views best show Ken’s idealized dimensions. [The front view shows both the chest and waist best.] 4. Now have the students put dimensions on the sketches for the length of the arms, the length of the legs, the length of the neck, and the length of the head. 5. Discuss the proper placement of the dimensions (front view for arms and legs; either view for neck and head). 6. Show some simple engineering drawings with dimensions and discuss the placement of each dimension in its most descriptive view, relating this back to the work with Ken as appropriate.

© 2010 Sheryl Sorby. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes. This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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Using Everyday Examples in Engineering (E3) Pattern Development Sheryl Sorby Michigan Technological University Concept: Basics in creating a pattern development with fold lines and tabs Distribute empty boxes that food came in to student groups. Examples of the types of boxes that you could use are: granola bars, cereal, macaroni and cheese, pasta, frozen dinners, Pop-tarts, etc. Try to have at least two boxes of the same brand in the collection you bring to class. Distribute rulers to the groups and pieces of card stock. 1. Ask the students to estimate the amount of cardboard (in sq. inches or in sq. centimeters) used in the box. 2. Ask the students to calculate the volume of the box. o Discuss the concept of shipping costs for cardboard manufacturers. Do the students think it would be more cost-effective for the manufacturer to ship the boxes “already assembled” or as flat pieces that the food manufacturer assembles? 3. Have the students attempt to create a box out of the card stock. o Successful groups will likely discover the need for scoring for fold lines; other groups may not think of this technique. 4. Student groups can now carefully “unfold” their boxes. Have them measure the area of the cardboard and compare it to the estimate they made earlier. o Discuss where the differences in the actual versus the estimate came from. Likely sources are the tabs for the side seams or the overlap on the top and bottom. 5. Have a general discussion regarding characteristics of a pattern development features you should be sure that students note are: o Fold Lines o Tabs o Scoring on the fold lines 6. For groups that have the same brand of box, have them compare boxes with one another. They should discover that they fit together—the way they were cut by the cardboard box manufacturer was done to minimize waste.

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www.EngageEngineering.org Using Everyday Examples in Engineering (E3) Curve Principles: Spaghetti Ramy Harik Lebanese American University

Photo Credit: “Spaghetti” by Tim Bartel, available under a Creative Commons Attribution- ShareAlike 2.0 Generic License.

Engage Bring in three bowls of spaghetti. Ask students how they like their spaghetti (under cooked, medium cooked, or overcooked). Start distributing plates across the class. You can also ask them if they have ever been to Italy and put some show photos from Italy on an overhead. Explain

We need to generate a curve that represents the above set of points.

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not 19 necessarily reflect the views of NSF.

We have three ways: ‐

Interpolation: the curve will pass exactly by each point

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Approximation: the curve will pass by the points with a margin of error to preserve a certain smoothness

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Form: the curve will be guided by a certain polygon allowing it for extra flexibility

Explore and Explain Now have groups of 5 or 6 students work with a plate of spaghetti and ask them to analyze the type of the spaghetti they have in front of them. Ask them to identify which type of curves they can construct out of their spaghetti. You can distribute the graphs on the next page with two sets of examples (to compare): 1. P0(-1,0), P1(0,2), P2(1,1), P3(3,0) 2. P0(-1,1), P1(0,0), P2(1,2), P3(0,0) Once they have played with the spaghetti, you can end this part of the activity by discussing the following points: ‐ Overcooked spaghetti represents interpolation. The spaghetti has lost all of its smoothness to pass through all the points given. The shape of the curve is not smooth and is very hard to be manufactured. ‐ Medium-cooked spaghetti represents free-form. The spaghetti is locally flexible and can follow different directions yet still it still holds on all together. This is a very important representation for computer-aided design. ‐ Under-cooked spaghetti represents approximation. The whole curve is welded together with a minor flexibility. It is practically impossible to follow the points without ‘breaking’ the spaghetti.

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Evaluate Here you solve the exercise for the first set. a) Determine the Lagrange polynomial that passes through the points The general Lagrange Polynomial is:

The polynomial is then

b) Determine the spline polynomial (with q=2) that passes through the points

The equation should be eligible for the 4 points: P0(-1,0)  P1(0,2)  P2(1,1)



P3(3,0)



The equation should continue with no second/third degree elements after M3: (x3)  (x2)



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Matlab Solution:

Other potential questions: c) Determine the least squares parametric polynomial (2nd degree in y, 1st degree in x) that passes next to the points d) Calculate the Bezier polynomial having the points as control points. Evaluate Students can repeat the above steps for the below sets of points: 1. P0(-2,-2), P1(-1,0), P2(0,2), P3(1,0), P4(3,9) 2. P0(-3,0), P1(-1,5), P2(1,-5), P3(3,0) 3. P0(-2,3), P1(-1,-3), P2(0,0), P3(1,-3), P4(2,3) © 2012 Ramy Harik. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes. 23

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Using Everyday Examples in Engineering (E3) Creating Suitable Criteria for Option Evaluation: Apples and Oranges Francis J. Hopcroft Wentworth Institute of Technology

Photo Credit: “Comparing” by Kate Andrews, available under a Creative Commons Attribution NonCommercial ShareAlike 2.0 Generic License

Background: Freshmen, in particular, often have a very difficult time grasping the concept of criteria appropriate to the evaluation of two or more engineering options. They typically tend to evaluate each option using a different set of criteria – none of which they have defined. The objective of this exercise is to raise the awareness of the student to the need for carefully selected, objective criteria for the evaluation of options. Engage: Place an apple and an orange on the desk, or ask students to imagine the presence of an apple and an orange. (Any other two fruits will also do, if the instructor happens to favor others). Then ask the students, working in groups or individually, depending on class size and the convenience of the instructor, to determine which fruit is the “best” and to indicate why that choice was made. Explore: After students have had a few minutes to develop their answers, ask each group to explain which fruit they chose and why they chose that particular one. There should be both a good mix of choices and a wide range of reasons for making that choice within the various groups. Some groups may find it difficult to actually select a “best” fruit. This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not 24 necessarily reflect the views of NSF.

Then ask students to define the criteria their group used to determine which fruit is best and after providing adequate time for that to happen, write the selected criteria from each group on the board. Explain: The selection of an option from among available options is a common problem in engineering. The selection of the best available option requires a common set of criteria for evaluating those options. It is not possible, in fact, to select a “best” fruit unless a suitable set of common criteria is developed first and applied equally to each of the fruits in the set. Moreover, criteria such as “taste”, “or color appeal”, or similar criteria are too subjective and are not measurable or reproducible. Criteria need to be very objective and easily measurable to be useful. They should lead to repeatable results from a variety of investigators and that can only happen when the criteria are not subjective. Criteria such as “edible mass”, “edible mass percentage”, “edible mass per dollar of cost” and similar criteria are both measureable and repeatable. Ask the students to rethink their selection criteria and to develop five objective criteria per group that can be used to determine which is the “best” fruit. Write those criteria on the board for each group, set aside the duplications for further discussions, and then discuss the remainders to see why they are or are not ‘objective’ enough to properly evaluate the fruit. In the end, there should be a suitable list of criteria that can be used to make the desired determination. An explanation of how that all applies to any other general engineering problem, and the selection of options for solving that problem, should follow. © 2011 Francis J. Hopcroft. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be reproduced for educational purposes.

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www.EngageEngineering.org

Wanted: Everyday Examples in Engineering (E3s) •

Who: ENGAGE (Engaging Students in Engineering), funded by NSF, is developing a library of Everyday Examples in Engineering (E3s). E3s are needed for 1st and 2nd year engineering courses, especially Introduction to Engineering, Circuits, Engineering Graphics, and Fluids.

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What: E3s are objects that are familiar to students that can be used to demonstrate/teach engineering concepts. Examples of E3s include using hula hoops and wooden rulers to teach free or forced vibrations; sausages to demonstrate Mohr's circle of stress; or iPods that demonstrate many things. E3s aren’t design challenges or projects. Examples of E3s can be viewed at www.EngageEngineering.org in the resources section.

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Why: Integrating E3s into 1st and 2nd year engineering courses is one of the research-based strategies ENGAGE is using to improve retention of engineering students. Studies indicate that students are motivated to learn when they can make a connection to something familiar to them which illustrates the concept they are trying to understand. How: The initial submission does not need to be in any particular format. It can be a description of the Everyday Example and the specific concept that it demonstrates. More extensive write-ups are also welcome. E3s will be reviewed and if accepted, project staff will work with the author on revisions so there is sufficient detail to be used effectively and easily by faculty. Honorarium & Recognition: The author(s) of each accepted E3 will receive a $150 thank you honorarium. In addition, a letter will be sent to the author’s department chair and dean. The letter will include reference to the author’s contribution to ENGAGE by having their submission selected under a peer review process, and made available for national and international dissemination. Authors will maintain their E3 copyright and give permission to ENGAGE to use the materials. Where: For more information or to submit E3s, contact Dr. Patricia Campbell ([email protected], 978-448-5402) or Dr. Jennifer Weisman ([email protected]).

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any 26 opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

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