Motivating Engineering Students by UsingModern Case Studies

L. R. MUSTOE & A. C. CROFTDepartment of Mathematical Sciences, Loughborough University, LE11 3TU, UK. E-mail:l.r.mustoe@lboro.ac.uk

The use of case studies to motivate engineering students is long established. In order to enthusetodays school and college students it is suggested that case studies be presented which show theapplication of relatively simple mathematics to the forefront technologies. The first example is ofan aeroplane taking off where Newtons second law of motion is used in a modern context; a simpleand a more refined model offered. The remaining examples are in the area of digital imageprocessing and show the crucial importance of matrix algebra in this modern application.

INTRODUCTION

THE NEED to attract more of the most mathe-matically able school students to study undergrad-uate engineering degree programmes is ever moreurgent. Part of the difficulty in persuading moreyoung people to pursue their study of mathematicsbeyond the compulsory stage is their perception,which is shared by the general public, that much ofthe mathematics to which they have been exposedis irrelevant to modern life. A consequence of thefailure to attract sufficient numbers of more ablestudents into continuing with mathematics is thattoo few of them enter engineering degree pro-grammes with a sufficiently strong mathematicalbackground, yet such a background is required inorder to underpin many state-of-the-art engineer-ing applications.

Contrary to popular belief mathematics is rich inits applications in the modern world and much ofthe technology upon which we have come to relyhas been designed with mathematical methods atthe heart of the process. There is a need, then, toget this message across at a sufficiently early stageto convince school students that mathematics is arelevant and living subject, with exciting applica-tions in areas of activity which impinge on theirdaily lives. Armed with mathematical tools studentscan look forward to a wide range of careers inemerging technologies and modern applicationareas.

This paper aims to examine one way in whichthat message can be transmitted.

MAKING A CASE FOR CASE STUDIES

Case studies have long been used as a meansof motivating students at all levels to acquire

mathematical techniques and expertise. However,much of the effort has, unfortunately been lesseffective than it might have been. The difficultywith introducing case studies into the curriculum isgetting the balance right. If a case study relies ontoo much detailed mathematical technique then itcan detract from the application and weaken theimpact which it seeks to make. On the other hand,too simplistic a mathematical model can offer afalse prospectus and mislead students into believ-ing that the only mathematics which they requirein their studies is at a low level.

Further, if an unrealistic application of mathe-matics is offered to the students then they will seethrough the smokescreen and the exercise will havebeen counter-productive. If mathematics is appliedto a situation to which it has not been appliedhitherto, to which it is not currently being appliedand to which it is highly unlikely to be applied inthe future then the students will get the impressionthat the mathematicians are desperately seekingapplications in order to justify the presence of theirsubject in the curriculum.

On the other hand, a carefully-chosen case studycan act as a real motivator to students and help toconvince them that mathematics does have avaluable, indeed essential, role in the advancementof technology and is relevant to their highly tech-nologically-developed world. Every aspect of ayoung persons life is becoming more dependenton technology: games-stations, hi-fi and mobiletelephones are but three obvious examples of a lifedominated by the programmable silicon chip. If weare to interest todays schoolchildren in mathe-matics then the applications which we present tothem must relate to that world: cooling liquids andtrains going uphill are no longer good enough,however worthy they may be. Presenting modernexamples of the interface between mathematicsand engineering can have the effect of excitingthe more mathematically able students to considerengineering as a career and to maintain their appetite* Accepted 4 May 1999.

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Int. J. Engng Ed. Vol. 15, No. 6, pp. 469476, 1999 0949-149X/91 $3.00+0.00Printed in Great Britain. # 1999 TEMPUS Publications.

for mathematics in their engineering studies. It canalso convince intending students of engineeringthat mathematics is an important component oftheir studies, not a necessary evil to be endured.

Engineering colleagues can make a real con-tribution by suggesting forefront areas of theirsubject which can be amenable to relativelysimple mathematical models, to be presented ascase studies. They and the mathematics lecturerscan work together to produce a portfolio of suchcase studies for wider dissemination. The exampleswhich follow can be presented at various levels,from school to first-year university.

Example 1: Aircraft take-off distanceAs flying has become more popular and largeraeroplanes are developed, longer runways arerequired in order to accommodate the greatertake-off and landing distances. How long arunway is required for a particular aeroplane?Here, we concentrate on a simple take-off. Aircraftdesigners recommend that an aeroplane lifts offwhen its speed along the ground reaches a value1.2 times its stalling speed.

Figure 1 shows the forces acting on an aero-plane. T is the engine thrust, D the aerodynamicdrag, L the aerodynamic lift, W the gross take-offweight and is the coefficient of rolling frictionbetween the wheels and the runway. Note that thecombined normal reaction between the wheelsand the ground is W L. Applying Newtonssecond law of motion in the horizontal directiongives:

W

gv

dv

ds T D W L Fa 1

where v is the velocity of the aeroplane and s is thedistance travelled along the runway. Fa is the netaccelerating force. During take-off it is a reason-able assumption, based on experimental measure-ments, that T is constant, as is W. Both D and Lhave been shown to be proportional to v2 so thatwe may write

Fa F0 F1 F0v21

v2

where v1 is the speed at take-off. The acceleratingforce at the start is:

F0 T Wsince v 0. The accelerating force at lift-off is:

F1 T Dsince the reaction is zero as the aircraft leaves theground.

A simple model neglects the drag and frictionforces since these have been estimated to accountfor between 10 and 15 percent of the energyexpended. Then

T Wg

vdv

ds

Separating the variables and integrating from thestart to lift-off we obtain:s1

0

ds WTg

v10

v dv

where s1 is the distance travelled before the aircrafttakes off, and hence:

s1 W2Tg

v21

Remember that:

v1 1:2vswhere vs is the stalling speed. It can be calculatedfrom the formula:

vs

2W

SCLmax

swhere CLmax is the maximum coefficient of lift, isair density and S is the wing area.

Consider a typical executive jet aeroplane forwhich W 48000 N, T 14000 N, S 29 m2, 1:225 and CLmax 1:2.

The stalling velocity is:

vs

2 480001:225 29 1:2

r 47:45 ms1

Fig. 1. Forces acting upon an aircraft during take off.

L. R. Mustoe and A. C. Croft470

Hence the lift-off velocity is:

v1 56:9 ms1

Finally, the length of runway required before lift-off is:

s1 480002 14000 9:81 56:9

2 565:8 m

A more sophisticated model uses equation (1).Separating the variables and integrating, asbefore, producess1

0

ds Wg

v10

v dv

F0 F1 F0v21

v2

so that:

s1 W2g

v21F1 F0

ln

F1

F0

First, we use the following relationship, which hasbeen determined experimentally, to determine thedrag coefficient CD:

CD 0:02 kC2Lprofile drag lift-dependent drag

where

CL 2k

Then

F0 T W

F1 T D T 12SCDv

21

For the aeroplane already considered, we takek 0:04, 0:04, which are typical values. Then

CL 0:042 0:04 0:5

F0 14 000 0:04 48 000 12 080 N

F1 14 000 12 1:225

290:02 0:04 0:5256:92 12275 NFinally,

s1 48 000 56:92

2 9:8112275 12 080

ln 1227512080

650:5 m

This means that the distance estimated using thesimple model was about 13% too short. Parametervalues for other types of aeroplanes can beobtained from, for example, [1].

Example 2: Matrices and digital image processingIn this example we show how relevant and inter-esting examples can be used to motivate techniquesin linear algebra, and in particular matrix notationand matrix multiplication. These topics can appearrather abstract when they are met for the first timein a maths class, but with modern digital technol-ogies such as mobile phones and digital television,linear algebra has become as important a tool ascalculus for the engineer.

The Internet is now a part of many peoplesdaily lives. Young people, especially, are attractedto it and to related new technologies. They can usethe Internet to visit the web-sites of their favouritepop groups or sporting heroes and heroines, androutinely use it to obtain information for collegeprojects. It is an easy matter to download highquality graphic images to grace their bedroomwalls or to enhance their assignments. Their famil-iarity of the process involved in obtaining imagesmeans that many will have heard of file types suchas gif or jpg. They will recognise these as thesuffices appended to the filenames, as in spice.jpgand wimbledon.gif. Many home computers are nowsold with digital scanners so that the users can, forexample, scan their favourite photographs, displaythem on the computer, and e-mail them to theirrelatives around the world. What many will notknow is that a range of mathematical tools arerequired to produce these files. Furthermore, anumber of mathematical techniques to whichthey may have been introduced in their secondaryschool years, and which may have seemed ratherdull and lacking in application at the time, formthe basis of these tools. Armed with these tools,young people should be aware that careerprospects in companies involved in image process-ing are excellent.

An image stored by a computer is generally adigital image. This is an image which can be repre-sented by a matrix. Consider, as a very simpleexample, the mathematical representation of thesymbol . Very crudely we can picture the symbolas in Fig. 2.

We could represent this image of by a 3 5matrix such as:

B 0 0 0 0 0

1 0 1 0 1

1 0 1 0 1

0BB@1CCA

Note that we have divided the image into a numberof picture elements, or pixels, in this case fifteen ofthem. We have then assigned an integer to eachpixel, 0 if the field is to be shaded black, and 1otherwise. In this example there are just two levels,black or white, and so each element in the matrix isthe number 0 or 1.

An alternative way of thinking about each row isto draw a graph where the y values are either 0 or1. So the second row, that is the row vector(1 0 1 0 1), can be graphed as shown in Fig. 3.

Motivating Engineering Students by Using Modern Case Studies 471

This sort of representation will be helpful when welook at how matrices can be used to process theimage.

Images such as black-and-white photographscan be digitised in the same way. The image isdivided into m n elements where, in practice, mand n are large integers. Each element is assigned anumber which represents the intensity of the imagein that area. Often a 64-, 128- or 256-point greyscale is used so that a range of grey levels can berepresented ranging from black at one extreme towhite at the other. A digital image is then repre-sented by an m n matrix, whose elements arenumbers in the range 164, 1128 or 1256depending upon the number of grey levels used.There are computational advantages in selectingmatrices with sizes and number of grey levelswhich are powers of 2, as in 26 64 and27 128. For example a black-and-white televi-sion image is typically a 512 512 matrix whoseelements are integers in the range 1128. The 256-point digital image represented by the matrix

B

0 50 100 150

50 100 150 200

100 150 200 225

150 200 225 255

0BBBBB@

1CCCCCAis shown in Fig. 4. Each row could be graphed aswas shown in Fig. 3.

A very graphic way of bring home to studentsjust how useful such ideas can be is to refer them toinformation about how the American FederalBureau of Investigation, the FBI, is digitising andthereby storing electronically, its vast database offingerprints. They have been collecting fingerprintssince 1924 and now have over 200 million cardsand tens of thousands of new cards arrive daily.The only way that it is possible to search thismassive record in order to try to match fingerprintsfound at the scene of a crime is to use computersand to accomplish this each fingerprint image mustbe digitised. Information on the image codingmechanisms in use is readily available on theInternet. Students could start, for example, bylooking at the site [2] but other sites can befound by searching. It soon becomes evident that

this technology relies very heavily on the mathe-matical techniques of linear algebra.

Sometimes it is desirable to remove sharp edgesfrom an image. One way in which an image can besmoothed is to replace a matrix element by theaverage of some of its neighbours. For example thefour element row vector:

2 1 4 2 graphed in Fig. 5 can be smoothed to that shownwith a dotted line by replacing each element by themean of the four elements. For what will follow itis crucial to note how this can be achieved by post-multiplying the row vector by the matrix

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

0BBBBB@

1CCCCCAso that

2 1 4 2

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

0BBBBB@

1CCCCCA 94 94 94 94

This kind of operation is the basis for manytechniques so vital for effective digital imageprocessing.

Fig. 2. A simple digital representation of .

Fig. 3. A graph of the row vector 1 0 1 0 1 .

Fig. 4. A digitised image having several levels of grey.

L. R. Mustoe and A. C. Croft472

Example 3: mathematical techniques for imagecompression

We showed above how digital images are storedusing matrices. It is a simple matter to calculate thenumber of storage bits needed for different sizedmatrices and different numbers of grey levels.

For example, consider a single image consistingof 256 256 pixels. Suppose the image is made upof 256 grey levels. Noting that 256 28 we see thateach pixel requires 8 bits of storage making a totalof 256 256 8 524; 288 bits. If these are storedin 8-bit bytes, 65 536 bytes will be needed. It iseasy to see that storage requirements increase veryrapidly as we increase the size of the storagematrix, number of grey levels and the number ofimages. When colour images are considered thenumbers rocket further. Clearly a vast amount ofdata needs to be stored if the digital image is to berepresented directly. For example, a single colourpicture stored at the resolution viewed on a televi-sion screen requires around 1 million bytes forstorage. In order to view an image over theInternet it needs to be transmitted from the web-

site being visited to the users computer. This takestime, and the larger the file, the longer it takes totransmit it. No one likes sitting in front of acomputer screen waiting for what seems likeforever for an image to arrive. So, for the technol-ogy to be viable it is necessary to reduce the size ofthese files, or compress the image in some way.This is an exceptionally important branch ofmodern engineering, which like those alreadymentioned, relies very heavily on mathematics.

To see how compression might be attemptedconsider the following example.

The solid graph shown in Figure 6 has beendrawn by plotting the row vector

1 2 5 4 2 2 4 3 which could have been a row in a digital image asdescribed in the previous case study. The trickwhen trying to compress an image is to throwaway some of the informationthat which willcause the least deterioration in the quality of animage. We start by looking for locations wherethe variation in the image is least. We will nowprocess this row vector in the following way.First we find the mean of the first and secondelements, and get 1.5, the mean of the third andfourth elements, to get 4.5, and so on. Theseresults are placed in the second row of a matrixas shown:

1 2 5 4 2 2 4 3

1:5 4:5 2 3:5

!Now subtract each of the means just calculatedfrom the first of the elements used to calculate it.So we subtract 1.5 from 1, to give 0:5; wesubtract 4.5 from 5 to give 0.5; 2 from 2 gives 0and finally 3.5 from 4 gives 0.5. These differences

Fig. 5. A row vector can be smoothed by matrix multiplication.

Fig. 6. Graphical representation of an original image and a compressed version.

Motivating Engineering Students by Using Modern Case Studies 473

are placed in the second row of the matrix follow-ing the elements already placed there:

1 2 5 4 2 2 4 3

1:5 4:5 2 3:5 0:5 0:5 0 0:5

!It is crucial to recognise the importance of the zerohere. It represents a place in the image where thereis no change from the mean, that is a place of novariation. Note that this is consistent with theoriginal row vector where the value 2 is repeated.

The process is repeated, by averaging thepreviously calculated averages in pairs, and thenfinding the variation from the mean of the first ofthe pairs. This gives:

1 2 5 4 2 2 4 3

1:5 4:5 2 3:5 0:5 0:5 0 0:53 2:75 1:5 0:75 0:5 0:5 0 0:5

0BB@1CCA

To emphasize the distinction between the means,and the differences, the means have now beentypeset bold. Finally, one more application of theprocess yields:

1 2 5 4 2 2 4 3

1:5 4:5 2 3:5 0:5 0:5 0 0:53 2:75 1:5 0:75 0:5 0:5 0 0:5

2:875 0:125 1:5 0:75 0:5 0:5 0 0:5

0BBBBB@

1CCCCCAIn the last row, the number 2.875 is the mean of allthe original values and the remaining numbersrepresent variations from the mean. If the processis reversed we will recover the original row vector.However, and here is the trick, suppose that wethrow away one of the non-zero values represent-ing variation, and in particular we throw away theone with smallest magnitude, the 0.1250. Thisleaves the row vector:

2:875 0 1:5 0:75 0:5 0:5 0 0:5 Note that, whereas none of the original data valueswere zero, we now have two zeros. Very efficienttechniques exist for storing matrices with a signifi-cant proportion of zeros, so-called sparse matrices.So, we have thrown away some information andachieved a smaller set of data to be stored. Thiscould be transmitted, for example over the Inter-net, and it only remains to decode it at thereceiving end. This is achieved by reversing theprocess above. If this is done the restored imagevector is

0:8750 1:8750 4:8750 3:8750 2:1250 4:1250 3:1250

Both the original data, and the compressed data(shown as the dotted line) are depicted in Figure 6.Clearly there is some deterioration of the restoredimage compared with the original, but for manypurposes this sort of degradation will not be a

problem. If you had seen these images separately itwould be difficult to tell them apart. For example,when looking at a photographic image on a web-site, or a computer game, fine detail may not beimportant.

Techniques of linear algebra are seen to beparticularly important when it is realised that theoperations used to transform the row vector

1 2 5 4 2 2 4 3 to

2:875 0:125 1:5 0:75 0:5 0:5 0 0:5 can be achieved by matrix multiplication. Toproduce the second row the original vector mustbe post-multiplied by the matrix

M1

12

0 0 0 12

0 0 0

12 0 0 0 12 0 0 00 1

20 0 0 1

20 0

0 12

0 0 0 12

0 0

0 0 12

0 0 0 12

0

0 0 12

0 0 0 12

0

0 0 0 12

0 0 0 12

0 0 0 12

0 0 0 12

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCAPerform the matrix multiplication to convinceyourself that this is so. The third row can beproduced from the second by post-multiplying by

M2

12

0 12

0 0 0 0 0

12

0 12

0 0 0 0 0

0 12

0 12

0 0 0 0

0 12

0 12

0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCAFinally the fourth row can be obtained from thethird by post-multiplying by

M3

12

12

0 0 0 0 0 0

12 1

20 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

L. R. Mustoe and A. C. Croft474

Writing the original vector as P and the trans-formed vector as T we see that

T PM1M2M3or

T PM where M M1M2M3The image is then edited to add further zeros(throwing away variations of small magnitude)producing a new matrix T, and then the processis reversed to give P, say, simply by postmultiply-ing by the inverse of M:

P TM1

When it is necessary to compress a whole image,rather than a single row vector, the matrix opera-tions are exactly the same, although it is usual toapply the averaging and differencing both tocolumns and rows in a matrix. A very readableand thorough introduction to this process can befound in the excellent paper by Mulcahy [3].Matlab code to implement the process on acomputer is also available. A much wider rangeof material can be found in [4].

The importance of the underlying processcannot be underestimated because it encapsulatesthe essence of many other mathematical methods.What has happened is that the original data hasbeen processed or transformed. In engineering,mathematical transformations are applied toobtain new information about some data whichis not necessarily evident when the data is in itsoriginal form. We saw this when, by transformingthe image data above, we obtained informationabout the average value, and the variation of thedata from that average. There are a number oftransformations which can be applied to extractnew information from data, among which a groupknown as Fourier transforms are the most impor-tant. These have been around for a long time.Increasing demands from the industry to produceever faster, and higher quality, compression

mechanisms have led to a proliferation of othertransforms. One particular group presently at theforefront is a group known as wavelet transformswhich are based on the technique of averaging anddifferencing detailed above. These are used by theFBI to compress their hundreds of millions ofdigital images so that they can be stored efficiently,but nevertheless retain sufficient detail to be of usein fingerprint matching.

Example 4: jpeg compressionWe have noted already, in Example 2, thatstudents are often familiar with computer filescontaining digital images. When a still photo-graphic image is viewed over the internet the like-lihood is that this image is stored as a so-calledJPEG (pronounced jay-peg) file.

JPEG is a compression mechanism which worksespecially well on natural images like scenes orphotographs, although it is not designed tocompress images containing lettering and linedrawings. JPEG stands for the committee whodrew up the specification for this compressionmechanism: the Joint Photographic ExpertsGroup. When compressing an image using JPEGsome of the information in the original photo-graph or file is thrown away and lost forever.This degrades the quality of the image and if toomuch information is thrown away the effect will bevery noticeable. However, by carefully selectingwhich bits of information to lose the human eyecan be deceived into not noticing any change.

The original digital image is first subdivided intosmaller blocks of size 8 8. A transform called thediscrete cosine transformation is applied to eachblock. This transform provides additional infor-mation about the frequency content of the image.Loosely, this is information about where values arechanging slowly (low frequency) and where valuesare changing quickly (high frequency). Highfrequency values are discarded through an opera-tion involving matrix multiplication, resulting in acompressed image file. This can be transmitted

Fig. 7. A jpg image and a compressed version of the same scene.

Motivating Engineering Students by Using Modern Case Studies 475

efficiently over the Internet, and decompressed,albeit with some loss of quality at the receivingend. For example the first image in Fig. 7 is a jpgfile with file size of 40KB and the second is acompressed version of size 24KB.

CONCLUSIONS

The case studies offered in this paper aredesigned to show that applications of mathematicsin modern engineering settings are feasible forpresenting to young people in schools or collegesin order to excite their interest in mathematics as aliving subject with real, and increasing relevance in

the world of today and tomorrow. These exem-plars can, of course, be refined and made moresophisticated and are not meant to be the last wordin their area of application. A good case study canbe re-visited as students mature mathematicallyand this will help to encourage them to developcontinually their mathematical skills.

The generation of suitable case studies requires acloser collaboration between mathematics and en-gineering lecturers and school teachers. It is vitalthat we increase the number of mathematicallyable young people who pursue engineering as acareer and we believe that the approach suggestedin this paper is one way forward.

REFERENCES

1. Brasseys (UK), Brasseys World Aircraft and Systems Directory, London (1996) ISBN 1 85753 198.2. C. Brislawn, http://www.c3.lanl.gov/~brislawn/FBI/FBI.html3. C. Mulcahy, Plotting and scheming with wavelets, from http://www.spelman.edu/~colm/wav.html.4. Mathcom Wavelet Resources, http://www.mathsoft.com/wavelets.html.

Leslie Mustoe obtained his bachelors degree in pure mathematics and has a Ph.D. inmathematical education. He has taught mathematics to engineering undergraduates formore than twenty-five years. He has authored eight textbooks on engineering mathematicsand has written several papers on teaching mathematics to engineers. He advocated thewider use of numerical methods in the undergraduate curriculum and developed anapproach which taught these methods alongside their analytical counterparts. Dr Mustoeis concerned with improving the mathematical ability of entrants to engineering courses. Heis Chairman of the Mathematics Working Group of the European Society for EngineeringEducation (SEFI).

Anthony Croft has worked as Manager of the Mathematics Learning Support Centre atLoughborough University since 1996. Prior to this he taught mathematics to engineeringundergraduates at De Montfort University Leicester for eight years. He graduated from theUniversity of Leeds in 1981 and was awarded a Ph.D. from the University of Keele in 1990.Dr Crofts particular interest is the mathematical education of engineering undergraduatesand he has written several books aimed at this audience.

L. R. Mustoe and A. C. Croft476