Mechanics in the Engineering FirstCurriculum at Northwestern University*

T. BELYTSCHKO, A. BAYLISS, C. BRINSON, S. CARR, W. KATH, S. KRISHNASWAMY,B. MORAN, J. NOCEDAL, and M. PESHKINRobert R. McCormick School of Engineering and Applied Science, Northwestern University,2145 Sheridan Road, Evanston IL 60208, USA

A new core curriculum, Engineering First, has been developed at Northwestern University whichintegrates a subset of mathematics and science with engineering. The topics chosen for integrationin the course are those which most closely relate mathematics to the computer solution ofengineering problems: linear algebra and ordinary differential equations. The science of mechanicsis emphasized, although other areas such as circuits and communication networks are alsoconsidered. The analytical topics are taught in a four course sequence called Engineering Analysis,which begins with the first quarter of the freshman year. The familiarization of students withcomputer methods in these courses enables us to introduce them in the freshman year to designanalysis: the selection of design parameters by analysis. The sequence is currently being taught in apilot version to 80 students; pre-pilots were taught last year. A program of evaluation has beendeveloped and early results are very favorable. Next year the sequence will be taught to allfreshmen.

INTRODUCTION

ENGINEERING First represents a major revisionin the engineering core curriculum at North-western University aimed at increasing the student'scompetence in design and modern engineeringmethods. Subsets of the traditional mathematicsand science curriculum have been integrated withengineering to provide a sequence of courses whichintroduces the students to problem solving inengineering and spans the gap between theoryand application. In addition, these courses providean opportunity to pose problems in design analy-sis, which, combined with a two-course freshmansequence in design and communication, give thestudents an early grounding in design and anunderstanding of engineering.

As in many other engineering schools, the corecurriculum at Northwestern has been almostunchanged for the past three decades. The struc-ture of the curriculum was largely driven by theengineering science thrust which evolved afterWorld War II [1]. In the current curriculum, thefirst two years consists primarily of courses inmathematics, science, communications and elec-tives. Only a few scattered and disconnectedengineering courses are taken by the students inthe first two years (see Table 1).

This has several disadvantages:

1. It is difficult to start teaching design early in thecurriculum, because a context in which designcan be taught is not available.

2. A degree of student dissatisfaction was apparent

due to the absence of engineering in the first twoyears.

3. Skills in computer methods cannot be developedin the first two years.

Many students felt that the heavily mathematics/science-based curriculum provided no understand-ing of engineering in the beginning and stifledmotivation. Thus, students who were uncertain oftheir selection of an engineering career had littleexperience in their first two years on which to basetheir choice.

A highly innovative, fully integrated core curri-culum of mathematics, science and engineeringwas developed at Northwestern in the 1970s. ThisBlock Program was jointly taught by faculty inengineering, science and mathematics. The BlockProgram combined all of the core science andmathematics courses and featured sequential, con-centrated time blocks on each science and mathe-matics topic, motivated by the idea that studentsenjoy focusing on one subject at a time. However,the co-ordination of this program required a levelof effort which was not sustainable in a majorresearch university and it was discontinued after aperiod of 8 years.

Our recent re-examination of the curriculum wasmotivated by the urgent need to improve ourteaching of design and to adapt the curriculum tothe dramatically altered work environment oftoday's engineers. There have recently been manycalls by national organizations to improve theundergraduate engineering curriculum [2, 3]. Tra-ditional curricula, including that at Northwestern,provide few opportunities for teaching design untilthe last two years. Furthermore, the content of the* Accepted 2 June 1997.

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core courses in engineering topics does not reflectthe changes in the engineering workplace that havetaken place. The Engineering First revision has thefollowing objectives:

� introduce design in the freshman year so that afamiliarity with the approaches and issues indesign can be developed to a higher state ofmaturity;

� integrate certain parts of mathematics, scienceand engineering in order to improve the stu-dent's understanding of these topics and theirmotivation;

� improve computer literacy and problem solvingon computers by exposing the students to acontinuous series of exercises in design analysisbeginning in the first quarter of the first year.

To meet these objectives, two sequences ofcourses were developed: a four-quarter sequenceof courses called Engineering Analysis (EA) anda two-quarter sequence called EngineeringDesign and Communication. The engineeringdesign and communication courses are a novelcoordination between the engineering and com-munications faculty in which the basics of engi-neering design are taught in conjunction withwriting; the description of this course sequencecan be found elsewhere [4].

The engineering analysis sequenceEngineering Analysis is a four-quarter sequence

which integrates engineering with linear algebra,ordinary differential equations, programming, andmechanics as taught in Physics. Thus it is anintegration of selected fields in mathematics andscience with engineering. The topics in mathe-matics which have been integrated are those mostclosely related to computational applications. Theintegration of these fields serves several purposes.First, by combining mathematics and sciencewith its applications, the student's understandingand motivation is improved. Secondly, computerapplications of these methods are given through-out the courses so that the linkage between com-puter methods and the basic principles becomes

more apparent. Computer applications not onlyaid in teaching the student how engineeringproblems are formulated and solved, but providea means of teaching design analysis: the selectionof parameters for a design by means of analysis.Although this is only one aspect of design, it is acritical bridge between the analytical methodswhich are taught throughout the engineeringcurriculum and design.

While schools like Drexel University [5, 6],Rose-Hulman Institute of Technology [7, 8]have had considerable success with almost com-plete integration of engineering with science andmathematics, it was felt that a complete inte-gration was not possible at Northwestern. Theadditional teaching load in the EngineeringSchool which would result from complete inte-gration would be very burdensome for a facultyheavily involved in research, and our previousexperiences at Northwestern with integrationbased on cross-disciplinary teaching indicatedthat it was not sustainable. Moreover, there wasa strong consensus that all engineering studentsshould be exposed to the intellectual style ofscience and mathematics.

Some schools, such as Carnegie-Mellon andUniversity of Pittsburgh have instituted year-longfreshman courses in the major engineering area.While this is quite appealing, the number ofstudents at Northwestern who are undecided asto their major even towards the middle of thesophomore year made this option unworkable.Furthermore, since a course in a discipline typi-cally does not cover basic material, the number ofcore courses or advanced courses would have to bereduced to accommodate such major courses.

Therefore in the Engineering First curriculum,we elected to integrate the subset of the mathe-matics courses which is most closely associatedwith computing and engineering analysis: linearalgebra and ordinary differential equations. Thescience and engineering content of EngineeringFirst focuses on mechanics. Mechanics is generallytaught in two to three courses in an undergraduateengineering curriculum: the first course in physicsand subsequent engineering courses in statics and

Table 1. Current curriculum

Freshman Year Sophomore Year

Fall Winter Spring Fall Winter Spring

Math Calculus 1 Calculus 2 Calculus 3 Vector Series, Linear DifferentialCalculus Algebra Equations

Physics Mechanics Electricity & Waves, Optics,Magnetism Quantum Mech.

Chemistry General InorganicChem. Chem.

Engineering Computer Freshman Statics Major MajorProgramming Seminar

Other Elective Speech, Writing, Elective Elective ElectivesElective Elective

T. Belytschko et al.458

dynamics (Northwestern had an unusualarrangement whereby statics and dynamics werecombined into a single course for most disciplines).Mechanics is an attractive setting in which to teachengineering analysis because it is readily com-prehensible to freshmen and many of the methodsapply to other fields. Moreover, to give a studentan appreciation of the similarities of the mathe-matical fundamentals of various fields, examplesin other fields, such as circuits and telephonenetworks are included in the EA courses.

This paper describes the four-course EAsequence in the Engineering First program. Theoverall structure of the program is described andthen each of the courses is described in detail. Thecourses which have been eliminated from thecurriculum and the topics which have beende-emphasized or removed are discussed. Finally,we describe our experiences with these courses.Two of these courses have been taught in pre-pilot and pilot versions and our experiences withthose are described, including feedback from thestudents. In the coming years, these courses will berequired for all freshmen.

CURRICULUM CHANGES

The old and new curricula for the first two yearsof engineering are shown in Tables 1 and 2. As canbe seen, in our old curriculum, engineering waslimited in the first two years to three disconnectedcourses (with one or two courses taught in thechosen major at the end of the sophomore year,depending on the department). The first course iselectrical and computer engineering ECE A01,which is a course in FORTRAN programmingcombined with some numerical methods, such asnumerical integration (a C programming coursewas also an option). Students usually took noengineering courses in the second quarter. In thethird quarter, students had the option of taking aseminar related to a major, although only 30%

took this option. In the first quarter of the sopho-more year, students took a combined course instatics and dynamics, but this course usuallyinvolved no computer applications nor anydesign analysis and was quite unrelated to thecourse content of the previous two engineeringcourses.

In the new curriculum, students take a contin-uous sequence of engineering analysis courses fromthe first quarter in the freshman year. Skills devel-oped in the first course are used and furtherdeveloped in the subsequent courses, and thecourse material is highly interrelated. Computerimplementation of methods developed in thecourses play a central role, so the students canquickly appreciate how the concepts taught in thecourses are used in engineering analysis. Further-more, to develop an appreciation for design, pro-blems in design analysis are assigned in eachcourse.

In contrast to the original curriculum, computerprogramming in FORTRAN or C is no longer arequirement. Instead, the emphasis in the EAsequence is on solving problems with computers.For this purpose, programming is taught withMATLAB [9], a higher level language and computa-tionally oriented software package, in the EA1course and used for all subsequent courses in thesequence. Use of a higher level language dramati-cally reduces the workload of developing a com-puter solution, so that more problems can beassigned without overloading the student. In addi-tion, higher level languages include convenientgraphical display capabilities which aid the studentin understanding the results.

Our decision to eliminate programming in astandard language, such as FORTRAN or C, asa requirement in the curriculum was based onchanges in the engineering workplace. Whereas adecade ago, many engineers in industry oftenprogrammed in one of the standard languages,this is not the case today. Engineers today useprimarily general-purpose software and higher

Table 2. Engineering First curriculum

Freshman Year Sophomore Year

Fall Winter Spring Fall Winter Spring

Math Calculus 1 Calculus 2 Calculus 3 VectorCalculus

Physics Electricity & Waves, Optics,Magnetism Quantum Mech.

Chemistry General InorganicChem. Chem.

Engineering EA1: Linear EA2: Linear EA3: Dynamic EA4: Differential Major MajorAlgebra & Algebra and System EquationsProgramming Mechanics Modeling

Engineering ED&C-1 ED&C-2

Other Elective Elective Elective Elective Elective

Mechanics in the Engineering First Curriculum 459

level languages. Programming in the standardlanguages is done almost exclusively by softwareengineers today and requires many skills whichcannot be taught in a single course. Therefore wefelt that programming in standard languagesshould be an elective for students with a specialinterest in software. Teaching computer appli-cations in a sequence of courses using a languagelike C or FORTRAN would be prohibitivelyburdensome to the students.

The first course in the sequence is EA1, whichreplaces the course ECE A01. The EA1 coursealso covers the majority of the material from ourintroductory level course in linear algebra. How-ever, whereas linear algebra is usually taughtwithout many illustrations or applications, inEA1 the applications are an essential ingredientof the course. The concepts taught are developedto the point of engineering applications, so thatstudents can immediately grasp their significanceand usefulness.

EA2 partially replaces the first physics course inmechanics and the combined statics and dynamicscourse. Statics and dynamics are taught in bothEA2 and EA3. An important consideration in thedesign of the EA2 course was to build on thematerial taught in EA1. Therefore, emphasis isplaced on posing problems as linear systems ofequations, solving them using MATLAB and visual-izing results obtained by varying parameters. Toprovide a vehicle for the integration of linearalgebra and mechanics, matrix methods of struc-tural analysis are presented for simple cases.Students are thus introduced to finite elementmethods, a tool widely used in engineering. Severaldesign analysis problems are assigned in whichdesign parameters are evaluated by these methods.

EA3 focuses on dynamics, but expands thepurview of a typical dynamics course by con-sidering other dynamical systems, such as simplecircuits and hydraulic models. In this course,eigenvectors are introduced and immediatelyillustrated and applied through the evaluation ofnatural modes and frequencies of a system. The

concept of a differential equation is also intro-duced, although only numerical methods for theirsolution are used. The analytical solution and thedescription of the theory of linear differentialequations is taught subsequently in EA4.

EA4 is quite similar to a classical introductorycourse in differential equations except for theaddition of substantial material on numericalsolutions and many applications to engineeringproblems. The objective is to immediately givethe student an appreciation of the central roleof ordinary differential equations in simulationand to illustrate the powers of simulation inunderstanding the behavior of systems and indesign.

The four courses are taken in sequence, so thestudent completes the sequence by the secondquarter of the sophomore year. However, thecourses are offered in two rotations to accommo-date students that get off sequence. In addition tothe EA sequence, in the new curriculum the studentstake a design and communication sequence in thesecond and third quarters of the freshman year.

The list of the new courses and the ones whichare replaced are given in Fig. 1. It can be seen thatthe new curriculum, even with the addition of thetwo-quarter design and communication sequence,does not increase the number of courses, and forthose students who took the Freshman Seminar,the number of courses is decreased. A large part ofthis benefit was generated by the integration of themathematics with the engineering sciences.

DESCRIPTION OF ENGINEERINGANALYSIS COURSES

EA1: Linear algebra and programmingThe goals of EA1 are two-fold:

1. Teach basic linear algebra of square and rectan-gular systems with an emphasis on applicationsand computations.

2. Teach computer programming using MATLAB.

Fig. 1. Replacement of courses by new curricula.

T. Belytschko et al.460

Specifically the course is designed to teach basiclinear algebra (not including eigenvalues) withapplications and programming methods usingMATLAB. The linear algebra material includesmatrix notation, the solution of square systemsusing the LU decomposition, matrix inverses,rectangular systems, subspaces, spanning sets,linear independence and dependence, dimensionand rank, projections and least squares. Thematerial is first taught mathematically, and thenillustrated by applications. These applicationsinclude circuits, heat transfer and fluid flow. Over-determined systems are illustrated by applicationsfrom communication networks.

Concurrently with the mathematics the stu-dents are taught and expected to use computation.Students are first taught the concept of computerarithmetic (i.e., words of memory, storage of realnumbers using exponents and mantissas andround-off errors). After this background studentsare taught the basics of MATLAB (scalars, vectorsand matrices), arithmetic operations in MATLAB

and basic MATLAB functions. They are then able touse MATLAB to compute some of the linear algebraconcepts (e.g. matrix products, dot products andtransposes). As the course progresses more compli-cated concepts such as decision and loop structuresare stressed. Finally, the importance of modular-ization (in MATLAB through M-files) is empha-sized both in classroom presentations and throughthe assignments.

The linear algebra and computational materialare illustrated by computational exerciseschosen from different disciplines, including cir-cuits, heat transfer, fluid flow and communi-cation networks. In addition to the lecturematerial, a standard linear algebra text and aMATLAB text are used in this course [10, 11],although we will replace Strang with a moreintroductory linear algebra text next year. As thecourse progresses, students are expected to domore than just solve specific problems. Forexample, computational problems involving deter-mining parameters to satisfy certain criteria ordetermining sensitivity of solutions to differentparameters, are given.

We now list specific goals associated with thecourse. The overall goal of the course is to give thestudents an understanding of the following topicsin the theory of linear algebra:

� matrix and vector notation and operationsand recognize equivalence between systems ofequations and matrix notation;

� singular and nonsingular systems;� the mathematics underlying row operations;� the ideas behind the LU decomposition and the

importance of this decomposition in solvingsystems of linear equations;

� the concepts of inverses;� the differences between rectangular systems

and square systems in particular to illustrateboth in general and by example, differences

between square, overdetermined and undeter-mined systems;

� the concept of vector space and subspaces;� spanning sets, linear independence and concept

of basis and dimension;� the concept and meaning of subspaces asso-

ciated with a matrix namely, column space,null space, row space and left null space;

� the circumstances for which solutions do notexist and circumstances where solutions arenot unique students should understand thisboth mathematically and in terms of appli-cations (e.g. least squares for overdeterminedsystems and communication networks forunderdetermined systems);

� the concepts of orthogonality and orthogonaldecomposition;

� projections and overdetermined systems in thecontext of least squares;

� how to obtain orthonormal sets (Gram-Schmidt) and why these sets are important.

In teaching the applications of linear algebra,the goal is to develop an understanding in thefollowing basic concepts and ideas:

� the diversity of applications of linear systems ofequations;

� round-off error as a general phenomenonassociated with computation in general (i.e.not necessarily just for computation of linearsystems of equations);

� round-off error and ill-conditioned linearsystems, via specific numerical examples (e.g.Hilbert matrix);

� specific applications from circuits (Kirchhoff'slaws), heat transfer, fluid flow and communi-cation networks.

Our goal in the programming aspect of thecourse is to teach:

� decision (i.e. logic) and repetition (i.e. for loops)structures in the context of both formal syntaxand applications where these structures have tobe used;

� the importance of modularization for bothprogram development and debugging, in parti-cular, both the formal syntax of M-functionsand techniques to break up programming pro-jects into small modules which can be developedand debugged individually;

� how to generate graphs and plots in both oneand two dimensions, show how they interactwith MATLAB code and stress the importanceof graphics in interpreting the results ofcomputations;

� students to combine their programming andlinear algebra knowledge by working throughspecific programming projects in computationallinear algebra.

An example of a homework problem from thisclass is given in Fig. 2 and the complete assign-ment will be available on our website. This figure

Mechanics in the Engineering First Curriculum 461

represents a very simplified telephone networkwhere calls originating at city 1 and city 2 needto be routed to city 5 and city 6. The inputs tonodes 1 and 2 (i.e. 10 and 5) represent the numberof calls (in thousands) generated in each of thesecities. The outputs from nodes 5 and 6 representthe number of calls with these destinations. Theproblem is how to route the calls between thesetwo pairs of cities. The students analyze the net-work and reduce it to a system of equations thatcan be manipulated via linear algebra concepts.The model is then programmed in MATLAB with acost function to obtain an optimal solution.

EA2: Linear algebra and mechanicsEA2 is the second course in the Engineering

First freshman engineering sequence. This courseintegrates mathematics, mechanics and design ana-lysis and makes extensive use of computer analysisas a tool for exploring problems in mechanics. Italso reinforces the methods of linear algebrataught in EA1. A course outline is given in Table 3.

As can be seen from Table 3, the first part of thecourse starts with vectors, particle dynamics andequilibrium, but also includes programming andnumerical solutions of problems not amenable toclosed form analytical solutions. Students see hownumerical integration of the equations of motionof a particle can be used to solve complex problemsreadily. It should be noted that students havenot had a course in differential equations (this isnow covered in EA4) and the exploration ofparticle motion by numerical methods provesattractive at this early stage (finite-differencetechniques for integration of the equations ofmotion were covered).

In the first design analysis problem, the flightpath of a vehicle in the presence of air drag, thestudents are introduced to several ideas: namely,that analytical solutions of idealized problems(neglecting air drag) can form the starting pointfor a real problem (including air drag); thatnumerical solutions are indispensable in realsituations but at the same time not too difficultto obtain; that such solutions might have to beobtained through an iterative analysis (theproblem requires numerical integration using theso-called shooting method since restrictions on the

position and velocity are imposed on the far end ofthe trajectory).

In the second part of the course, EA2 departsfrom traditional courses both in content andphilosophy. The typical material covered in phy-sics courses at this point include work andenergy principles; these however are deferred toEA3 in our sequence. Instead, Newton's laws ofequilibrium are extended to rigid bodies of finitedimensions, i.e. the particle idealization is relaxed.

Fig. 2. Example homework problem from EA1: a communications network.

Table 3. Topics Covered in EA2

� Vectors� Scalars and vectors� Vector operations: addition, scalar multiplication, dot and

cross products; mixed triple products� MATLAB programming of vector operations� Kinematics and Dynamics of Particles:� Displacement, velocity, and acceleration� Newton's laws of motion for a particle� Analytical integration of simple motions; rectilinear,

circular and projectile motions� Numerical integration of the equations of motion (forward

Euler method):Motion of an electron in an electromagnetic fieldProjectile motion with air dragOrbital motions

� Design analysis: Determination of launch positions andvelocities such that pellets can be shot through air (takinginto account air drag) into a slot at a specified height withspecified restrictions on the entry velocity.� Mechanics (Statics) of Rigid Bodies:� Concept of force and moment equilibrium� Analysis of rigid trusses, frames and machines� Numerical solutions: graphical solutions to problems that

lead to transcendental equations� Design Analysis:

Device to raise a refrigerator using cables and pulleysDesign of simple weight balances

� Mechanics of Deformable Bodies:� Springs and deformable rods - Hooke's experiment: stress

and strain� Stretching of rods of uniform and varying cross-section� Systems of rods and springs: statically indeterminate 1D

and truss systems� Matrix methods of analysis of systems of rods and springs

Element stiffness matrix for rods and springsAssembly of global stiffness matrix of unconstrained

systemsImposition of constraints and applied forces

� Numerical Solutions: Determination of the stresses anddisplacements of a tapered observation tower under theaction of its own weight.

T. Belytschko et al.462

The concept of moment equilibrium is introduced,and analysis of simple rigid trusses, frames andmachines is presented, as is typically done in asophomore engineering mechanics course. Onceagain, numerical methods are emphasized, includ-ing determination of the roots of transcendentalequations using the bisection method and/or gra-phically. The students are assigned simple design

analysis problems which involve computer solu-tions and introduce them to the design of simplemechanical systems.

The last part of the course introduces thestudents to statically indeterminate problems, andforces the students to re-evaluate the limitations ofthe rigid body assumption. This naturally leads toan introduction to the mechanics of deformable

Mechanics in the Engineering First Curriculum 463

bodies, a topic typically covered in a sophomorecourse on strength of materials. Analysis of rod/spring systems and simple deformable trussesleads to matrix/finite element analysis (in onedimension) of such systems. The students areshown how this leads to a system of linear equa-tions, and how the method can be implemented ina MATLAB program. A final problem is assignedwhich requires the students to model a structureof varying geometry and varying loads as a one-dimensional system of rod elements.

In summary, the content of the course is drawnvertically from material typically offered in fresh-man physics and sophomore engineering mech-anics and strength of materials courses. Newton'sequations of motion (and equilibrium) form the

thread that logically connects the entire material.Within this contextual framework, the course:

� provides the students opportunities for tacklingopen-ended design analysis problems;

� introduces several powerful methods of mathe-matical and numerical analyses that haveapplications outside of mechanics;

� integrates computer programming at every level.

Sample problems.A typical computer homework problem is given

in Fig. 3 as an example of the integration ofmathematics, vector mechanics and computer pro-gramming. Students were previously required towrite a program to determine cross products. (The

Fig. 4. EA2 homework problem: matrix analysis of a tapered tower.

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MATLAB code has a built-in cross-product func-tion but this exercise is nonetheless useful inreinforcing students' understanding of program-ming and the use of determinants.) The problem inFig. 3 involves particle motion in three dimensionsand illustrates some interesting aspects of themotion of an electron in an electromagnetic field.The students are asked to visualize the resultsgraphically. The solution to part (f) of the problemis also shown. Other related homework problemsconsisted of explorations of orbital mechanics.Here, the students were required to explore theuse of time-steps and to comment on their attemptsto obtain a convergent solution. The students werealso asked to identify various types of orbits(circular, elliptic, parabolic and hyperbolic).

Another computer homework problem involv-ing mechanics, linear algebra and MATLAB pro-gramming is given in Fig. 4. Here, the studentsare asked to suitably modify a MATLAB code tosolve problems involving one-dimensional systemsof rods and springs. The students gain an appre-ciation of approximating a complex geometry(essentially a rod with tapered cross-section) asN-segments of uniform cross-section, and they

learn how to approximate distributed loads (theweight) as `lumped' loads at discrete points. Oncethey cast a real engineering structure into anapproximate model, they are required to obtainthe stress and displacements through the structure.They are then encouraged to improve upon thissolution by refining their model.

Text.The text used for the first part of the course is

Statics and Dynamics by Bedford and Fowler [12].It was chosen on the basis of its readability andits use of numerical methods, which fit in nicelywith the philosophy of this course. Supplementalmaterial [13] covering the latter parts of thecourse was made available to students.

EA3: Dynamic System ModelingEA3 focuses on the modeling of dynamic sys-

tems, the reduction of models to differentialequations of motion, and some exploration of thebehavior of the solution of those equations.Numerical methods of solution are emphasizedby using MATLAB to solve differential equationsand visualizing the results.

Fig. 5. An example of web-based homework problem for EA3.

Mechanics in the Engineering First Curriculum 465

EA3 is a heavily computer-based course. Thetext itself, as well as the homework problems, areon the web. This makes possible a highly inter-active text, in which students can downloadprogram `shells', adapt them for their simulatedsystem, and explore system behavior. An exampleof such a web-based exercise is shown in Fig. 5. Inthis exercise, students apply what they havelearned about electrical components and systemsto the circuit shown, which is representative of agraphic equalizer channel; they solve the systemnumerically via a MATLAB differential equationsolver and can then listen to the effect of the circuitapplied to a frequency sweep and to an excerpt ofHandel's Messiah.

The goal is to learn system modeling acrossseveral physical domains (mechanical, hydraulic,electrical), and in each:

� to understand the elements of each domain (e.g.spring, capacitor; or voltage, pressure, force);

� to recognize the elements in real life, so that realsystems can be abstracted into ideal systemscomposed of familiar elements;

� to express precisely the way in which the ele-ments interact (e.g. circuit diagrams, free-bodydiagrams);

� to reduce the idealized systems to equations,which describe their behavior quantitatively.

The topics covered are:

1. Behavior of the elements; modeling systems ofelements; free-body diagrams; masses, springs,and dampers; mechanical rotation; absoluteand relative pressure; flow rate; Kirchoff'slaws; capacitors, inductors, resistors, batteries.

2. Multiple degrees of freedom; simple harmonicoscillators (example: mechanical and electricaldomain); matrix formulation (example: trans-mission line treated as a 32-element lumpedparameter model); numerical solutions tomatrix equations; eigenvalues, normal modes.

3. Constants of the motion; energy/work inmechanical systems; elastic and inelastic colli-sions; energy/power in electrical systems; energyin hydraulic systems; linear momentum, colli-sions and angular momentum in mechanicalsystems.

EA4: Differential equationsEA4 is a one-quarter course on ordinary differ-

ential equations (ODEs) and the final course in theEngineering Analysis sequence. In many ways, acourse on ODEs is the perfect setting in which tosummarize and solidify the concepts that havebeen developed throughout the previous threecourses because of the many examples of engineer-ing systems that can be described using differentialequations. This has already been recognized atmany universities, of course, since a large numberof ODE courses are heavily populated with engi-neering students and the accompanying textbooks

are filled with numerous applications from scienceand engineering.

Nevertheless, ODE courses are often consideredby students to be `cookbook' in that much of thecurriculum appears to be driven by solution ofequations for which exact, analytic answers can befound. As a result, the course material often comesacross as a recipe: for one type of equation, use thismethod; for another type, use this one, and so on.

Analytic solutions are extremely important, ofcourse, but in practice it is no longer true that ananalytic solution is always the best answer. Often,a good numerical solution will be more illus-trative than a complicated analytic answer. Toverify computed solutions, however, it is importantto have some qualitative idea beforehand as to howa solution is expected to behave, and this isincreasingly the role of analytic solutions.

EA4 is designed around a three-tiered approachcombining:

� Exact (analytic methods): classification ofODEs; separable equations; integrating factors;second-order linear equations; constant coeffi-cient equations; forced equations; boundaryvalue problems; first-order systems; nonlinearsystems.

� Qualitative and approximate methods: directionfields; critical points and stability; linearizedstability; phase plane methods; perturbationmethods; bifurcation theory.

� Numerical solutions: Euler, improved Euler andRunge-Kutta methods; stiff equations; implicitmethods; graphical representation.

Students will be taught to understand the advan-tages and disadvantages of each solution methodand when each is likely to be appropriate. Inaddition, students will learn how to combinesolution techniques to gain a better understandingof a solution's behavior. A differential equationtextbook will be used for this course [14], withseveral other texts as reference sources [15, 16].

Analytic solutions, for example, will be used toelucidate the basic theory and to explain what oneshould expect for more complicated problemswhen numerical solutions are used. Time will bedevoted to non-dimensionalizing or scaling equa-tions and how to identify terms which are smalland easily neglected, or are large and which mightcause problems when numerical solutions areattempted (i.e. stiff equations). The concept ofperturbation methods will be introduced as amethod for finding corrections when small para-meters are present. For nonlinear equations, cri-tical points and their stability, and qualitativetechniques such as phase line and phase planemethods will be discussed. Bifurcation conceptswill be introduced to explain how solutionschange as parameters are varied.

MATLAB will be used to provide the numericalsolutions and the graphical representation ofresults. In addition, various numerical solutionmethods (i.e. forward Euler, improved Euler,

T. Belytschko et al.466

Runge-Kutta, etc.) will be explained. Implicitschemes for handling stiff equations (e.g. thosearising in chemical reactions) will also be discussed.

Throughout EA4 problems examples from real-world applications will be used to illustrate theconcepts. Considerable discussion will be includedregarding the principles behind the design of theODE models of physical systems and the useful-ness of differential equations. Some of the topicswhich can be included are:

� heating and cooling� growth and decay� birth and death processes (elementary queuing

theory)� mechanical vibrations and damped oscillations� forced vibrations and resonance� electrical circuits� nonlinear mechanical systems� chemical reactions.

As a specific example, a simple model of theoscillations produced in a multistory building byan earthquake can be constructed by connectingmasses using springs, as shown in Fig. 6. Studentswill first discuss the pros and cons of such models.(How good is the mass-spring model likely to be?Is it reasonable to consider only horizontal dis-placements? What effects have been left out?)Students will then solve the problems using differ-ent methods, and will discuss the advantages anddisadvantages of each method (e.g., expanding inthe natural eigenmodes of the system gives betterconceptual information but looking for periodicsolutions directly is more practical). These prob-lems will be followed by open-ended design analysis

questions (e.g. how one would build an earthquakedamping system to protect the building).

Web and network useTeaching Engineering First has been helped

immensely by use of the Web throughout thecourses. In addition to homework problems andtextual material on the Web for several of the EAsequence, an electronic conference system dedi-cated to the engineering analysis sequence hasbeen set up and used to communicate with thestudents. Faculty and teaching assistants involvedin the course (as well as some faculty interested inseeing this resource in action) and all students inthe course have access to the conference. Areas (orchannels) for general class discussion are providedand specialized areas for discussion of homeworksare also available. Participants can post messages,comments and questions to the group in e-mailstyle formats, or alternatively can chat in real timewith other participants who are logged on. Groupdiscussion can be held and private communication(messages or chat) with individuals can be held. Anespecially useful feature of the conference is thatdocuments (such as a sample MATLAB program fordirect use or modification by students) can beposted for the students to download and a specialfolder (drop-box) is provided for students to turnin their completed programming assignments.These programs can be accessed and tested bythe teaching assistants.

The general consensus on this type of conferencefacility is that it is extremely useful and will likelybe increasingly used in other courses not part ofthe present curriculum development. The con-ference provides a valuable means of obtainingtimely feedback from students as the course pro-gresses (as the they give vent to their thoughts onthe difficulty or otherwise of an assignment, forexample) and, consequently, problems can often benipped in the bud.

The Engineering First website, www.engfirst.nwu.edu has been used in the pilot courses andwill be further developed and even more heavilyused in the full implementation of EngineeringFirst next year. The website contains both generalinformation on the curriculum (of interest toprospective students, parents, colleagues at otheruniversities) as well as course material for use ofthe students in the classes. Use of the website forsupplementary text material, homework problems,and examples has proven extremely useful, bothfor the faculty and the students.

IMPLEMENTATION AND EVALUATION

The first two EA courses were first taught aspre-pilots in the academic year 1995±6. These pre-pilot classes were taught as special sections of thecorresponding engineering courses in the originalcurriculum. The classes were designated as `specialsections with computer enrichment'. The students

Fig. 6. Homework from EA4, modeling oscillations in abuilding due to an earthquake.

Mechanics in the Engineering First Curriculum 467

were self-selected and there was considerableinterest in participating in this experiment. Theseclasses enabled the instructors to develop andrefine the computer assignments which are a keypart of the EA courses and to test the pedagogicalsoundness of the material. The response of thestudents was strikingly favorable, which pro-vided a major impetus for proceeding with thedevelopment of the new courses.

In 1996±97, the entire sequence of EA courses isbeing offered as pilots to a group of 80 students.These students were selected by a lottery afterindicating an interest in taking this curriculum;we did not have enough slots to accommodate allstudents who were interested. The students weresubdivided into two sections of 40 taught bydifferent instructors. The two instructors colla-borated closely to organize the syllabi and developadditional new material.

In 1997±98 all incoming freshmen will take thenew curriculum. The Northwestern freshmanengineering class typically consists of about 320students, so eight sections of the engineeringanalysis courses will be offered. Although goingto a full-scale implementation immediately afterthe pilots limits the time and iterations to tunethese courses, a continuation of the pilots phasewas deemed undesirable because of the resultinginability of advanced courses to take advantage ofthe new material and the difficulties of managinga dual-track curriculum. Revisions of the courseswill be carried out in the summer quarter by teamsled by the instructors of the pilot courses.

The implementation of these courses willrequire substantial additional resources. Whilethe number of courses taken by freshmen has notbeen changed, two courses have been shifted fromthe College of Arts and Sciences. Since all fresh-men will be taking these courses, the combinedeffect of the engineering analysis and engineeringdesign & communications courses has been to addmarkedly to the teaching load of the engineeringfaculty.

In addition, the shift of part of the freshmantraining to the engineering school requires asignificant increase in physical resources. WhileNorthwestern requires that students have accessto a personal computer and most students meetthis requirement by purchasing their own com-puter, a series of computer-intensive coursesrequires substantial computer laboratories wherestudents can be instructed and where they can gowith questions. Although we have primarilyinvested in workstation laboratories in the past,it has become clear that PCs are now more appro-priate in these laboratories. Since students do mostof their work on their own PCs, laboratories usingworkstations are quite awkward. The operatingsystem and the look and feel of many softwarepackages differs between PCs and workstations,which hampers the students' learning. Further-more, the speed and software availability for PCsis now on a par with workstations. In view of the

much lower costs of the hardware and softwareassociated with PCs, their advantages lookoverwhelming. We believe a similar shift willgradually occur in the engineering workplace, sofamiliarity with workstations will be of decreasingimportance. Some of the advanced courses willcontinue to use workstations.

For purposes of evaluating and tuning thesecourses, an extensive program of concurrentevaluation has been instituted. The programconsists of questionnaires filled out by students,discussion sessions with the students, and testingadministered to students in both the pilotcourses and a control group in the traditionalcurriculum.

Prior to beginning the freshman year, all stu-dents were given a questionnaire dealing withgeneral expectations and attitudes. At the end ofthe freshman year, a more comprehensive ques-tionnaire to examine student satisfaction, theirperceptions of course difficulty and workloadand their evaluations of the courses they havetaken will be administered. By comparing feedbackfrom students who took the pilot courses withthose who took the regular curriculum, we hopeto measure how the new curriculum meets thestudent's perceived needs.

As part of the evaluation program, at the mid-point of each course, a meeting is held with thestudents in which the instructors are not present.The objective is to find out whether the level of thematerial matches the abilities of the students, theintellectual challenge and interest of these courses,and the suitability of the workload. Such dialoguesare very useful in ascertaining weaknesses in acourse. For example, we found out quite earlythat the original text in EA1 was felt to be toodifficult. Weaknesses in organization and motiva-tion are also intensely discussed in such sessions.These meetings with the students are also veryuseful in determining the merits and shortcomingsof these courses. In general, students have beenvery receptive to the integration of mathematicswith engineering applications by computer. Forthe majority of engineering students, this approachincreases their interest in the mathematics, for theycan appreciate its importance when they see itapplied.

The evaluation of the impact of the new courseson learning poses a difficult challenge. One impor-tant skill which is stressed by the new curriculum,the ability to develop computer models for designanalysis and relate mathematical tools to computermethods, cannot be readily assessed by a test.However, one of our key goals was not to overlookthe objective of the core courses: to provide afirm understanding of fundamentals. Therefore,we intend to administer similar test questions tothe students taking the pilot version and the oldcurriculum at several points in the first two years.These questions will be administered as part ofregular examinations, since the use of tests whichdo not count towards a grade is often misleading

T. Belytschko et al.468

because of the diminished motivation and theburden on the students.

In addition to this, a standard questionnaire isadministered at the end of each course whichmeasures the student's perception of the qualityof the course, its organization and the instruction.These surveys were designed and given to thestudents at the end of the courses along withthe usual teaching evaluation questionnaires. Inthe pre-pilot and pilot EA classes taught thus far,the responses from students have been over-whelmingly positive. The students had friendsenrolled in the `standard' curricula, yet theresponse to the key question `I would still registerfor this special section if I could start the quarterall over again,' was a 3.4/4.0 for EA2 pre-pilot. Forthe EA1 pilot class, students agreed that the`course provided a good integration of linearalgebra and computing' 4.6/6.0. The students alsofound the design portions of the computer assign-ments useful and interesting and perceived learningMATLAB as useful with rankings of 3.1/4.0 and 3.2/4.0 in EA2. There were many helpful commentsfrom students in the written portion of the formsfor all the pre-pilot and pilot courses taught thusfar. A few selected student responses are:

� `I . . . liked the MATLAB homework as an alter-native to the other more mundane homework.'

� `I learned the most from the design projects.'� `The course helped me learn, however . . . a lot of

extra effort was put into this class the pro-gramming was especially difficult as I had noexperience.'

� `. . . helped me learn how linear algebra can beused in engineering.'

� `I learned how to think systematically and howto solve real life problems. This class was highlybeneficial. It was most excellent.'

� `I learned a lot, but in my mind it took toomuch effort. Strang [textbook] was too vagueand obscure.'

� `I learned a lotÐa great crash course into thefuture!'

The most important effects of the program willbecome apparent as the students take their coursesin the junior and senior years and go on topositions in engineering and other fields. Whenthe current class enters the last two years of theengineering program, instructors will be able toevaluate the adequacy of the preparation and theproblem-solving and design skills of the students.While we do not plan to formally measure thesechanges, meetings will be set up with faculty toshare their experiences and to determine anyshortcomings or drawbacks of the program.

Evaluation after graduation is largely anecdotal,although it is often very revealing. Unfortunately,this feedback is only available many years after thestart of a new program. The Engineering Schoolpolls all graduates three years after graduation.Questions aim to identify the strengths and theweaknesses of the engineering curriculum. Gradu-

ates are asked which courses and material theyhave found most useful in their work environmentand what skills are inadequately developed. Theyare also asked to rate the usefulness and theintellectual challenge of courses they have takenin their undergraduate program.

Thus, while our evaluation is primarily quali-tative, we should have substantial material bywhich to evaluate this program. Our attitudetowards this program is that it is a continuousexperiment. If feedback from the students andfaculty indicates that something is not working,the preparation is not adequate, or new material isdesirable, we will revise the courses accordingly.

DISCUSSION AND CONCLUSIONS

The Engineering First program at NorthwesternUniversity has been developed to effect a partialintegration of mathematics and science withengineering and to facilitate the development ofdesign skills early in the curriculum. A series ofcourses have been designed that integrate linearalgebra and ordinary differential equations withmechanics and engineering. These courses enablethe applications in engineering to be combinedwith the mathematics; thus the application ofmathematics to engineering problems can be illus-trated at the same time it is taught. In addition, atwo-course sequence for freshmen that combinesdesign with communication has been developed;this is described elsewhere [4]. These coursesshould provide students with a better founda-tion in modern engineering methods and design,starting from the freshman year.

The curriculum is currently in a pilot stage inwhich it is being taken by a group of 80 students,but it will be taken by all freshmen in the comingyear. We have organized an intensive program ofstudent feedback and a moderate program oftesting to evaluate these courses. In particular, wehave carefully watched the tendency of faculty tooverreach in designing new courses by adding toomuch material or overburdening the freshmen.Nevertheless, we have not avoided this problemcompletely and will have to tune the coursesfurther.

While a substantial part of the content of thesecourses is computer-oriented, many importantconcepts in mathematics and science have to betaught as part of these courses. While the computerimplementation gives the opportunity to show thelink between theory and application, there is apersistent danger that insufficient stress will beplaced on the theory. By carefully evaluating thestudents' competence at various stages of thisprogram and by stressing to the faculty the needto thoroughly teach and drill the fundamentals, webelieve that this integrated program will improvethe students' understanding and knowledge of thefundamentals.

It should be pointed out that while these courses

Mechanics in the Engineering First Curriculum 469

look markedly different from existing core courses,in a sense the changes have been evolutionaryrather than revolutionary. Three of these courseare built on traditional core courses and standardtexts are available for teaching the bulk of thematerial. The supplementary material is providedto the student through notes which are posted onthe Web. Thus the course content did not need tobe developed from scratch. Instead the newcourses represent mostly the rearrangement andenrichment of existing core courses.

We hope that the impact of these courses willecho throughout the entire curriculum: the mid-point of the sophomore year, students will have asolid background in use of mathematical andnumerical models for engineering problems. Inaddition, the design analysis aspect of the courseswill start the students on the learning process ofhow to reduce a complex real-world problem to itsessential elements necessary for a model and solu-tion. Since they have learned material in an inte-grated fashion, they should be ready to integrateadvanced subjects such as fluids and heat transfer,where they can tackle interesting design problems.Other upper level courses will be able to benefitfrom this preparation: for example, the Strength ofMaterials class can now incorporate introductorymaterial on finite elements, since matrix methodsin one dimension were introduced in EA2 [17]; inthe junior chemical engineering course, Kineticsand Reactor Engineering, a project for reactordesign involving solution of differential equationswill be readily understood and tackled by the

students. Moreover, individual departments maybe able to begin some of their major courses earlierthan is now possible.

A rewarding part of this experience is theenthusiasm that has been rekindled amongfaculty when challenged to develop new coursesand teach students with new approaches. Thefaculty have devoted substantial time to thiseffort. Furthermore, their enthusiasm has spreadto the students. The students' ratings of thesecourses have been very high and the entire processhas been invigorating.

Engineering education is now at a crossroads:while the workplace changed dramatically, engin-eering education was almost static for the pastthree decades. It is clear that the content of ourcore courses must be changed to better prepare ourstudents. In particular, with the increasing em-phasis of employers and ABET on design, afoundation for design must be built beginning inthe very first year. The engineering analysis coursesdescribed here are part of a two-pronged effort atNorthwestern for meeting these needs. The stu-dents' response to our initial effort has been veryfavorable. Although we realize that adjustmentswill have to be made, the needs of students andemployers will be better met by a program whichteaches engineering skills from the first year.

AcknowledgmentsÐWe would like to acknowledge theencouragement and support of the Dean, Jerome B. Cohen,who initiated this project and provided significant support, bothin time and resources.

REFERENCES

1. L. E. Grinter, `Report on Evaluation of Engineering Education,' Int. J. Engr. Educ., (1995)pp. 25±60.

2. J. Harris, E. M. DeLoatch, W. R. Grogan, I. C. Peden, and , J. R. Whinnery, `Journal ofEngineering Education Round Table: Reflections on the Grinter Report,' J. Engr. Educ., 83 (1994),pp. 69±94.

3. K. S. Pister, Engineering Education: Designing an Adaptive System, National Academy Press,pp. 79.

4. P. Hirsch, J. E. Colgate, B. Shwom, D. Kelso, J. Anderson and G. Olson, Engineering Design andCommunication: Jump-starting the Engineering Curriculum, Proceedings of the 1998 ASEE AnnualConference, Seattle, WA.

5. R. Carr, D. H. Thomas, T. S. Venkataraman, A. L. Smith, M. A., Gealt, R. Quinn and M. Tangel,Mathematical and scientific foundations for an integrated engineering curriculum, J. Engr. Educ.84 (1995), pp. 137±150.

6. Drexel University, E4: Enhanced Educational Experience for EngineersÐSummary Report, Phila-delphia, PA, Drexel University Press.

7. Rose-Hulman Institute of Technology; http://foundation.ua.edu/http://www.rose-hulman.edu/Users/groups/Assessment/Public/html/IRA/index.html, 1997).

8. G. J. Rogers, and B. J. Winkel, Integrated, First-Year Curriculum in Science, Engineering,and Mathematics at Rose-Hulman Institute of Technology: Nature, Evolution, and Evalua-tion, Proceedings of the 1993 Conference of the American Society for Engineering Education,(1993).

9. The Mathworks Inc., (http://www.mathworks.com, Natick, Mass., 1995).10. D. Etter, Engineering Problem Solving with MATLAB, Simon & Schuster, Upper Saddle River, NJ

(1997).11. G. Strang, Linear Algebra and Its Applications, Harcourt Brace Jovanovich, New York. 3rd ed.

(1988).12. A. Bedford and W. L. Fowler, Statics and Dynamics, Addison Wesley, New York (1995).13. T. Belytschko, C. Brinson, B. Moran and S. Krishnaswamy, Introduction to Finite Elements

(Matrix Methods), for EA2, Evanston, IL, supplementary notes (1997).14. C. H. Edwards, and D. E. Penney, Differential Equations: Computing and Modeling, (1995).

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15. M. Braun, Differential Equations and their Applications, Springer-Verlag, New York. 3rd ed.(1983).

16. J. C. Polking, Ordinary Differential Equations Using MATLAB, (1993).17. L.C. Brinson, T. Belytshko, B. Moran and T. Black, Design and computational methods in basic

mechanics courses, J. Engr. Educ., April, pp. 159±166 (1997).

Alvin Bayliss was born in New Jersey in 1945. He graduated from CCNY in 1967 with a BSdegree in physics. He received a Ph.D. in mathematics from New York University in 1975.He worked at ICASE, New York University and the Exxon Research and EngineeringCompany prior to joining the faculty at Northwestern where he is currently a CharlesDeering McCormick Professor of Teaching Excellence with joint appointments in theDepartments of Engineering Sciences and Applied Mathematics and Electrical andComputer Engineering. During his tenure at Northwestern he has developed courses incomputer programming and numerical methods at both the undergraduate and graduatelevels. These include courses in Fortran, numerical modeling and simulation, numericallinear algebra and numerical methods for partial differential equations. He has activelyworked on the development of Engineering Analysis I. He has worked extensively indeveloping numerical methods for partial differential equations modeling physicalphenomena. In particular, he has worked in combustion, acoustics, fluid dynamics,plasma physics and wave propagation. alvin@ece.nwu.edu

Ted B. Belytschko is Walter P. Murphy Professor of Computational Mechanics at North-western University. His expertise is in computational mathematics, nonlinear simulationand finite element methods, including crashworthiness, weapons effects, nuclear reactorsafety and simulation of prototype tests and manufacturing processes. Among the awardshe has received are the Walter Huber Research Prize of the American Society of CivilEngineers (ASCE), the ASCE Structural Dynamics and Materials Award, the Pi Tau SigmaGold Medal from the American Society of Mechanical Engineers (ASME), the JapanSociety of Mechanical Engineers Computational Mechanics Award, the ICES Gold Medal,and an Honorary Doctorate from the University of Liege. He is editor of Engineering withComputers, Engineering and Design, managing editor of Computational Mechanics, asso-ciate editor of Computer Methods in Applied Mechanics and Engineering and of AppliedMechanics and on the boards of many other journals. He is past Chairman of the AppliedMechanics Division of ASME, Past Chairman of the Engineering Mechanics Division ofASCE, and past President of US Association for Computational Mechanics. He is amember of the National Academy of Engineering. t-belytschko@nwu.edu

L. Cate Brinson is currently an assistant professor in the Mechanical EngineeringDepartment at Northwestern University. After receiving her Ph.D. in 1990 from Caltech,Dr. Brinson performed postdoctoral studies in Germany at the DLR (equivalent to NASA)and since 1992 she has been on the faculty at Northwestern University. Her primaryresearch focus is on the modeling and characterization of advanced material systems,including high performance composites and intelligent materials. Dr. Brinson has receivedseveral awards, including the 1995±99 NSF CAREER Award, the ASEE New MechanicsEducator Award, an ASEE Summer Faculty Fellowship and an AAUW PostdoctoralFellowship, and she held the June and Donald Brewer Junior Chair at NorthwesternUniversity from 1992±1994. She has made numerous technical presentations on herresearch and has co-organized several symposia at recent ASME and SES conferences.In recognition of her teaching, she was elected to the Northwestern Associated StudentGovernment Faculty Honor Roll in 1996. She is a member of many professional societies,was recently elected to the SES board of directors and is an associate editor of the Journal ofIntelligent Material Systems and Structures. cbrinson@nwu.edu

Stephen H. Carr is Associate Dean of Undergraduate Engineering at NorthwesternUniversity, Evanston, IL. He also is Professor of Materials Science and Engineering andChemical Engineering at Northwestern. In addition, he teaches courses in materials andmanufacturing, and he directs graduate research in areas of electro- and opto-activepolymers and network polymers. Dean Carr serves as a consultant to a wide variety ofindustries. His education was a BS in chemical engineering from the University ofCincinnati and MS and Ph.D. degrees from Case Western Reserve University. He hasbeen at Northwestern since 1970. Prof. Carr is a fellow of ASM International and theAmerican Physical Society; he is recipient of the Distinguished Alumnus Award from theUniversity of Cincinnati and a Ralph Teetor Award from the Society of AutomotiveEngineers. He is licensed in the State of Ohio. s-carr@nwu.edu

William L. Kath is a professor of Engineering Sciences and Applied Mathematics atNorthwestern University. He received his SB degree from MIT in 1978 and a Ph.D. fromCaltech in 1981. He was an NSF postdoctoral fellow from 1981±1982 and Von Karman

Mechanics in the Engineering First Curriculum 471

Instructor of Applied Mathematics from 1982±1984, both at Caltech. Since 1984 he hasbeen on the faculty at Northwestern. In 1985 he developed the course, `Honors Calculus forEngineers,' and in 1987 he received the McCormick School of Engineering's Award forTeaching Excellence. Dr. Kath was also a Presidential Young Investigator from 1985±1990.His main area of research is concerned with the modeling of high bit-rate optical fibercommunication systems and technologies. kath@nwu.edu

Sridhar Krishnaswamy obtained his Ph.D. in Aeronautics in 1989 from the CaliforniaInstitute of Technology. After a post-doctoral stay at the University of California at SanDiego, he joined Northwestern University in 1990 where he is now an Associate Professorof Mechanical Engineering. His teaching interests are in mechanics of solids, and hisresearch interests focus on experimental mechanics. s-krishnaswamy@nwu.edu

Brian Moran is an associate professor of Civil Engineering at Northwestern University. Hereceived his BE degree in Civil Engineering in 1980 and his M.Eng.Sc. degree in MechanicalEngineering in 1982 from University College Galway, Ireland. He received an Sc.M inApplied Mathematics in 1986 and a Ph.D degree in Solid Mechanics in 1988 from BrownUniversity. Following completion of his doctoral studies, he spent a year at Caltech as anIBM Research Fellow. He joined the faculty at Northwestern University in 1988 and hasbeen there since. His research interests are in computational mechanics of materials withemphasis on fracture mechanics and constitutive modeling of materials including phasetransformations. He has been recognized on several occasions for his teaching excellenceincluding nomination to the McCormick Faculty Teaching Honor Roll in 1990 and theAssociated Student Government Faculty Honor Roll in the same year. He received the W.M. Keck Foundation Award for Engineering Teaching Excellence in 1994. b-moran@nwu.edu

Jorge Nocedal received a B.Sc. degree in Physics from the National University of Mexico(UNAM) and his Ph.D. in Mathematical Sciences from Rice University. In 1978 he becameAssistant Professor at the Institute of Applied Mathematics (IMAS) of the NationalUniversity of Mexico. In 1981 he moved to the Courant Institute of Mathematical Sciences(New York University) as visiting assistant professor and research associate. He has been atNorthwestern since 1983, where he is now Professor of Electrical and Computer Engi-neering, and Deputy Director of the Optimization Technology Center. Professor Nocedal'sresearch interests include methods for nonlinear optimization; the solution of inverseeigenvalue problems; analysis of quasi-Newton methods; and methods for large-scalenonlinear optimization. He has studied applications in meteorology, weather forecastingand seismology. He is co-editor of Mathematical Programming and an associate editor forSIAM Journal on Optimization. He has also served as co-chair of the 1992 SIAM Meetingon Optimization. nocedal@ece.nwu.edu

Michael Peshkin is an Associate Professor of Mechanical Engineering at NorthwesternUniversity. His current research interests include computer-assisted surgery and passivecollaborative robotsÐ`cobots'. A paper he co-authored on the subject of cobots won the`best conference paper' award at the 1996 International Conference on Robotics andAutomation. Dr. Peshkin also serves as Chair of the Conference Editorial Board for theIEEE Control Systems Society, and was an associate editor of the IEEE Transactions onRobotics and Automation. Dr. Peshkin received a Bachelor's degree from the University ofChicago, a Master's degree from Cornell University, and a PhD in Physics (1986) fromCarnegie-Mellon University, where he worked in CMU's Robotics Institute. He is a 1988recipient of the National Science Foundation Presidential Young Investigator award.peshkin@nwu.edu

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