Problem-Solving Tools for Engineering Students Bruce R. Dewey Mechanical Engineering University of Wyoming Laramie, WY

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1 Problem-Solving Tools for Engineering Students Bruce R. Dewey Mechanical Engineering University of Wyoming Laramie, WY Abstract - The common first course in the engineering curriculum has evolved from drafting/graphics to one in computer applications. Goals for such a course include the introduction of engineering methodologies, stimulation of interest in the study of engineering, and development of skills in using certain tools. Candidate computing tools should solve equations such as those in engineering applications and produce various presentation graphics. High level programming languages have been in engineering curricula since the 97s. More recently, various solution programs have become available. The intuitive graphical user interface makes the use of spreadsheets appealing. As features have been added, we have found increasing use of spreadsheets for problem solving. An introductory course emphasizes basics, while subsequent courses introduce applications that work well on spreadsheets. ntroduction n addition to the usual calculus, chemistry, and English, beginning engineering students traditionally have a first technical course that focuses on problem-solving aspects. Over time, such a course has evolved from board-based graphics to one in computer applications. General goals for this course encompass aspects of appreciating engineering methodologies, creating interest in the study of engineering, and providing some common tools that will be useful in subsequent courses. General-purpose problem-solving tools work engineering problems that have been reduced to mathematical terms. Useful in typical engineering courses as well as engineering practice, a list of abilities for such tools might include: Solving systems of linear and nonlinear algebraic equations Producing d and d presentation graphics including regression to arbitrary functions Performing repetitious calculations involving large amounts of data Finding optimum solutions problems subjected to various constraints Numerically integrating and differentiating arbitrary functions Supporting creative endeavors where what-if scenarios can be tested Being inepensive, easy to obtain, and simple to learn. A little over thirty years ago, Fortran was the only practical way to solve problems with the computing resource. Subsequently, other high level languages have been developed with significant advantages over Fortran. The ubiquitous and intuitive graphical user interface, such as the one in spreadsheets, provides even more productivity. Should only one software tool be chosen to meet the above requirements, spreadsheets are the clear choice for a common curriculum. Over the past eight years, an approach has evolved for incorporating spreadsheet-based problem solving in the curriculum. Our introductory course emphasizes problem-solving methodology, starting with formulation of practical problems and using various solution techniques in spreadsheets. Sophomore courses assign occasional problems that are appropriate for spreadsheet solutions. To show how spreadsheets support the above requirements, we will first detail some of the built-in functions available in the industry-leader Ecel. Net, we briefly discuss the available general-purpose numerical equation solvers. Some eamples follow, illustrating how the listed goals are met with various Ecel features. Functions in Ecel Using one of nearly three hundred built-in functions in Ecel may be the quickest way to arrive at the solution to some problem. A paperback [] is available that gives a summary of how to use the built-in functions and formulas. While many of the functions handle esoteric financial calculations or do some housekeeping item, a surprising number find direct application in engineering. About built-in statistical functions in Ecel support practically all that is needed by engineers in this realm of work. A number of functions cater to the common needs of finding central tendency statistics for data sets and of calculating best-fit straight lines, log-log, semi-log, and polynomial curves. Furthermore, Ecel determines several different probability distribution functions. The robust set of financial functions handily supports the computation needed in engineering economics. While depreciation rules are not built-in, schedules of depreciation can easy be computed with row and column formulas. The rate-of-return calculations that can be veing to students are trivialized in the Ecel environment. A set of functions labeled engineering is of some interest. ncluded here are functions to convert numbers between binary, decimal, octal, and headecimal as well as to perform comple arithmetic. Bessel and error functions can be computed.

2 A group of mathematical functions provides trigonometric functions, including inverse and hyperbolic. Matri functions provide multiplication, inversion, and determinant computation. Although not listed as a mathematical function, Ecel is very adept at sorting data. Logical and information functions allow branching as various conditions are met. The most used of these is the F function. Date and time functions find use in acquisition of eperimental data and in financial calculation. The built-in plotting of presentation graphics produces plots of high quality. Plot types include -y (Ecel calls these scatter plots ), pie, column, bar, radar (polar), combination (such as column and line), -d surface, and -d column. A best-fit line and R are very easy to add to an -y plot. Numerical Equation Solvers Ecel contains four different, useful schemes for numerical solution of transcendental equations and systems of linear and nonlinear equations.. Matri nversion Linear systems of equations of the form [A][X] = [B] are solved with the MMULT and MNVERSE functions that are built into Ecel. The typed function that returns [X] = [A]? {B] =MMULT(MNVERSE(A),(B)) is entered in the array of cells [X] intended to receive the solution. The matrices [A], [B], and [X] are naturally adapted to the row/column structure of the spreadsheet. Although the author has not tested the limitations of the built-in inversion algorithm, the routine seems to be robust for a moderately large number of equations. Ecel returns an error message if the system is singular, etc.. Goal Seek The Goal Seek feature, a command in the Tools menu, lets you vary the contents of one cell in order to make another cell equal some arbitrary value. n other words, a cell containing the value of could be changed with an iterative scheme until another cell containing f() becomes equal to a chosen value such as. Goal Seek is faster, easier to use, and more bulletproof than Solver, below.. Solver Solver, also accessed under the Tools menu is considerably more versatile than the Goal Seek command. Solver causes the contents of one cell (the target cell) to take on a certain value (or a maimum or a minimum value) by varying the contents of one or more other cells (Changing Cells). Optionally, one or more constraints can be applied to any or all of the Changing Cells. Solver deals with large-scale optimization problems, the solution of nonlinear systems of equations, and other decision problems. Solver uses proprietary nonlinear optimization code, simple methods, and branch and bound algorithms that are supplied to Microsoft by Frontline Systems, ncline Village Nevada. n use, Solver has proven to be fairly robust and versatile. mprovements have been made in Solver as Ecel has gone up in versions. Sometimes solutions require good starting guesses to converge, an advantage in teaching engineering. n the problem solving process, students should be encouraged to estimate answers and to question whether results are reasonable. n nonlinear problems with multiple solutions, users need to provide starting guesses over the range of acceptable solutions.. terative Solution of Circular Reference Formulas While this solution method is by far more arcane than the other choices, it is ideal from some problems, such as the partial differential equation solution below. A circular reference is where the contents of two or more cells depend on each other usually causing an error message because of the undefined nature of the information. To use this method of solution, the user selects manual calculation and iteration from the Tools:Options menu. Additionally, the number of iterations and the maimum change (the convergence criterion) can be controlled from the same menu. Eamples When asked what computer software they use most, a very informal survey of perhaps two dozen mechanical engineers in industry placed word processing and Ecel as nearly universal choices. AutoCAD, Ansys, and Nastran had a few mentions. Ecel found use for such diverse tasks as Optimizing production schedules in a multi-product manufacturing environment with hundreds of variables Doing load and balance calculations on a commercial airplane Assembling data arising from various sources as input to engineering analysis codes; passing data files in a universal form among various groups Doing budgets and keeping accounts (a no-brainer!) Compiling and analing quality control statistics Chosen to demonstrate application of one (or two) of the Ecel equation solvers, some typical engineering problem solutions typical of those done in engineering classes are presented below. Eample Simultaneous Linear Equations n many engineering analyses, simultaneous linear equations of equations of the form [A]{} = {B} arise. As a numerical eample, consider

3 ? =? 9?? () difference between the values in the third and fourth columns. Ecel s Solver is called to minimize the sum of the residuals squared (the last value in the fifth column) by changing the guesses for σ and µ. The resulting values appear in Fig. ; the data in column four have been computed with these values. On the spreadsheet, the input for [A] will be in cells A:D5; {B} will be in F:F5. Descriptive titles are recommended. The solutions, appearing in the column with cells for {}, have the underlying code =MMULT(MNVERSE(A:D5),F:F5) () where the cell ranges references provide the location of the input data. With Ecel, the results are recalculated automatically whenever any of the input data are changed. f a user does this code sequence frequently, the macro language in Ecel permits the construction of a new function that replaces Eq. () with a user-chosen name and a call such as = SMEQ(A:D5,F:F5) () The presentation of this solution on a spreadsheet (not shown) can look almost identical to Eq. (), with the enclosing brackets being added using the drawing tools. Eample Nonlinear Regression The widely used regression process, finding unknown parameters in the equation of a line or curve to best fit a set of data, aids in the interpretation eperimental data. Linear regression is built into Ecel as the Trendline function. The Trendline gives the choice of fitting the data to a straight line, to a semilog line, a log-log line, or to a polynomial up to order five. The coefficient of determination R can be reported in addition to the equation of the line. Ecel s Solver provides an intuitive method to perform nonlinear regression by minimizing the sum of squares of the residuals (the same criterion used in linear regression.) The following eample from hydrology involves fitting an observed distribution of rainfall to the normal probability distribution function. The starting point is the frequency distribution of the amount of rainfall over 55 years of observation as shown in the first two columns of the worksheet in Fig.. The normalized frequency distribution f() is given in the third column. The model to which the data are regressed is given by the equation Annual Rainfall - Teton County f() f() model Resi Amount Frequency Frequency squa in (no. yrs) fraction Parameters for normal distribution function σ µ Figure. Worksheet for nonlinear regression. The results are plotted in Fig., using one of the supplied line chart types in Ecel. The shaded bars show the given histogram, while the curve shows the normal distribution function computed with the solved values of σ and µ. Of course, features such as color, shading, titles, font, line styles are under user control. The foregoing method fits data to any arbitrary curve. Criteria other than minimizing the sum of the squares of the residuals can be used with this method. For eample, it is not difficult to regress with the criterion that the sum of the squares of the perpendicular distances to a straight line is to be minimized. where the unknown parameters σ and µ will be determined with nonlinear regression to best fit the data. With guesses for σ and µ placed in the cells provided at the bottom of the sheet, Eq. () coded in the fourth column of the worksheet. The fifth column is the square of the

4 Not shown in this eample is the built-in Trendline function that very easily performs linear regression. Frequency Annual Rainfall - Teton Cty Figure. Histogram and best-fit normal distribution curve for Eample. Eample Solution of a Nonlinear System of Equations Mass balance problems, hydraulics problems, and others frequently yield systems of nonlinear equations. Solution of such systems is the forte of programs like TK-Solver and EES. n Ecel, each of the equations need to be arranged in the form f( i ) =. As an eample, consider the contrived system Amount (in). = +.77 =. = + = where the i are the unknowns. (The contrived roots are.,.,., and..) The worksheet of Fig. provides four cells to hold the initial guesses and the subsequent results of the iterative solution. A solution criterion that works well, avoiding problems with negative signs, is that (5) Figure. Worksheet for solution of nonlinear system of equations. The result, which depends on the quality of the initial guesses, is good to four places without tweaking the default precision in Solver. Eample Eigenvalue Problem Goal Seek, Solver, and matri commands are useful in the iterative solution of problems involving the etraction of eigenvalues and eigenvectors. As an eample, the orientation of the principal aes and the values of the principal mass moments are determined for a -d solid. The worksheet, Fig., contains space for the input of the moments and products of inertia. The user provides a reasonable guess for one of the three principal moments in the cell for. Goal Seek is used to make the iteratively find such that y z where the value ( Det ) of the determinant is computed with Ecel s MDETERM function. Goal Seek is appropriate for this solution, since only one variable is changed. The corresponding direction cosines (l, m, n) must make y z subject to the constraint l y y yy + m + n = = The user puts guesses for the direction cosines in {d}, with the constraint of Eq. (9) coded in the shaded cell labeled Σ (di^). The three shaded work cells for the product {P} are each squared and added in the shaded cell labeled Σ(Pi^). zz z z l m? = n? (8) (9) (7) ( ( )) = (6) f i j Solver is used to minimize the contents of the cell labeled sum of squares, which contains the code for Eq. (6). The results for the s appear in place of the initial guesses.

5 .677 Figure. Worksheet eample for determination of one of three principal moments of inertia. Solver is used to change the {d} to minimize Σ(Pi^) with the constraint that the contents of Σ(di^) =. The numbers shown in Fig. represent one of the solutions. The user is epected to enter reasonable guesses for the other solutions good pedagogy because some understanding of the problem is required. Eample 5 Partial Differential Equation Steady state heat conduction is governed by the equation T =. The row-column arrangement of the worksheet is intuitively used for nodes in PDE problems that are reduced to difference equations. An interior node with temperature T ij is related to its adjacent nodes by the equation T ij yy zz 6 y z Det Det.67 {d} {P}=[M]{d} Σ(di^) Σ(Pi^)..79E-7 T = i+, j + Ti, j+ + Ti, j + Ti, j () Boundary nodes may have temperature specified, be insulated, or have other conditions such as convection. n a simple numerical eample, a symmetrical rectangular chimney is considered that has outside dimensions of ft by ft with.5-ft thick uniform walls. The outside temperature is 7 F and the inside temperature varies linearly from F at the center to 5 F at the corners. To make a compact illustration, nodes are selected. ft on center, and symmetry permits modeling one-fourth the cross section. The Ecel worksheet provides one cell for each node point, arranged eactly as in physical space. Because we are going to use iterative solution of circular reference formulas, we select manual calculation and iteration under the Tools: Options:Calculation menu. The specified temperature distribution is placed on the inner and outer boundaries. The interior nodes and the insulated (symmetry) boundaries are coded with relative cell references in the manner of Eq. (). Copy and fill features make this a very quick process. After calculation, the worksheet of Fig. 5 appears Figure 5. Worksheet for heat conduction. Cells pertain to one quadrant of the chimney. Ecel s surface plot feature works very well for illustrating the result, Fig. 6. To improve visualization, the user can rotate plots around vertical and horizontal aes, and change the perspective viewing distance parameter. Conclusions The eamples presented show a sample of the versatility of Ecel in solving engineering problems. Engineering students seem to have no trouble catching on to doing operations like these. Beginning users of Ecel will find getting started with the ecellent tet by Gottfried [] much less tedious than using references aimed at non-technical/business users. For the past eight years, our first-year students have been introduced to Ecel from an engineering standpoint. Students find Ecel fun and easy to use. n the introductory course, students have far more challenge in formulating problem solutions than in using computer software. Some instructors in other core courses reinforce Ecel instruction by assigning the "computer problems" found in various tetbooks. Very little use of Ecel macro functions seems to

6 be needed. The supported programming language, visual basic, fills in when needed. n our curriculum, a junior-level course, taught by math/computer science faculty, presents structured programming in a high-level language (either Fortran or C as determined by the department or advisor.) Programming eercises cover various topics in numerical methods. Because this course follows the traditional differential equations class in the mathematics department, a more rigorous treatment of the nuances of various algorithms is possible. On an as-needed basis, other engineering courses introduce software such as Matlab, Mathcad, C++, AutoCAD, finite element analysis, etc. [] Gottfried, B., Spreadsheet Tools for Engineers, Ecel 97 Version. WCB/McGraw-Hill T 5 5 Figure 6. Surface plot of temperature distribution in chimney. When given assignments and projects that require a computing resource, our students will nearly always pick Ecel. This software is unsurpassed for reducing laboratory data and for making all sorts of data plots. The built-in presentation graphics of Ecel encourages structure, e.g. proper labels and "even" calibration of aes, of students' work. Plots and tables produced by Ecel can be seamlessly pasted into Word documents. None of the other computing tools enjoy the wide availability (many computers come with Ecel bundled) and versatility of Ecel. Furthermore, these other tools are less intuitive and more difficult to use than Ecel. For many reasons, Ecel is a clear choice for the first software learned, and the one software system needed, if there can be only one. References Cited [] Microsoft Ecel 97 Worksheet Function Reference, Microsoft Press, 997.