Engineering Problem Solving and Excel

EGN 1006 – Introduction

to Engineering

Mathematical Solution Procedures Commonly Used in Engineering Analysis

� Data Analysis Techniques (Statistics)

� Curve Fitting techniques (Looking at Graphs)

� Interpolation techniques � Interpolation techniques

� Single and multiple algebraic equations

� Evaluating Integrals (Evaluate effects over time)

� Economic Analysis

� Optimization (Finding the best solution)

Applicable Engineering Fundamentals

Most engineering problems are based upon

one of three underlying principles:

Equilibrium – Force, Flux, and Chemical1. Equilibrium – Force, Flux, and Chemical

2. Conservation Laws – Energy and Mass

3. Rate Phenomena – How something

changes over time.

What is a spreadsheet?

� A spreadsheet is basically a table containing NUMERICALand/or ALPHANUMERICAL values.

� Individual elements are known as CELLS.

� Each CELL can contain a single value or a STRING (sequence � Each CELL can contain a single value or a STRING (sequence of characters)

� The cells are arranged in columns and rows are referenced by a CELL ADDRESS ( For example, B3 refers to the cell in COLUMN B, row 3.

� The collection of cells is referred to as a WORKSHEET.

� A cell can have a manually entered number or be assigned a FORMULA EVALUATION such as C7 being =(C3+C4+C5)

The Excel Window

Entering Data

There are two ways to enter data into Excel

� A simple numerical value called a number constant.

� A string, called a text constant.� A string, called a text constant.

When you are finished entering a number in a cell hit

ENTER or click the “checkmark”.

Using Formulas

In Excel, a formula MUST always begin with an equal sign (=), followed by an expression involving:

Consider: =(C3+B2+5)involving:

•Constants

•Operators

•Cell Addresses

Consider: =(C3+B2+5)

�C3 & B2 are cell addresses

�5 is the numerical constant

�The (+) sign is the operator

This formula could be entered in D7 where the formula

would be applied. Note: Any change in C3 or B2 will

automatically change D7!

Arithmetic Operators

Operator Purpose Example

+ Addition A1+B1

- Subtraction A1-B1

* Multiplication A1*B1

/ Division A1/B1

^ Exponentiation A1^3

% Percentage A1%

Operator Preference

Since some formulas include more than one operator, the question arises as to which one is carried out first. The

Operator

Preference

Operator

1 %one is carried out first. The order is outlined to the right. If any formula has two operators from the same group, the order is carried out from left to right.

1 %

2 ^

3 * and /

4 + and -

For example, in the formula =(C1/D2*E3), the division

would be carried out first then multiplication.

A Simple Spreadsheet Application

A small machine shop has the following parts on hand:

Item QuantityItem Quantity

Screws 6500

Nuts 9000

Bolts 5400

Start by creating a worksheet that includes this

information, plus the total number of parts on hand.

Answer the questions on the worksheet provided.

Using Functions

Excel includes many different functions which can carry out a wide variety of operations.

They include:They include:

� Mathematical and statistical operations

� Process financial data

� Process AND return text information

Each function has a specific name followed by an ARGUMENT enclosed in parenthesis.

Function Examples

� =Sum(C1,C2,C3) This will add the numbers

in the three cell addresses. The ARGUMENT

is inside the parenthesis and separated by is inside the parenthesis and separated by

commas.

� =Sum(C1:C50) the use of a COLON

indicates a RANGE and will add up ALL cells

between the two cell addresses.

Other function examples

� =SQRT(x) Takes square root of “x”

� =Min(x1:x20) Returns the minimum # in the set

� =Max(x1:x20) Returns the maximum # in the set� =Max(x1:x20) Returns the maximum # in the set

� =Round (x,n) Rounds “x” to n decimal places

� =Average (x1:x15) Returns the average

Example: =sum(A1, SQRT(A2/2),2*B3+5,D7:D12)

This example has FOUR arguments as

evidenced by the commas

Example #2 – Student Exam Scores

Create the following worksheet: See paper worksheet

Student Exam 1 Exam 2 Final Exam Overall Score

Davis 82 77 94Davis 82 77 94

Graham 66 80 75

Jones 95 100 97

Meyers 47 62 78

Richards 80 58 73

Thomas 74 81 85

Williams 57 62 67

Moving things around!

You can:

� Select and Highlight a block of cells

� Copy a block of cells� Copy a block of cells

� Move a clock of cells

� Delete rows or columns

� Create grids

� Change font color, fill in backgrounds, etc

� Adjust column widths

Creating Graphs

The easiest way to create a graph in Excel is to use the Chart Wizard!

Follow these steps:Follow these steps:

� Select the block of cells containing the data to be plotted. You may include headings!

� Click on the CHART WIZARD icon

� Choose the graph type

� Type in a title

� And select Chart on this worksheet or “As object in”

More on Graphs

� Graphs done on a SEPARATE sheet can

easily be copied or pasted into a WORD

document.document.

� Graphs embedded into worksheet can be

edited even after they have been inserted.

Creating and Editing a Line Graph

The voltage within an electronic device varies with time in accordance with the formula:

Seconds Volts

0 10

1 6.07

with the formula:

teV

5.010 −

=

1 6.07

2 3.68

3 2.23

4 1.35

5 0.82

6 0.50

7 0.30

8 0.18

9 0.11

10 0.07

Prepare an Excel

worksheet and line

graph (scatter) with

the data to the right

Editing the graph

Once the graph is embedded into the worksheet, click on the graph until the chart toolbox appearstoolbox appears

Select the drop down menu to add a title

and label each axis with units. Also,

choose LEGEND then RIGHT CLICK on

the small box and choose CLEAR!

Analyzing Data

Engineering analysis usually begins with the

analysis of data! Engineers gather data to

measure VARIABILITY or CONSISTENCY.measure VARIABILITY or CONSISTENCY.

Measured Data can tell you a great deal if you

know how to interpret the results. Let Excel

do the tedious work for you so that you can

focus on the interpretation of results.

Data Characteristics

There are several commonly used parameters that allow us to draw conclusions about the characteristics of a data set. They are:

Mean Median Mode Max

Min Variance Standard Deviation

Mean, Median, and Mode

� Mean – is the arithmetic average of a data set. It represents expected behavior. AVERAGE( ) is used in Excel

� Median – the value where half of the data falls above and half the data falls below. MEDIAN ( ) is used in Excel

� Mode – the value that occurs with the greatest frequency with in data set. Mode ( ) is used in Excel. If a tie results it will always list the FIRST frequent number it encounters

Min and Max

The min and Max simple represent the

extremities of the data set. In Excel ,the MIN

( ) and Max ( ) functions return these values. ( ) and Max ( ) functions return these values.

NOTE: The MIN and MAX functions return

the values that are the smallest and

largest ALGEBRAICALLY. They do not

return values in terms of MAGNITUDE.

Example: ( -5,-2, 1) ; Min = -5 & Max = 1

Variance

The variance provides an indication of the degree of SPREAD in the data. The greater the variance, the

)(1

1

1

22−

−

= ∑=

xxn

sn

i

igreater the variance, the greater the spread. It is determined by the following formula:

Excel uses the VAR( ) function

mean x

valuedata individualx

valuesdata of # n

variance

1

i

2

1

=

=

=

=

−∑

=

s

n i

Standard Deviation

The standard deviation is simply the square root of the variance.

deviation standard2== ss deviation standard2== ss

So why bother with the standard deviation?

The variance is a much more practical value to

have but its UNITS are NOT consistent with the

mean, median, or mode. Excel use the stdev ( )

command.

Analyzing a data set

A car manufacturer wishes to determine how

accurately the cylinders are being machined

in several engine blocks. The design in several engine blocks. The design

specification call for a cylinder diameter of

3.500 inches, with a tolerance of +/- 0.005

inches. See paper worksheet

Histograms

Thought the previous statistical characteristics

can prove useful in interpreting data, it is

often more desirable to the plot the data in a often more desirable to the plot the data in a

manner that illustrates how the values are

distributed within their range. This is called a

HISTOGRAM or RELATIVE FREQUENCY

plot.

More on Histograms

To create a histogram, you must first subdivide the range of the data into a series of adjacent, equally spaced intervals. The first adjacent, equally spaced intervals. The first interval must begin at or below the smallest value (the min) and the last interval must extend to or beyond the largest data value (the max). These intervals are called CLASS INTERVALS. Then you determine HOW MANY values fall within each interval

The car manufacturer continued’

The histogram feature is found under TOOLS then choose DATA ANALYSIS. The choose histogram. There are two things the

histogram needs:

An INPUT RANGE – this comes from your data. Click on the � An INPUT RANGE – this comes from your data. Click on the Input range box then click on the first cell of the data, hold, and highlight until the last cell is chosen

� An OUTPUT RANGE – this is the interval bounds. Do the same click and hold

� To see the chart you must click OUTPUT RANGE under OUTPUT OPTIONS and specify where you want the chart located. Click on the cell where you want it to be placed.

The car manufacturer continued’

Your histogram

should look like

this!this!

Relative Frequency

Now we can go back and LOOK AT the percentage of

values that fell into each interval. These values were

found using the following equation:found using the following equation:

set datain valuesofnumber total

intervalin that valuesofnumber

interval in thefrequency relative

=

=

=

=

n

n

f

n

nf

i

ii

Cumulative Distribution

A histogram can provide a great graphical illustration of

how a data set is distributed. A CUMULATIVE

DISTRIBUTION is equally important. It provides us DISTRIBUTION is equally important. It provides us

ANOTHER graphical way to view the

data….BUT….it allows us to determine the

LIKELIHOOD of a RANDOM VALUE being less than

or greater than a specified value. It is almost like a

percent chance and is the graphical representation

of the calculated relative frequency.

Cumulative Distribution Cont’

To find the cumulative

distribution: 11

ffF

fF

+=

=

So basically you just sum

up the relative

frequency using the

interval prior.....3213

212

fffF

ffF

++=

+=

Create a cumulative distribution column

using your relative frequency data.

Plotting the Cumulative Distribution

Cut the original

histogram and

Choose TOOLS Choose TOOLS

then data analysis.

This time under

OUTPUT OPTIONS

click on cumulative

distribution and

chart output

Notice that the calculated CD values will appear!

Fixing The Cumulative Distribution Plot

The problem with this type of graph is that the

bar graph has GAPS in it.

Right Click on one of the bars in the graph� Right Click on one of the bars in the graph

� Choose FORMAT DATA SERIES

� Click on OPTIONS

� Change GAP WIDTH to ZERO!

Drawing inferences

A engineer can now look at the cumulative

distribution and randomly pull a part off of the

manufacturing line. The plot will tell him the manufacturing line. The plot will tell him the

% likelihood that an arbitrary cylinder

diameter within a randomly selected engine

block WILL NOT exceed a certain length. For

example, what is the likelihood that a cylinder

will NOT exceed 3.503 inches?

Fitting Equations to Data

The data an engineer collects could reveal:

� Spatial profile

� Time history

� Cause and effect relationship

� System output as a function of input

Mathematical expressions are then used to

CAPTURE the relationship shown in the data

Fitting a straight line to a set of data

Data is usually represented by values that show some SCATTER, which is due to fluctuations or errors in measurement.fluctuations or errors in measurement.

Therefore, we NEVER connect the dots on a graph! We pass the points through an AGGREGATE or TRENDLINE. In science, this is probably referred to as a “line of best fit”.

Force exerted by a spring

Data point

#

Distance (cm) Force (N)

1 2 2.01 2 2.0

2 4 3.5

3 7 4.5

4 11 8.0

5 17 9.5

Making a Trendline

� Using the data you just entered into EXCEL, use

chart wizard to construct a SCATTER plot. Place in

WORKSHEET 2. Add title and units to axes. WORKSHEET 2. Add title and units to axes.

� RIGHT CLICK on one of the data points (all should

highlight) and choose trendline.

� Since this plot looks linear, choose LINEAR.

� The choose OPTIONS and click on ADD equation to

chart and ADD “R” value.

Regression Statistics

Excel can also provide a great deal of built in

statistics. But they may prove MORE than

what you need.what you need.

� Choose TOOLS then DATA ANALYSIS

� Choose Regression

� Highlight the appropriate cells and where to

place the stats.

Assessing Quality using r2

The r2 value helps and engineer assess the QUALITY

of the curve fit.

Any number close to 1.0 is a good fit. You can think of Any number close to 1.0 is a good fit. You can think of

this value as a %fit. A 1.0 would represent 100%.

If the r2 value is too low, right click on the trendline and

change the type to LOGARITHMIC or other type of

curve fit. The largest r2 value is the one that fits the

data the best.

The “OTHER” fitting functions

� Exponential

� Logarithmic

� Power Function

� Polynomial ( NOTE: By INCREASING the

order, you can increase your r2 value)