engineering problem solving and excel
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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)