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					Class notes EG31 2                                                         M. Brienza
Engineering Calculations (ref. Introduction to Engineering by Paul H. Wright)

Engineering Calculations
While you might be able to sketch out a new idea or concept without worrying a lot about
dimensions or calculations, the first definitive presentation or demonstration of feasibility will
calculations that will be needed to examine all aspects of the concept. To name only a few,
such feasibility demonstrations will quickly deal with such issues as size, weight, efficiency,
cost, materials, heat management, physical strength, power source, compatibility to other
devices, etc. In these early phases of concept development, the ability to clearly sketch and
layout your idea with its calculations, is a skill you must develop to accurately communicate an
engineering idea. You will have this opportunity to practice this skill in both homework and
exam problems.

Especially in exams, a clear demonstration that you understand the problem and some path to
its solution will be rewarded with a large percentage of credit for the problem. Eventually,
when you are a practicing engineer, the time between your first documentation of a calculation
or concept and its use in whatever capacity can be a considerable interval. If your first
documentation is not clear and well structured, you or someone else may not be able to
discern exactly what you did.
If you are dealing with a homework or exam problem a reasonable outline would include:
       A sketch with annotations of important details showing what the problem is. Such a
sketch should almost be able to “tell the story” without a text description. This presentation of
the problem should also include a “time-line’ of events including a definition of events that
define the end of the calculation. For example, if we were solving how long a safety ramp is
needed to be to safely bring an out of control 18 wheeler to a stop, then the answer is defined
when and where the vehicle’s velocity is zero.
       The next step would be to now state the problem in mathematical terms showing the
appropriate operative equations, e.g. Newton laws of motion, and tools such as free-body
       At this point, the use of appropriate equations and solution techniques are applied e.g.
the use of free-body diagrams and equilibrium equations for linear and rotation motion could

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Class notes EG31 2                                                       M. Brienza
be used to generate the solvable equations. When the problem is now reduced to inserting the
numerical parameters of the problem into the final equations two final issues remain.
          First, the parameters must be in compatible units. This means that all dimensions of
length are in the same units, e.g. meters, feet, yard, light-years, whatever. Secondly the
number of significant figures allowed in the calculation must be determined. (More on this
At some point in the presentation of the problem, you have an estimate of the order of
magnitude of a reasonable answer. (More on this later)
Now, and not before, you can reach for your calculator and determine the numeric
answer. This calculation must now be modified by appropriate rounding of the calculator
display to the number of significant figures and annotated with the correct dimensions. A
number describing anything physical is worthless without units.

Numbering systems
Without trying to delineate any significant percentage of numbering systems that have been
developed and used throughout history, we will mention only a few, in particular, those that are
in some manner still used today. The simplest is perhaps the binary system that merely yields
a yes or no answer to a question. In a mathematical system we often apply the yes answer to
the digit 1 and the no answer to the digit 0. This of course is the base level system used by
computer processors and their memory systems and is called a binary system. Thus numbers
are formed by a series of 1’s and 0’s appropriately placed in the series.
The numbering system depends on the delineation of how many values each position has
available. A binary system has two values, 1 or 0. Each location has a value of 2 n x the value,
1 or 0 in the location. n is the number of the location where the first location is zero. The
value of the number represented by the entire string is the sum of the values of all the location.
                Digital processing discriminates on the two valued system of yes a voltage
                is present or no, voltage is not present. If we could invent a digital
                switching device with three possible states, then the counting system
                would be a three valued system with assignments of 0, 1, and 2.

For most of our counting needs (which include analogue measurements like how many feet
and fractions thereof is the length of a board) we prefer unit dimensions that are reasonable for
the measurement being made. For most of our counting and measurements we use a decimal

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Class notes EG31 2                                                        M. Brienza
system that assigns 10 values, 0 – 9, in each position of the string of numbers used to define a
number. As we move from one position to the next, the value of the digit changes by a factor
of 10. So the first place has the value of the digit there, the second place has the value of 10 x
the digit, etc. The final number is the sum of all the location values.

Not all of our systems are decimal, i.e. have ten values available for each place. Most notable
is our time system. We have seconds, minutes, and hours. For these three, the base count
changes depending on the unit. First we have 60 seconds for each minute, and 60 minutes for
each hour. Beyond that it changes its base to 24 hours in a day, and then 365 and a fraction
days each year. It gets worse since the hours in a day and days in a year keep changing.
These later two of course were originally based on astronomical data, the earth’s rotation and
period around the sun. A serious debate is now going on as to whether we should establish an
absolute clock/calendar based on an absolute time interval with no regard for the calendar we
currently use that keeps changing. Airline systems already use such a clock/calendar out of
For the general population, we may not be happy about having Christmas in July!!
The other notable counting systems involve our measurement of angles. The base for angles
is 360, i.e. there are 360 degrees in one complete rotation. These are then divided into 60
minutes for each degree and then 60 seconds of arc for each minute. Our designation of
location on the earth are given by angular numbers of degrees, minutes, and seconds of arc
from agreed upon zero lines, the equator for latitude and the line passing through Greenwich,
England for longitude.

Calculations about ANTYHING will require dimensions. Dimensions are the definition of the
units of any science. You can’t describe much of anything without units. In particular
Engineering would not exist without them. Units are definable, measurable entities by which
the magnitude of physical quantities can be determined. The most common and fundamental
units are length, time and mass and charge or current. Other units are contrived from
these and are used as operation units such as temperature, pressure, energy, etc. and are
most often renamed after a famous scientist who did extensive and original work in the

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Class notes EG31 2                                                        M. Brienza
particular discipline. The unit of force in the current international standard for units is named
after Sir Isaac Newton.

Dimensions therefore tell us the magnitude of the unit measure while the numeric tells us how
many of the unit measures there are. For example, saying the length of a rod is 14.73 long is
meaningless unless you indicate the dimensions. 14.73 mm is very different than 14.73 feet!
It is therefore absolutely necessary to use the same units for all parameters that go into the
calculation of a particular result.

Dimensions can be treated as another algebraic quantity in a calculation and when they are
carried along with their numeric “partner”, they will act as a check on your algebraic result as
they indicate the correctness of the units for the answer.

There are cases of dimensionless numbers that are often used but they are ratios of numbers.
In engineering such dimensionless numbers are often used as relative measuring descriptions
of certain phenomena. For example a decibel is a dimensionless number that represents a
comparison of two numbers of the same dimensions. Used as a measure of sound intensity,
the intensity is compared to an intensity of 10-12 watts/m2, the minimum detectable intensity of
the average, undamaged, young human ear. This db ratio is used extensively in engineering,
particularly in electronic engineering where as an example it used to compare two signal

So now that we know how to count, and we recognize the need for dimensions, the most
challenging aspect of dimensions is to define them so that, for example, a meter is physically
the same for everyone using that dimension. This sounds simple enough but it is in reality a
very difficult task for reasons we will discuss shortly.

In science and engineering the number of fundamental dimensions is limited to the half dozen
from which all others can be derived. For example length and time are recognized as
fundamental dimensions while velocity is a derived dimension made up from the definition of
velocity in terms of length and time. This leaves us with the half dozen or so that need to be

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Class notes EG31 2                                                      M. Brienza
defined by some physical entity. Before we go looking for a standard for length as an
example, we need to characterize the needed characteristics of a good unit standard. The
need for standards seems to be obvious, but that obviousness depends on the use of the
standard. For example, if five or six hundred years ago you day’s wages was ten apples and
four potatoes and each day you were satisfied with the quality and quantity of your wages then
everything was OK. But one day you noticed that the basket of apples and potatoes seemed
lighter than usual and you observed that the apples and potatoes seemed smaller. Your
complaints to the lord of the castle were not received well as he told you in no uncertain terms
that you got your agreed ten apples and four potatoes. The issue here of course is that the
changing size of the food changes the wages considerably. Thus you calmly suggested to the
lord that we needed to define an apple and a potato in terms of a standard. Now they routinely
used balances for comparison of the weight of two objects. So the lord of the castle stepped
outside and picked up two stones and said that this one in my right hand shall define the
weight of ten apples and this in my left hand four potatoes. In this manner we can determine
the fairness of the “take-home” wages. That may be OK for you but how about the rest of the
workers, especially those at the next castle? Is there pay measured with the same two rocks
or do they have a different set, probably of different weights. Well without belaboring the
obvious, it should be clear that a good standard should have some fundamental
characteristics. We will suggest a few below.

      1. The physical entity of the standard should be agreed upon by all who are going to
          use it.
      2. The accuracy of the entity should be greater than the accuracy required by all users.
      3. The standard should be reproducible, i.e. secondary standards can be produced
          from the primary in such a way that they can be used for the intended measurement.
      4. The construction and storage of the primary standard should be capable of
          establishing and maintaining the accuracy of the standard.
      5. The standard should be part of a system of measurement standards that together
          are capable of measuring any physical quantity with units that are clearly and
          precisely defined and that possess a logical relationship between units to facilitate

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Class notes EG31 2                                                         M. Brienza
The Most Commonly Used, Fundamental Units of Measurement

Time Like other basic units, the unit for time has some obvious sources. Clearly the day, the
lunar month, the seasons and the year have all been historic units of time. The day was
divided into smaller units, the hour. Eventually, the second was defined as 1/84,600 of the
mean solar day. Time keeping, or secondary standards has a very interesting history that we
could spend a semester talking about. Recently, the second has been defined as a specified
number of oscillations (9,162,631,770) of a cesium transition in an atomic clock.

This is a really beautiful standard since the clock is easily duplicated and all cesium atoms are
same and therefore the oscillations are always the same.

Length Up until a relatively short time ago, the primary standard unit of length was
established as the distance between two scratches on a bar of platinum-iridium kept in Paris.
Before that it might have been the length of the King’s foot (1 ft), or the length of stride or
distance from the King’s nose to his outstretched fingertip (1 yard), or some fraction of the
circumference of the earth (1 meter) or distance from the North pole to the equator. As you
can readily see there are problems with such standards. For a standard to be useful it must be
easily and accurately reproducible into secondary standards that are transportable and
accurate for the purposes at hand. If you want to measure a length, you don’t what to have to
take a trip to Paris every time! Various rulers (e.g. yard-sticks, tape measures, etc.) could fill
these requirements within certain limits. As our need for more and more accuracy grows so
does the difficulty with both the primary and secondary standards.
About a half century ago, a nice reproducible standard for length was specified as a specific
number of wavelengths of the orange light emitted by an excited atom of krypton-86. This
standard stood for a while but our needs soon required yet a better one.
Currently, length is defined by how far light travels in a known length of time. In order to do
this we need a specification for the speed of light and a very good clock. The clock is the
atomic clock we mentioned above.

What has been done more recently is to define c, the speed of light, to be EXACTLY EQUAL
to the best current measurement, c = 299,792,458 m/s. As has been experimentally verified,

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Class notes EG31 2                                                        M. Brienza
the speed of light is constant, regardless of the motion of the frames of reference in which the
measurements are taken. This is one of the basic and proven tenants of Special Relativity.
Currently we are very comfortable that the speed of light does not change regardless of how it
is measured. With that confidence, we now redefine the meter to be the distance light travels
in a vacuum in a specified number of “ticks” of our atomic cesium clock. Notice that the
standards for both length and time are now standardized by nature itself, namely specific

Thus both the time and distance standards are defined by unchanging properties of nature and
not by scratches on a metal bar or the ever changing rotational rate of the earth. In effect
there are no “primary” standards and “secondary” standards the standard can be duplicated
whenever and wherever they are needed. You might think that we physicists are “gilding the
lily” but such systems as our global GPS systems could not function without them! In fact if
special relativity were not accounted for, the GPS systems would not work as the errors of
classical calculations could be off literally “by miles”.

Mass Since we have yet to measure the mass of a specific atom or collection of atoms, we
still use a cylinder of platinum-iridium kept in Paris as the primary standard for mass (1 kg.)
We do define an atomic unit based on an isotope of Carbon but that has yet to be applied to
masses that we mortals can “handle”.

We like most of the scientific world will use the so-called S.I. units of kilograms – second --
meters as our primary units.

Good News – Bad News
The good news is that there are internationally recognized fundamental standard units. The
bad news is that we humans don’t like to give up old habits. A few years before most of you
were born, the U.S. had decided to go metric. We actually started putting up road speed and
distance signs in metric units. That’s about as far as it got. In spite of government trying very
hard to make the case for the metric system the effort failed in the lay community. The
scientific community, including health care professionals, continued to use the metric system.
The public in general loudly objected and refused to accept it so we stayed with our nice

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Class notes EG31 2                                                        M. Brienza
decimal system of money but continued with our well worn system of inches, feet, miles,
quarts, gallons, yards, acres, and pounds. The non-decimal ratios between these units are a
computational nightmare to the man on the street. Calculating the exchange of all the liquid
volume standards is enough to make a chef cry. Would you know how many teaspoons there
are in a pint let alone a liter?
A sizeable problem still exists in both the European metric system and the English/American
system. The problem is in the interchange of mass and force. In the our system we define the
pound as a force based on the gravitational attraction while the European metric system uses
the kilogram as a unit of force. In that way they specify a weight by comparing masses. So we
tend to talk about pounds as an unchanging quantity (mass) and the European’s speak of
grams and kilograms as a weight (force). So both systems have their problems mostly fueled
by the intransitive nature of the public. At least the metric system is completely decimal and
therefore far more covenant for calculation.

This problem exists between the sciences and engineering as well. The sciences have
essentially gone completely metric, including standardizing and using a unit for force (the
Newton) and a unit for mass (the Kilogram). The engineers, probably because of the
necessarily close relationship with the public, have and are still using essentially the
English/American system. Confusion of units as well as mistakes in converting between metric
and English units often lead to unnecessary, and often times expensive, mistakes.
Unfortunately, it makes your task as an engineer all the more difficult as we rapidly move into a
global society.

Conversion of Units
With different units being used throughout the world, with different relationships between units
of the same gender (units of liquid volume, for example) it is important to take extreme care in
converting between measuring systems. The differences between English and metric derived
units can be especially confusing. These include common units like pressure, speed, energy
that are commonly used in each system. It will be important to you to understand and perhaps
remember the ballpark conversions of derived quantities that are often used in your
engineering specialty. For example, mechanical engineers, who deal with heat management,

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Class notes EG31 2                                                       M. Brienza
pressure, and stress and strain will have to have a command of the conversions so that
estimates can be understood in engineering communications, oral or written.
As we have already stressed, units must be used with physical numbers. A number is useless
without units. Often times it is necessary to convert from one unit to another and you should
use units with each algebraic element or number of a calculation and then deal with the unit
conversion of numbers with their units as algebraic quantities.

E.g. To convert 5 miles/hr to ?? meters/sec. we might have

5 m/h= (5mi/hr)(1.6km/mi)(103m/km)(1h/3600s) = 2.2 m/s.

Note that all but the desired units cancel leaving the final, desired units as the known correct
m/s. While this may seem to be a trivial exercise, you wouldn’t believe how many errors are
made by not going through this simple exercise to assure the conversion is correct. Bad unit
conversion is almost always a result of algebraic mistakes. Remember do the algebra first
then the last step of the problem that is the arithmetic exercise of plugging the numbers into
your calculator. As we have and will continue to say, resist the temptation to pick up
your calculator earlier in the problem. If you do, you will loose the ability to see the
algebraic units of your answer and therefore miss this convenient check on your algebra.
Using dimensions in this way throughout a problem will keep you from making silly errors.
Unfortunately, calculators use only numbers so you are more likely to make such an error if
you use a calculator “too early” in a problem. Work your way algebraically with units through to
the end of a problem and then use the calculator for the final arithmetic calculation will help
assure you that your algebra is correct. If you get some unreasonable combination of
dimensions for your answer, you will know you have a problem. Dimensional analysis will
ensure the dimensional integrity of your problem.
Naturally, the numerical answer should always be rounded to the correct number of significant
figures. (More on that later)

Order of Magnitude
Besides doing simple dimensional analysis to check an answer, you should also do an order-
of-magnitude approximation for the answer. You should practice reducing numbers to a single

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Class notes EG31 2                                                          M. Brienza
significant figure and a power of ten. For example, a given number like 9.8 m/s 2 should be
approximated at 1x101 m/s. With numbers in this form, you can quickly estimate the
magnitude of the answer and assess whether it is reasonable. If it is not, then look for an
algebra mistake, usually involving the handling of exponents of 10 in a calculation. In this first
semester we will be dealing with physical areas where you have a great deal of life experience.
Since both the text and we instructors usually try to use reasonable parameters for problems,
you should always try to apply an experienced based, reasonability test to the answer. In
short, if the answer doesn’t make sense then more than likely the answer is probably wrong.
One such experience I always like to relate is when a student calculated the tension in an
eighth inch cotton string she was actually using in an experiment at about 2.5 MILLION

Significant Figures
“A significant figure in a number is defined as a figure that may be considered reliable as a
result of measurements or calculations.” In the scientific or engineering literature you may
often see a after the last figure of a +/- sign with another number following. This is a further
specification as to the value of the precision of the last significant figure in the number. It is a
specification on the statistical probability that the true value is probably between the two values
specified. I know the above statement sounds like a cop-out on the answer but we will see a
bit later the significance of this estimate of validity. It essentially defines the error bars on the
data point. If this were plotted as part of a graphical presentation, those error bars would be

Since most measured values are only known to a small number of significant figures, usually
limited by our ability to precisely measure them, you should not express an answer to more
significant figures than are warranted by the available data. If you are adding or subtracting,
the answer should only show a figure in the place of the last figure of the least significant
figure. When you are adding or subtracting figures, the decimal points are necessarily lined
up. Thus the least significant figure is the one to the right. For example if a force is given as
3.2 Newtons, and it is being added to another force, say 1725.463, then the sum would be
1728.663 Newtons. Examining the two values being summed, we note the least known
number, i.e. the one with the fewest figures to the right, is the 3.2 N force. The final answer,

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Class notes EG31 2                                                         M. Brienza
1728.663N, should then be rounded to 7 and the final answer would be 1728.7 N. In other
words, the answer should only show its least significant figure as far to the right as shown in
the least precise number in the calculation.

If you are multiplying or dividing two numbers that the answer should contain the same number
of significant figures as the number with the least number of significant digits. The that digit
should be rounded also. When rounding, look at the digit in the answer and if equal to or
greater than 5 round up, otherwise round down.

When we multiply the two numbers above rather than adding them, the answer would be
5520., after rounding the result in the least significant location. Read the introductory section
in the PS 15 lab manual on this subject. Get in the habit of showing answers to the correct
level of significant figures even in homework problems.

In the lab, you will have to judge the number of significant figures that a particular
measurement is capable of producing. Just because you see a digital readout, do not assume
the measurement is that precise. There may be other things limiting a reading. For example,
if you are releasing a car on a frictionless incline, the timing of the position gates at the top and
bottom may produce very precise measurements of the time down the track. But the accuracy
of that reading may be significantly altered by your release technique (you might give it a little
push without realizing it, or a systematic error might be introduced by your limitation to
establish the angle of the track itself. The lab, a gradable exercise, must capture and discuss
errors and their significance.

Scientific Notation
More often than not, engineering calculations can contain very large or very small numbers.
Image that we are multiplying the following two numbers:

2,340,000,000. x 0.000,000,000,041 and find the correct answer to be 0.096.

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Class notes EG31 2                                                        M. Brienza
This type of presentation makes it very difficult to use and maintain the proper number of
zero’s. Missing a count of only one zero results in an order of magnitude error, i.e. you answer
would be off by a factor of ten.

These numbers should be written in scientific notation as
       (2.34 x.109) x (4.1 x 10-11) = 9.6 x 10-2
In doing so, there are no blurry vision problems keeping track of all the zeros, and the correct
level of significant figures is easily established.

Bottom line, I really don’t want to see strings of zero on homework or test problems. By using
this scientific notation, multiplication and division is very easy and you can quickly estimate the
answer without even using the calculator. For example it is now obvious that the answer is
approximately 2.5x4 or about 10. The exponent is 9 – 11 = -2, even I can do that in my head.
When you multiply two numbers as we have here, you add the exponents and when you
divide, you subtract the exponents of the denominator from the exponents in the numerator,
using the signs of the exponent as presented. This adding and subtracting of the exponents of
the powers of 10 does not require the two numbers are the same animal. In our example
above the first number might have the units of Newtons, the second meters, and the answer
joules if in fact this happens to be a force acting over some distance.

Mathematics that we assume you are OK with, but if you are not then let’s review it now.
The areas we expect some reasonable amount of fluency with are:
Algebra – Geometry – Trigonometry – and Graphing
       If you are not comfortable in these areas then get some help quickly. Use the tutoring
       service, the math center, or I can help you.

We expect that you have taken or are now taking a calculus course and are becoming
knowledgeable of differential and simple integral calculus.

Finally, we will introduce you to some statistics as they apply to engineering data.

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Class notes EG31 2                                                                M. Brienza
Engineering Statistics
While engineers use statistics for many applications, the major applications include:
       1. Understanding, control, and accounting for errors in measurement.
       2. The facilitation of the collection of adequate and reliable data planning and control of
           engineering projects.
       3. To improve the understanding and accounting for uncertainties in the demands
           placed on engineering structures and products.
       4. The analysis and control of the quality of workmanship and materials in the
           manufacturing processes.
It is not our intent to teach a course in statistics, although you will more than likely be required
to take one later in your academic training. We hope to use some fundamentals of statistics to
introduce you to the variability of data, how it is measured, how it is used in an engineering
context, and perhaps demonstrating the power of statistical tools.
Besides the referenced source above, much of this material can be found in text #1, Fundamentals of
Engineering, Chapter 13.

There is that very old saying that figures don’t lie but liars can figure. The weapon of choice is
in that discussion is usually the use or misuse of statistics. Let’s start by considering a
relatively simple measurement. You are the quality control person at the local marble factory.
Your product, marbles, are used in pinball machines and must be made so that they pass
through the holes in the table. Since these pinball machines are in constant use at the local
arcade, if an oversized marble gets into a machine, the ensuing downtime to fix it carries a
serious financial penalty. Needless to say, your boss, an executive in the marble factory will
get a nasty call from his customer. You, as head of quality control will quickly get a call from
your boss, who always has you on his speed dialer. Your job, as the quality control engineer,
is to find out if there really is a size problem and if so determine where in the production
process the problem lies.

So you head for the end of the production line and randomly select 100 marbles. The term
randomly is a key term. In this case it means that every time a marble is to be selected, every
marble has an equal probability of being selected from the very large bin full of marbles. You
have brought you calibrated calipers that have been certified by your standards department to
have an accuracy and precision (we will define these terms later) much greater than any

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Class notes EG31 2                                                           M. Brienza
reasonable expectation of variation in the marble diameter. The point here is that your
measuring tools must be capable of measuring the target measurement without substantially
introducing its own measuring variability as a significant factor in the data.

The raw data is entered into a spreadsheet, where it is analyzed. The data is first ordered by
magnitude and grouped into classes. The classes are numerical bins, e.g. > 17.0 mm to <
18.00 mm. A The number of bins is selected, usually on the order of the square root of the
number of measurements. For our example the measured sample population is 100, so the
number of bins would be 10. We then assign the number of marbles in each bin as the value
of that bin. The table of this data in this ordered fashion is a frequency distribution of the data.
It gives some information but it is more useful to plot the frequency distribution as a chart.
Each bin of the frequency distribution is ordered along the horizontal axis and the magnitude of
each bin, i.e. number of data points with each bin, is plotted as the height of each bin column.
The center of each bin is often indicated along the horizontal axis. In this way we plot the
magnitude of the bin as its particular population against center value for each bin. The third
bin might be from 17.30mm to 17.40 mm and we would plot its value vertically. (There is an
assumption that no data points fall on into two bins by having a diameter exactly equal to the
boundary value of neighboring bins.) This chart is a frequency distribution chart or
histogram. It illustrates the distribution of diameters by size.

The important utility of the frequency distribution chart is the analysis of the shape of the
distribution. In many cases, the distribution will be symmetric and well behaved and can be
represented by an exponential function whose various parameters can be adjusted to fit the
actual data, as represented in the histogram. The parameters that represent the best fit to the
data will then yield values of the various properties of the data distribution.

The so called normal distribution for a random variable, x. is defined by the following equation:

                                                       ( x )   2
                                       f ( x)       e                2 2

                                                 2

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Class notes EG31 2                                                            M. Brienza
Where x is the independent variable, is the center of the distribution, called the mean, and 
is a measure of the width of the distribution, called the standard deviation. The normal
distribution is also written as
                                                       z2
                                     f ( z)     e     2

Where the mean is set a zero, and the normalized variable z is given by

                                           x   
The integral of the normalized distribution curve from – infinity to z gives the probability that a
value of x is will be less than the desired z value. Since this type of calculation is needed so
often, tables of the integral values as a function of z values are commonly published.
Obviously preprogrammed calculators and computers would make these tables obsolete
today. This calculation, for example, could tell use the probability that our pin ball marbles will
be too large for the holes in the table.

The standard deviation when we are dealing with the entire population N (or a large sample) is
given by
                                                   xi   2          2

                                                i
                                                               
 is the average value of the entire population. If we were using only a small sample of the
entire population, then we would use the notation

                                                            
                                                  i
                                                              2           2
                                                     x x
                                                i
                                                  N 1           
                                                                 
Where x is the average value of the sample size.

The standard deviation is therefore a measure of how tightly the data points are grouped
around the mean value.

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Class notes EG31 2                                                           M. Brienza
Analysis of the best fit of the normal distribution to the data, i.e. the frequency distribution, will
indicate important information about the data. To facilitate calculations, the normal distribution
is often centered so the mean value, representing the position of the maximum value of the
function, is at zero. The inflection points on either side of zero, of this bell-shaped curve occur
at the + or – values of  the standard deviation. As the standard deviation becomes smaller,
the more likely the random data points are closer to the center, the mean value. By integrating
the Normal distribution function from -  to +  , that is to say finding the area under the normal
distribution curve between – and + one standard deviation, the result will be that the probability
of the data points being between them is approximately 68%.

Therefore in our example of the diameter of the pin ball marbles, assuming a completely
random manufacturing process, we will find 68% with a diameter between the diameter values
represented by the +/_ one sigma points. A good manufacturing process for these marbles
would be indicated by a small value for sigma so that the majority of the marbles would have a
diameter within a useable range of values.

If the actual frequency distribution function, (or histogram) is not symmetric, or double peaked,
or skewed to the right or left of the mean, the normal distribution function is really not of much
value. However the best fit curve to the actual frequency distribution can still be used as an
analysis tool, to analyze the data but the statistical tools will be more difficult to use and

There are many parameters that are commonly used in analyzing frequency distributions that
give more detailed information about the data. Many of these are tabulated and published or
are available as standard software analysis tools.

A surprising amount of qualitative information can be “read” from the shape of distribution
curves. Clustering of data around two or more data values could indicate two different
processes affecting the data simultaneously. For example if the data points were classroom
grades on a test, clustering of grades around two different values might indicate that the class
had two distinctly different groups of students. That might be due to the fact that one group
had more relevant background training than another group in the same class. This often

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Class notes EG31 2                                                        M. Brienza
happens in a science course where you might have freshman and juniors taking the same
course. The juniors, having had more training in math, as an example, as group could be
expected to do better in a course that requires a significant fluency in math.
Quality control engineers will use both rigorous statistical analysis and qualitative analysis to
solve quality issues in a manufacturing process. An experienced engineer, i.e. knowledgeable
about the various processes in his particular plant, might be able to easily spot the process
that is the major contributor to a manufacturing problem by just examining frequency
distribution graphs of data. For engineers, experience in their field is a very valuable asset and
such experienced engineers can expect to be compensated accordingly. When we discuss
the process by which products are designed and manufactured by engineering teams, we will
see the value of experienced people. (This is not to imply that only experienced engineers
should be on design teams. Much more on this later)

Let’s go back and look at what are called “Measures of Central Tendency”. These, like the
standard deviation that measures the spreading of the distribution curve, measure
characteristics associated with various averages of the data values. Unlike the standard
deviation, we must analyze the actual data, since we will be looking for measures that depend
on the actual data values. The thee measures well will discuss are the mean, the median,
and the mode.

The mean is nothing more than the average value of the data points. Numerically it can be
expressed as

                                             x     i
                                    mean    i 1

If N is the entire population the mean is commonly denoted by  and if N represents a smaller
sample size, the mean is denoted by x . The mean is therefore an average value of a
collection of data points.

The median is the location of the center value of the data points, i.e. if the data points are
arranged in order of value then the median is the center value. More simply, the median is that
value that has as many data points above it as below.

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Class notes EG31 2                                                        M. Brienza

The mode is the one or more data values that occurs with the greatest frequency. For
example if a particular data set were 3,4,2,2,2,8,7,9,9,1,5,6; then the mode would be 2. The
mode, depending on its size can be an indicator of either non-random selection of samples
from the population or some systematic error in the process.

Needless to say, statistics in general represent a set of powerful tools for data analysis. It
doesn’t much matter what the field of study, it is hard to image statistics not being used in
some manner. Like any mathematical tool, it is only as good as the “hand” that is guiding the
tool. Unfortunately, statistics is often misused or not understood and that can do more
damage than good. As engineers, no matter the subset, you must acquire a firm grounding in
statistic. Statistical data analysis tools are usually a part of any computer data analysis or
design tools. We will be introducing you to EXCEL, a spreadsheet program with many built-in
tools, including the common statistical analysis tools.

Graphical Analysis
As we have alluded in the previous section, significant information can be obtained form
graphical data; that is the dependence of data information as a function of some independent
variable. In many cases, the functional dependence of one set of data on another, can be
approximated by a known function. Depending on how well you can do that, various
parameters about the data can be ascertained.

For example, suppose we wish to determine the tensile strength of a steel rod. We put in
testing device that grasps the rod at each end and literally stretches it. The instrument records
the amount of stretch as a function of the tension in the rod. It is not unusual for the rod to be
stretched to the point of rupture. When the data is plotted as force as a function of the stretch,
we observe a direct linear proportion of the stress, the force divided by the cross-section of the
rod, to the strain, the deformation (the strain). As the strain (the percent change in the length)
increases, so does the required stress (the tension per unit cross-section area). This linear
behavior continues to the end of the “elastic” region (the region of strain where if the strain
were removed, the sample would return to its original, unstressed length). However, if the

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Class notes EG31 2                                                          M. Brienza
strain is increased beyond this so-called elastic limit, the rod will continue to stretch but now
with permanent deformation of the sample. If released from a point in this none-elastic level of
strain it will NOT return to its original length. It might recover to some degree but it will be
permanently deformed. As the strain increases through this region, the stress-strain plot
becomes irregular as it moves through this “plastic deformation” region until it finally breaks.
The linear region for most materials is limited to about 1 – 2%.
We would now like to graphically analyze the elastic region where there is a direct proportion
between the stress and strain. Observing its more or less linear relationship from the plotted
data points, we assume the actual result is linear, that is the stress = (a constant) x strain. The
actual data points are analyzed in what is known as a least squares fit to produce the best
straight line fit through the actual measured data points. The analysis minimizes the total sum
of the square of the distance of the actual data points from the “best – fit” line. This graphical
curve fitting can be done manually, on a spreadsheet, or by a software tool specifically
designed to rapidly execute the analysis.

Now that you have the best fit line you can now determine the slope of the line. That slope is
of course the constant in the stress-strain relationship. Note that the linear analysis is only
valid in the elastic region. The constant is known as Young’s modulus and is noted by “Y”.
The values of Y have been tabulated for essentially all construction materials. You might also
be familiar with the proportionality constant that relates the actual tension force to the elastic
deformation and is known as the constant in Hooke’s Law, namely F = Kx.

It should be noted at this point that if the data is reasonably good, i.e. our visual observation of
the data plot indicates that the actual data points do in fact are on a fairly “good” straight line,
the best fit straight line can often be “eye-balled” and drawn in with a straight edge on the data
graph. In doing so, it is relatively easy to establish a pretty good approximation of both the
slope and y intercept of data function. With the general equation for a straight line as
                                     y( x)  mx  b
Where m is the slope and b the y intercept.
The process we have just described is linear, least square curve fitting. If, when we plot the
data points, it is obvious that the data is not linear, then other functions are used instead of the

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Class notes EG31 2                                                       M. Brienza
straight-line, linear process described. If we have some expectation that the curve should be
exponential, then we might use an exponential function of the form
                                           f ( x)  Ae Bx
Where A and B are adjustable constants to achieve the least squares fit. If indeed the process
being studied is exponential the analysis can be quickly linearized and make suseptable to a
reasonable eye-ball, linear fit. This can be done by graphing the data on logarithmic graph
paper. If the data is of the form for f(x) above, then the data is plotted on semi-logarithmic
graph paper where only the dependent values are on a logarithmic scale, the y axis, while the
independent variable is on a linear scale, the x-axis.

If both the dependent and independent variables cover more than one or two orders of
magnitude, then so-called log-log paper is used.

Engineers for many years have designed graph paper specific to certain data analysis
processes. These specialized graphing sheets facilitated the analysis of data that had to be
done “manually” back before the tasks were computerized. It might be said that each
engineering specialization created its own graphing paper to facilitate analysis of common
problems in their particular area. Today of course, computers have eliminated the need for
most of these specialized graphical analyses. Never-the-less it is still important for you to
understand the utility of graphs and the analysis that can be quickly visualized and therefore
understood by these techniques. Naturally, graphical representations of functions and
processes are also powerful tools for teaching, communicating, understanding and problem
solving. We have already seen how graphical presentations and analysis are important tools
in statistics.

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