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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 diagrams. 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 Page 1 of 20 Printed 9/13/2012 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 later). 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 Page 2 of 20 Printed 9/13/2012 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 necessity. 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. Dimensions 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 Page 3 of 20 Printed 9/13/2012 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 strengths. Units 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 Page 4 of 20 Printed 9/13/2012 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 calculations. Page 5 of 20 Printed 9/13/2012 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, Page 6 of 20 Printed 9/13/2012 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 atoms. 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 Page 7 of 20 Printed 9/13/2012 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, Page 8 of 20 Printed 9/13/2012 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 Page 9 of 20 Printed 9/13/2012 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 POUNDS!! 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 shown. 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, Page 10 of 20 Printed 9/13/2012 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. Page 11 of 20 Printed 9/13/2012 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. Page 12 of 20 Printed 9/13/2012 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 Page 13 of 20 Printed 9/13/2012 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 1 f ( x) e 2 2 2 Page 14 of 20 Printed 9/13/2012 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 1 f ( z) e 2 2 Where the mean is set a zero, and the normalized variable z is given by z 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 1 xi 2 2 i N 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 1 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. Page 15 of 20 Printed 9/13/2012 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 interpret. 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 Page 16 of 20 Printed 9/13/2012 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 N x i mean i 1 N 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. Page 17 of 20 Printed 9/13/2012 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 Page 18 of 20 Printed 9/13/2012 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 Page 19 of 20 Printed 9/13/2012 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. Page 20 of 20 Printed 9/13/2012

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