Uncertainty Analysis of Experimental Data and Dimensional Measurements Introduction The primary objective of this experiment is to introduce analysis of measurement uncertainty and experimental error. The data that will be analyzed are dimensional measurements done using Vernier and micrometer calipers. The accuracy and precision of these tools are determined by comparisons with precision gage blocks. The dimensions of two groups of two machine bolts are then measured and described by statistical methods. The measured bolt diameters are compared to ASTM standard specifications and tolerances. The application to dimensional measurements of machine parts is relevant to component design, manufacturing, and statistical quality control. Vernier Calipers, Micrometer Calipers, and Gage Blocks Vernier calipers (calipers) and micrometer calipers (mics) are commonly used tools for making precise measurements of dimensions of machine parts [1]. Calipers are used for measuring dimensions of up to 6 or more depending on the calipers span. They commonly have one of three types of read-out devices: a Vernier scale, a circular dial indicator, or an electronic LCD display. The accuracy of calipers is approximately ± 0.005 in (± 0.1 mm). Micrometers are normally used for measurement of smaller dimensions with greater precision than calipers. The accuracy of micrometers is approximately ± 0.0005 in (± 10 µm). Measurements of such high precision (a human hair is approximately 0.002 in) are sensitive to applied pressure, temperature, and surface cleanliness. These factors must be controlled for precision measurements. Micrometers have a ratchet tightening screw to provide a standard pressure between the tool and the part. Due to thermal expansion, a one-inch long gage will change length by approximately 6 µin for each F of temperature change. Hence, micrometers are calibrated at a standard reference temperature of 68 F. Measurements made at other temperatures are still accurate provided that the test part is also steel (same thermal expansion coefficient) and both tool and part are at the same uniform temperature. Gage blocks represent precise industrial standards for the inch. They are small steel blocks with two polished parallel faces. There are three classifications of gage blocks: AA, accurate to ± 2 µin for size, flatness, and parallelism; A, accurate to ± 4 µin; and B, ± 8 µin. Gage blocks are normally used in sets having a range of thickness that make it possible to stack them such that any dimension between 0.100 and 8.000 in can be obtained in increments of 0.001 in. The blocks are stacked through a process called wringing, whereby clean surfaces are brought together in a sliding fashion while steady 1
pressure is exerted. The surfaces are sufficiently flat that when the wringing process is correctly executed, they will adhere together. Gage blocks are used most frequently in two ways: for calibration of adjustable gaging instruments such as dial indicators or calipers, mechanical and electronic comparators, and as standards for production and checking of wear on inspection instruments. Measurement of Bolt Dimensions Machine bolts follow standards for specification of dimensions, materials, and tolerances, as described in the Standard Handbook of Machine Design [2], among others. Standard screw threads consist of the Unified inch series and the Metric series. Unified inch series, dominant in the US, are based on the major (outermost) diameter and the number of threads per inch. The three standard series are designated UNC (coarse), UNF (fine), and UNEF (extra fine). Typical specifications are 1/4-20 UNC or 1/2-20 UNF, where 20 threads per inch on a 1/4 inch bolt is coarse and 20 threads per inch in a 1/2 inch bolt is fine. The Unified coarse-thread series provide the most resistance to internal thread stripping, fast assembly, and tolerance to nicks or dents. The Unified fine-thread series are most useful where a high strength fastener is required and where vibration may be a problem. The thread profile for Unified threads is shown in Fig. 1. Deviations of thread dimension from specifications can affect interchangeability and reliability of bolts. In high performance applications, such as for aircraft structures, tight specifications must be placed on critical fasteners. The two most important dimensions of screw threads are the major (outermost) diameter and the pitch (center) diameter. The most accurate and universal method of obtaining measurement of pitch diameter, taper, and out-of-roundness of external threads is the three-wire method [3]. In checking screw threads by the three-wire method, three wires of the same diameter for the particular pitch being checked are placed in contact with the threaded part. Two wires are placed with the thread on one side, and a third wire on the opposite side as illustrated in Figure 2. When the micrometer anvils are in contact with all three wires, the measurement obtained has a definite relation to the pitch diameter. Any wire diameter may be used as long as the wires are small enough to enter the thread and are large enough to project above the top crest of the thread. The best size wire is one that touches the mid-slope of a perfect thread. The following formula defines the pitch diameter of an American Standard screw E = M + 3 p 3W. (1) 2 2
E is the pitch diameter, M is the measurement over the wires, p is the pitch of the threads (inches per thread), and W is the wire diameter. Equation 1 may be derived using the geometry of screw threads shown in Fig. 1. Experimental Errors, Measurement Uncertainty, and Statistical Analysis All measurements are affected by errors; hence all measurements contain some uncertainty. In order to assure the integrity of a measurement, it is very important to identify all sources of experimental errors and to quantify uncertainty. There are many different types of errors, such as personal, instrumental, or accidental errors. 3
Personal errors include blunders, such as mistakes in arithmetic, or in recording an observation. Another important kind of personal error is known as personal bias, such as trying to fit the measurements to some preconceived idea. Instrumental errors are due to limitations of instruments or to imperfections in their manufacture and calibration. Accidental errors are deviations that are beyond the control of the observer, such as errors due to jarring, noise, or fluctuations in temperature. Errors are either systematic or random. Systematic errors are characterized by their tendency to be in one direction only, either positive or negative. For example, if a ruler is worn at one end, it will introduce a constant error. Improper instrument calibration causes systematic errors. Systematic errors are usually more important than random errors and they are generally much more difficult to deal with. Although systematic errors cannot be eliminated entirely, they can be minimized by careful experimental design and by assuring that measurements can be traced to known standards. Random errors, on the other hand, are produced by unaccounted-for variations in experimental conditions or from small errors in making observations, such as estimating units of the smallest scale division. They tend to be scattered over a finite range of values. If assumptions are made about the distribution of random errors it is possible to use statistical methods to deal with them. The terms accuracy and precision are often used to distinguish between systematic and random errors. If a measurement has small systematic errors, it has high accuracy; if the measurement random errors are small, it has high precision. A primary task is to determine just how uncertain a measurement is. Experimental errors are always unknown; uncertainty refers to a quantitative estimate of the amount of variation in a measurement. More precisely, uncertainty refers to the size of the range of values that is very likely to contain the true value. For example, from repeated dimensional measurements that exhibit scatter, we might specify that the actual dimension is X = x ±u x. X is the true value, x is the best point estimate, and u x is the estimated uncertainty. One way of determining x and u x is to say that u x is the difference between the highest and the lowest values of a set of measurements and x is halfway in between. Although this method is better than a single measurement, it is not very satisfactory for several reasons: (1) only two measured values are used, the rest are discarded, (3) x and u x are overly sensitive to these two particular measurements, (3) u x will tend to increase if more observations are made, (4) and the estimated uncertainty is larger than necessary. A better technique for determining x and u x will be given later. Propagation of errors refers to the fact that if one employs various experimental observations to calculate a result, and if the observations have uncertainties associated with them, then the result will have an uncertainty which depends on the uncertainties of the individual observations. Let R be the result of a calculation with n variables 4
R(x 1,x 2,...,x n ). Then the uncertainty u R in R is related to the uncertainties in the variables u 1,u 2,...,u n by [4] 2 R u R = u 1 + R u 2 +K+ R u n. (2) x 1 x 2 x n Statistical methods are very important in any analysis of experimental data. Descriptive Statistics is used to summarize a group of data. The complete data group of interest, such as all widgets manufactured on an assembly line or all possible measurements of a particular dimension, is called the population. The central tendency of a set of N elements X i is given by the population arithmetic mean [5] X = 1 N 2 N X i i=1 2. (3) The amount of scatter in the population is indicated by the population standard deviation σ= 1 N N ( X i X ) 2 (4) i=1 and σ 2 is called the population variance. Generally the entire population cannot (and should not) be measured. Instead, we draw conclusions about a population based on a subset of the population called a sample. A sample consisting of n<n observations x i taken at random from a population, has a sample arithmetic mean given by n x = 1 x n i. (5) The best (unbiased) estimate for the population standard deviation is given by the sample standard deviation, s = i=1 n 1 (x n i x ) 2. (6) 1 i=1 Using statistics it is now possible to give a better estimate of the uncertainty in experimental measurements. Assuming that the experimental errors follow a normal distribution, if a group of observations has mean x and standard deviation s, then the interval x -2s < x < x +2s includes 95% of all the observations. For very large sample sizes, the uncertainty in a measurement can be expressed in terms of an interval estimate using the sample standard deviation x = x ± 2s. (7) In other words, there is a 95% probability that any new observation will fall within this interval. Clearly, how well a sample represents a population depends on the sample size. The mean or standard deviation of a sample of size n=1 is likely to be a poor estimate of the population mean or standard deviation. Likewise, for small sample sizes, the uncertainty 5
expressed as ±2s may be a poor estimate. We can estimate the uncertainty for a population based on measurements from small sample (n<30) with the aid of the Student s t distribution. Thus there is a 95% probability that the next measurement will fall in the range x -t υ,95 s < x < x + t υ,95 s where ν = n-1, the number of degrees of freedom of the system. A 95% confidence interval (interval estimate) for the population mean (true value) may be written as s X = x ± t n. (8) t is the critical value for the student-t distribution (t=2.78 for n=5, t=2.26 for n=10, t = 2.15 for n = 15 ). This equation states that there is a 95% probability that the population mean falls within this interval about the sample mean. If there are no systematic errors present, then this interval represents the uncertainty of the measurement by accounting for the random errors. NOTE that the standard deviation is divided by the square root of n, reflecting the fact that a larger sample size reduces the uncertainty in the estimate of the population mean due to random measurement errors. Systematic errors, of course, will still exist and need to be minimized. A Note on Significant Figures In carrying out computations and presenting results, only significant figures should be retained. Significant figures are those known to be reasonably trustworthy. It is very poor practice to report results with the eight significant decimal figures shown on your calculator if your original data only contained two significant decimal figures. In this case, an interval estimate of a certain value should be specified as d = 5.23 ± 0.04, not as d = 5.2287 ± 0.0376 ; the additional figures beyond the second decimal place have no significance. The uncertainty has only one significant figure so the value of d should be rounded off to this place. References 1. Holman, Experimental Methods for Engineers, 5th Ed., McGraw-Hill, New York, 1989, p. 181. 2. Shigley and C.R. Mischke, Standard Handbook of Machine Design, Eds., McGraw- Hill, New York, 1990, p. 21.3. 3. Grohe, Precision Measurement and Gaging Techniques, Chemical Publishing Co., New York, 1960, p. 17. 4. Holman, Experimental Methods for Engineers, 5th Ed., McGraw-Hill, New York, 1989, p. 40. 5. Ibid., p. 49. 6. R.E. Green, Machinery s Handbook, 24th Ed., Industrial, New York, 1992, p. 1546. 6
Equipment Micrometer caliper Vernier caliper Wooden block with nut to hold the screws 9-piece gage block set (will need to share with other groups) 3 steel gage wires, approximately 0.032 inch diameter Ten 5/16-18 UNC 1 alloy steel bolts, ASTM A354 (6 marks on head, color does not matter) Ten 5/16-18 UNC 1 medium carbon steel bolts, ASTM A449 (3 marks on head, color does not matter) Procedure PLEASE READ THESE INSTRUCTIONS CAREFULLY AND SLOWLY SO THAT YOU DO NOT MISS ANYTHING. REMEMBER TO RECORD ALL MEASUREMENTS IN YOUR LAB BOOK. ALSO, READ SHORT-ANSWER AND DISCUSSION QUESTIONS BEFORE YOU START YOUR EXPERIMENT, SO THAT YOU WILL BETTER UNDERSTAND THE DATA YOU NEED TO COLLECT. NOTE: The micrometers and gage blocks are precision instruments. Follow instructions for their use and treat them with care. You will be asked to perform calculations on your measurements, so you may want to enter them directly into a spreadsheet. Be sure to document the contents of each file in your laboratory notebook. One possibility is to attach a print out of the data into your notebook. Make sure you have a personal copy of all files when you leave the laboratory. A. Gage Blocks, Micrometers, Vernier Calipers, and Measurement Uncertainty 1. Using the Vernier caliper, measure the gage block dimensions of 0.500, 1.000, 1.500, 2.000, 3.000, and 4.000 by stacking the blocks using the wringing method. Make five repeated measurements at each step. 2. Using the micrometer caliper, measure the gage block dimensions 0.500, 0.5625, 0.6625, 0.7875, and 0.9875 by stacking the blocks using the wringing method. Make five repeated measurements at each step. 3. Wipe the gage blocks with a paper towel when you are finished to clean them. 7
B. Measurement of Major Diameter and Pitch Diameter of Bolts 1. The best size wire diameter for the three-wire method on bolts with 18 threads per inch is 0.03207 in. Using the micrometer, measure the diameters of the three gage wires 10 times at various locations along the wires. 2. Measure the major diameter of one bolt from each sample group 10 times at various locations and rotations of the bolts. 3. Measure the dimension across three wires (three-wire method) on the same two bolts 10 times at various locations and rotations of the bolts. 4. Measure the major diameter of each bolt in both sample groups once. 5. Measure the dimension across three wires (three-wire method) on each bolt in both sample groups once. 8
Assignment Questions Uncertainty Analysis of Experimental Data and Dimensional Measurements Turn in one set of answers to the following questions per group. Please type up your written responses NOTE that you do not have to type up your equations or calculations, but make sure your writing is legible and the steps that you took can be followed easily. 1. Estimate the uncertainties in your measurement methods: measuring dimensions with the Vernier and the micrometer caliper. (1 point) 2. Are there any measurable systematic errors present in your measurement methods? Are there any measurable random errors present? How do the two measurement methods compare? Which instrument would you use in future measurements? (3 points) 3. Locate a website that will give you the critical value, t, for the student-t distribution for different values of n for 95% confidence. Use this website to determine the t value for n = 14 and n = 30. Print out a copy of the web site along with URL and hand in. (1 point) 4. What is the 95% confidence interval estimate for the range of possible measurements you could make when measuring one bolt? (1 point) 5. Evaluate the means and standard deviations for your measurements of (a) wire diameter, (b) bolt major diameter, and (c) diameter by the three-wire method, for the two sample groups of bolts. (3 points) 6. Give intervals indicating the range of that 95% of measurements are estimated to fall within for each of the two groups of bolts, for (a) wire diameter, (b) bolt major diameter, and (c) diameter by the three-wire method. Are there any systematic differences between the two groups? (3 points) 7. Evaluate the mean pitch diameter E and estimate the uncertainty in the range of values for E for the two sample groups. Use the method for propagation of uncertainty and your estimates for the uncertainties in the measured quantities. How does this uncertainty compare to the uncertainty in major diameter? (3 points) 8. Give 95% confidence interval estimates for the means of the two sample groups, for both the means of the major diameters and the mean pitch diameters of all the bolts in the two sample groups. (3 points) 9
Discussion Questions Uncertainty Analysis of Experimental Data and Dimensional Measurements Review these questions in your lab group during the first two hours of the discussion lab period. Try to read at least every question and get an idea of what it is asking and what your approach would be to answer 1. Discuss the difference between (a) the range that contains 95% of all the observations in major diameter (or 95% confidence interval for the observations) and (b) the 95% confidence level uncertainty in the mean major diameter. 2. Why is the student t factor used in calculating uncertainty? 3. Are the dimensions of the two groups of bolts the same? What is your definition of the same? Use a number line sketch to illustrate. 4. Do the average dimensions of the two groups of bolts fall within the standard tolerances as specified below? 5/16-18 UNC Class 3A Major diameter: maximum = 0.3125 in, minimum = 0.3038 in Pitch diameter: maximum = 0.2764 in, minimum = 0.2734 in 5. What percentage of bolts would fall outside the standard tolerances? Hint: use the TDIST function in Excel. 6. How would an engineer decide whether to reject a shipment of bolts based on dimensional measurements? 7. If you stack five gauge blocks via the wringing process and measure the total distance across the stack, how would you correctly determine measurement uncertainty? How good is the wringing process? 10