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Foundations of Measurement and Instrumentation

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Foundations of Measurement and Instrumentation d " NASA Reference = Publication 1222 April 1990 Foundations of Measurement and Instrumentation =.t,? r, C_CL ],._ Uncl .s _J NASA Reference Publication 1222 1990 Foundations of Measurement and Instrumentation Isidore Warshawsky Lewis Research Center Cleveland, Ohio National Aeronautics and Space Administration Office of Management Scientific and Technical Information Division Preface This text is intended to provide the user of instrumentation with an understanding of the factors that influence instrument selection, instrument application, the interpretation of results, and the subsequent planning of future operations. In selecting the tools and techniques of measurement, consideration must be given to factors like their reliability, accuracy, speed, cost, and role in contributing to the ultimate goal of the measurement operation. Because the relative importance of these factors varies with each application, the balance among them requires a human judgment. A mathematical formula may be used to describe this judgment, but only after the judgment has been made. The text also seeks to provide that intuitive understanding of the capabilities and limitations of measurement methods that will permit avoidance of an excessive precision and detail beyond that needed by a practical engineering problem. Such matching of effort to need not only saves time and labor, but also attests to the soundness and maturity of the judgment that has been exercised. The chain of variables, components, or steps that comprise a measurement operation can proceed from the physical variable to be measured, to a sensor and its installation, to transmission of the signal from the sensor to a signal transducer, conditioner, or modifier, to an indicator or recorder, to data analysis or manipulation, and to the interpretation and understanding of the data. This text, being directed to the instrument user rather than to the instrument designer, treats, in categorical terms, only those items in the chain that principally control the accuracy, reliability, and utility of the information acquired; these items are the first two or three items of the chain, and the very last item. The quality, sophistication, and pace of development of modern instruments for data acquisition and manipulation are such that the intermediate items of the chain rarely limit the ability to acquire and to understand the information that is desired about the physical variable being studied. This material is the substance of a set of thirty 2-hour lectures presented to members of the senior engineering staff of what is currently known as the NASA Lewis Research Center. The subjects selected for treatment in this text are those which have been found to be most significant in the practical application of instruments and of measurement techniques by this staff. The more basic principles are presented in the main text of each chapter; additional details appear as appended notes following each chapter. Illustrative examples are chosen, as far as possible, to represent practical applications, sometimes in several engineering disciplines. Many of these examples are in a form that makes them suitable for use as problems in a formal course of instruction. An understanding of the language of the differential calculus is necessary; but facility in its operations usually is not. To facilitate cross reference, and the text's use as a handbook, figures and specialized tables bear the same number as the section in which they appear. iii _L PRECEDING PAGE BLANK NOT FILMED "C_'. * V l, Contents Page Preface ...................................................................................................... iii CHAPTER 1. DIMENSIONS ......................................................................... 1 1.0. Physical quantities. Symbols .................................................................. 1 1.1. Relationships among physical quantities .................................................... 1 1.2. Psychophysical quantities ...................................................................... 3 1.3. The Pi theorem. Dimensionless numbers .................................................. 3 1.4. The Pi-zero theorem ........................................................................... 4 1.5. Dimensional analysis ........................................................................... 6 1.6. Empirical coefficients .......................................................................... 8 Notes for Chapter I .................................................................................. 8 CHAPTER 2. UNITS ................................................................................... 9 2.0. Terminology .................................................................................... 9 2.1. Coherent systems of units ................................................................... 9 2.2. Traditional systems of mechanical units .................................................. 10 2.3. Traditional systems of thermal units ....................................................... 10 2.4. Traditional systems of electric and magnetic units ..................................... 10 2.5. Rationalization ................................................................................. 11 2.6. Later systems of units ........................................................................ 11 2.7. The SI units .................................................................................... 11 2.8. Dimensionless units ........................................................................... 11 2.9. Dimensionless numbers ...................................................................... 14 2.10. Coherent equations ............................................................................ 15 2.11. Numerical equations .......................................................................... 16 2.12. Conversion of numerical equations ........................................................ 17 Notes for Chapter 2 .................................................................................. 18 CHAPTER 3. THE ACCURACY OF MEASUREMENTS ..................................... 19 3.0. Introduction. Symbols ........................................................................ 19 3.1. True and indicated values ................................................................... 20 3.2. Sources of measurement error .............................................................. 20 3.3. Correction and error .......................................................................... 22 3.4. Range and span ................................................................................ 22 3.5. Methods of describing the accuracy of a single measurement ....................... 22 3.6. Methods of describing the accuracy of an instrument ................................. 23 3.7. Limit of error .................................................................................. 23 3.8. Nonlinear indications ......................................................................... 23 3.9. Accuracy of an indirect measurement .................................................... 24 3.10. Systematic and random errors ............................................................. 24 3.11. Systematic errors ............................................................................. 25 3.12. Random errors ................................................................................ 26 3.13. Most probable value of a quantity. Means. The median ............................. 27 3.14. Residuals ....................................................................................... 29 3.15. Statistical descriptions of the past and the future ...................................... 29 3.16. The random-error distribution ............................................................. 29 3.17. Concise idealized measures of random-error dispersion .............................. 30 3.18. Gaussian, uniform, and median error distributions .................................... 31 3.19. Practical measures of random-error dispersion ......................................... 34 3.20. Percentiles ..................................................................................... 38 PRECEDtI'_G P_._E _LAN!( N3T F[LI',,I,ED Lt_ _.,_.L.L_,.-.-,,..,_-,,.,_ B_ 3.21 . Weighting of observations .................................................................. 38 3.22. Uncertainty in the practical measures of dispersion .................................... 39 3.23. Rejection of observations .................................................................... 40 3.24. Complete statement of the value of a physical quantity. .............................. 40 3.25. Truncation and rounding ..................................................................... 41 3.26. Average rate of random events, by counting ............................................ 41 3.27. Average rate of random events, by integration ......................................... 42 3.28. Combination of errors. Single measurement of a function of several variables. 43 3.29. The central limit theorem .................................................................... 46 3.30. Several measurements of a function of one or more variables ....................... 47 3.31. Indirect measurements of several quantities .............................................. 48 3.32. Application to an experimental measurement ............................................ 50
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