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ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 5 Propagation of Error T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University 4262011 ENGR201 Lecture 5 Spring 2011 1 Announcement 1 There will be no weekly quiz due to the midterm Next week s quiz will cover material from Lecture 4 and lecture 5 and accompanying problems and reading You will have 7 days from the time the midterm is scored to address any errors After that the statute of limitations is exceeded and no action will be taken 4262011 ENGR201 Lecture 5 Spring 2011 2 Announcement 2 Online Midterm will be given this week Week 5 Coverage reading and notes from lecture 1 2 and 3 Homework problems for lectures 1 2 and 3 Quizzes for lecture 1 2 amp 3 including this week s quiz Test will be 90 Minute duration Test will be open from 1201 noon Thursday thru 11 59 PM Friday containing multiple choice truefalse vocabulary match and computational problems note problems will be weighted differently It is suggested that you have prepared a formula sheet and have a calculator or spread sheet program available There is NO MAKEUP 4262011 ENGR201 Lecture 5 Spring 2011 3 Homework and Reading for Week 5 o Read text Chapter 5 Section 56 skip paragraph on Monte Carlo analysis for now Problems fromtextbook 513 514 o It is suggested you look up some additional material on Taylor Series and partial derivatives see last slide for suggested websites 4262011 ENGR201 Lecture 5 Spring 2011 4 Propagation of Error Uncertainties in your measured independent variables propagate to uncertainties in the dependent variable You can calculate the magnitude of the uncertainties in the dependent variable Using derivatives if you know the mathematical relationship between the independent and dependent variables such as a calibration curve Through a numerical approach that looks at the sum of the individual contributions of each uncertainty on the dependent variable 4262011 ENGR201 Lecture 5 Spring 2011 5 Propagation of Error Case of 1 Independent Variable Let y fX de ne the relationship between dependent variable y and a measured independent variable X We measure the independent variable X a number of times and establish its true value lies within the interval 37 i tSI C Then there must also be an uncertainty in true value ofy r5y f xitS We want a formal way to represent this relationship 4262011 ENGR201 Lecture 5 Spring 2011 Slight Diversion for Background Material Approximation of a function by Taylor Series A power series is an infinite sum ofthe form fxcl J af cll cl1 ac Jr ai cx a1 cx a1 A type of power series is the Taylor series It represents a function fx as an infinite sum of terms defined about a single point a Note we need knowledge of the derivatives of fx 4262011 ENGRZOl LectureSSpnrig 20M 7 An Example of Taylor Series Expansion for eX about the value 2 Approximate the function fx eX for values near 2 We write the Taylor Series expansion for ex about the value at using 4 terms see graph next page as fx fa f lsaxiasaxia2 lxea3 Plugging in our values we obtain the approximation 4262011 ENGRZOl Lecture 5 Spring 2011 Visualizing the Taylor Series Approx for eX or series mansion Er eq x abu 18 The approximation gets closer to the actual function as the number of terms increase E 12 We may want to use the least number of 7 terms to simplify the expression rm or approximation ar rx i i i n5 1 15 2 25 4262011 ENGR201 Lecture 5 Spring 2011 9 Back on track Applying Taylor Expansion to error pnopagation For the function y i6 f 415 the Taylor Series about x is feyirglgts grjgl 65 iii Where L represents higher order terms HOT Repeated general expansion for comparison to above 4262011 ENGR201 Lecture 5 Spring 2011 10 Matching terms in the expansion From the expansion 5yft tSf tSjL We identify corresponding terms 5 f6 6y g 13 4262011 ENGR201 Lecture 5 Spring 2011 11 Propagation of Error We now use a linear approximation for 6y by neglecting terms of power of 2 or greater The approximation is valid for small ts NKdy The derivative evaluated at the sample mean value ofx is a measure ofthe sensitivity ofy to changes in x See plot next page Relating the uncertainty in x to the uncertainty in y My 2 a ux 9xux XZX where Uy is the uncertainty in y and Ux is the uncertainty in x and 6X is more compact notation for the derivative term 4262011 ENGR201 Lecture 5 Spring 2011 12 Illustration of error in x propagating to an error in y y unitsl 4262011 ENGR201 Lecture 5 Spring 2011 13 Propagation of Error Case of gt1 Independent Variable When there are more than one independent variables the result R is R f1x1x2xL Each variable Xi contributes to the uncertainty in R R39 1 iuRP o where R f1x1x2xL HR f1u u 2u L 4262011 ENGR201 Lecture 5 Spring 2011 14 Propagation of Error gt1 lnd Var As in the case for a single variable a sensitivity term can be derived 07306 deli And the uncertainty in the result R is uR quTz 12 Does this lookfamiliar Maybe RSS of something t x i12L 4262011 ENGR201 Lecture 5 Spring 2011 15 What is 912 i12L anyway Called the partial derivative Important in expressing the differential of multivariable functions such as l R f1 x1x2L ij We define the change in R as dxl dcl xztoxL Look at equation on prior slide for ur and these terms d d 32 xlx3toxL dxz dc L xltoxH de 4262011 ENGR201 Lecture 5 Spring 2011 16 efault htmDESCR 4262011 ENGR201 Lecture 5 Spring 2011 18 Ex 53 Displacement Transducer For a transducer with inputoutput relationship of y KE estimate the uncertainty in displacement y for E 500 V if K 1010 mmV with UK t010 mmV and HE t001 V at 95 confidence f uy f uEuK Based on eq 513 y i6Equ l91lt 1lt11 Z 9 1K and 9 E E 6E K 6K uy Kugj Eulj In 4262011 ENGR201 Lecture 5 Spring 2011 19 Ex 53 Displacement Transducer mm uy 1010 V ir010mmj050mmj12 i051mm With the assumption that both terms in R88 are 20 the con dence is 95 2 2 12 wow 5V010m 4262011 ENGR201 Lecture 5 Spring 2011 20 Sequential Perturbation The derivative required to solve the error propagation problem is not always easy to perform An alternative method that is much easierto implement in spreadsheet programs is called sequential perturbation The technique is based on a nite difference method to approximate the derivatives 4262011 ENGR201 Lecture 5 Spring 2011 21 Sequential Perturbation 1 Under fixed operating conditions measure the independent variables and calculate a result Re where R0 fX1 X2 XL This becomes the Operating Point for the numerical approximation 2 Increase the independent variables by their respective uncertainties R1 fx1ux1x2xL R3 fx1x2 ux2xL RL 2 fx1x2xL uxL 3 In a similar way decrease the independent variables by their respective uncertainties Call these terms R 4262011 ENGR201 Lecture 5 Spring 2011 22 Sequential Perturbation 4Calculate the differences 5R and 5in for i 1 2 L 6R R 7 R0 6R R 7 R0 5Evaluate the approximation of the uncertainty contribution from each variable 5R 7 5R7 5Rx 2 m xux Then the uncertainty in the result is L 12 uRiZ5Rj 13 11 4262011 ENGR201 Lecture 5 Spring 2011 23 Revisiting the LVDT Example 52 Example 52 bySequential Perturbation xi value ui Ri Ria delrt delrt delR 1 E 2 K on ill ll Ro m nominal operating point uy hence y 5050 051 mm ostudents should review posted excel file for details of computations Will the Sequential Perturbation answer always equal that based on Partial Derivatives 4262011 ENGR201 Lecture 5 Spring 2011 24 Example Thermocouple Measurement with HandHeld Multimeter A measurement system consists of a hand held digital multimeter and an icepoint referenced type T thermocouple The smallest resolution of the multimeter is 01 mV so the uncertainty due to the instrument resolution is 005 mV If the voltage EMF produced lg the thermocouple is 07896mV it indicates a temperature of 20 obtained from a calibration curve Using the calibration curve we determine the temperatures corresponding to the uncertainty in the multimeter EAJFuV 07896 mV 005 mV08396 mV 3 T 212 C EAlFi uV 07896 m V7 005 m V 07396 m V 3 T 188 C ME 449549 Thermal Management Measurements Gerald W Recktenwald Portland State University 4262011 ENGR201 Lecture 5 Spring 2011 25 Example Thermocouple Measurement with HandHeld Multimeter Therefore the uncertainty in temperature due to the resolution of the instrumentation is 212 C 7188 C uT i il2 C my 2 Summary of uncertainties for measurement system uT 0 C No observed uctuations rand uT il2 C See calculation above uT 0 1 C Maximum error indicated on the thermocouple spool 02 0F at 200 OF ME 449549 Thermal Management Measurements Gerald W Recktenwald Portland State University 4262011 ENGR201 Lecture 5 Spring 2011 26 Example Thermocouple Measurement with HandHeld Multimeter Combining the uncertainties RSS gives uT 1122 012 12 C which is the overall uncertainty in the temperature measurement for an icepoint referenced thermocouple read with the handheld multimeter with voltage resolution of 1005 mV For the assumed values in this example the measurement error due to the inherent properties ofthe thermocouple wire are negligible The uncertainty in the measurement for this example is due to the limited number of significant digits available on the handheld meter ME 449549 Thermal Management Measurements Gerald W Recktenwald Portland State University 4262011 ENGR201 Lecture 5 Spring 2011 27 References Uncertainty Estimation and Calculation Gerald Recktenwald Portland State University Department of Mechanical Engineering httpwebcecspdxedugergclassM E449lectures lpdfuncertainySIides 2uppdf viewed 102409 4262011 ENGR201 Lecture 5 Spring 2011 28 Matlab Code corresponding to slide 8 Taylordemo October 16 2010 for ENGR201 Lecture 5 Taylor series approximation about zero close all clear all clc de ne values about zero x 501 5 fx expx exact values Tayor series expansion of expx about 2 fx4 exp21 x2 x2quot22 x2quot36 fx3 exp21 x2 x2quot22 fx2 exp21 x2 potx fx 39k39 x fx2 39r39 x fx3 39b39 x fx4 39g39 axis0 3 1 20 egend39expx39 392 terms39 393 terms39 394 terms39 39Location39 39best39 tite39Tayor series expansion of expx about 239 xabe39x value39 ylabe39fx or approximation of fx39 4262011 ENGR201 Lecture 5 Spring 2011 Interesting math websites cut and paste Partial Derivatives wwwyoutu be oomwatohvSbeDBmyAMl mathdemosgosuedumathdemospartial demopartialgallerypartialgalleryhtml Taylor Series wwwyoutu be oomwatohvojPoEZOl5WQ tutorial math lamar ed uClassesCalcl lTaylorSeri esaspx enwikipediaorgwikiTaylorseries 4262011 ENGR201 Lecture 5 Spring 2011 ENGR 201 Homework 1 bLuNi x NbI UJ 7704 513 Iquot Problem A Problem A 3 97900 or 979x104 4 154000 or 154 x 105 Problem B 9 gt Uquot UJJgtUJJgt A temperature probe is used to measure the temperature in an experiment Before using the probe in the experiment a calibration curve is needed to ensure the correct value to within the instrument accuracy Using known temperature sources to calibrate the temperature probe the following measurement results over multiple trials were conducted True Te m pe rature 625 127 182 306 366 422 543 606 642 127 183 303 363 426 545 603 663 125 184 304 362 423 543 60 3 641 122 187 30 5 364 424 544 603 659 124 183 303 363 427 545 602 632 124 186 302 363 425 545 604 1 Determine the bestfit calibration line of the data Display the results graphically as well as the calibration equation The data and bestfit line is shown below and plotted 4411 page 1 of 9 ENGR 201 Homework 1 Temperature Calibration Chart 70 LC A y 0999x 0450 9 60 l R2 09999 g 0 a 50 o E 0 OJ 9 40 E 0 30 Q Measured 395 o a 20 Calibration Flt G E 10 O O V O 20 4O 60 80 True Temperature C 2 What is the overall static sensitivity of the temperature measurement system The static sensitivity of the system is the slope of the calibration curve Reading from the calibration equation this is 0999 deg measureddeg true If the calibration were perfect the slope would be 1 3 Determine the accuracy limits of the temperature probe The accuracy limits are determined by creating a deviation chart displaying the real value minus the truth calibration valuefor each data point The deviation chart is shown below Deviation Chart 04 i 03 O o O 02 z Q Q C o 01 o o E 39 gt w 0 A o O O r O 9 0 10 20 30 40 50 60 70 01 g 0 O o 0 02 9 o O 03 Temperature C 4411 page 2 of9 ENGR 201 Homework 1 From the deviation chart the limits ofaccuracy are 0238 C and 0298 C 4411 page 3 of9 ENGR 201 Homework 1 4 Determine the systematic error of the probe The systematic error is the average of the reading minus the true values In this case each nominal temperature has its own systematic error Temperature Systematic C Error C 6 0437 12 0483 18 0417 24 0450 30 0383 36 0350 42 0450 48 0533 54 0417 60 0350 5 How could the systematic error estimation be improved The systematic error estimation could be improved by recalibrating the instrument or by using a more accurate temperature measurement system 4411 page4of9 ENGR 201 Homework 1 Text Problem 12 FIND Why calibrate What does calibrated mean SOLUTION The purpose of a calibration is to evaluate and document the accuracy of a measuring device A calibration should be performed whenever the accuracy level of a measured value must be ascertained An instrument that has been calibrated provides the engineer a basis for interpreting the device s output indication It provides assurance in the measurement Besides this purpose a calibration assures the engineer that the device is working as expected A periodic calibration of measuring instruments serves as a performance check on those instruments and provides a level of confidence in their indicated values A good rule is to calibrate measuring systems annually or more often as needed ISO 9000 certifications have strict rules on calibration results and the frequency of calibration 4411 page 5 of9 ENGR 201 Homework 1 Text Problem 13 FIND Suggest methods to estimate the accuracy and random and systematic errors of a dial thermometer SOLUTION Random error is related to repeatability how closely an instrument indicates the same value So a method that repeatedly exposes the instrument to one or more known temperatures could be developed to estimate the random error This is usually stated as a statistical estimate of the variation of the readings An important aspect of such a test is to include some mechanism to allow the instrument to change its indicated value following each reading so that it must readjust itself For example we could place the instrument in an environment of constant temperature and note its indicated value and then move the instrument to another constant temperature environment and note its value there The two chosen temperatures could be representative of the range of intended use of the instrument By alternating between the two constant temperature environments differences in indicated values within each environment would be indicative of the precision error to be expected of the instrument at that temperature Of course this assumes that the constant temperatures do indeed remain constant throughout the test and the instrument is used in an identical manner for each measurement Systematic error is a xed offset In the absence of random error this would be how closely the instrument indicates the correct value This offset would be present in every reading So an important aspect of this check is to calibrate it against a value that is at least as accurate as you need This is not trivial For example you could use the ice point ODC as a check for systematic error The ice point is formed from a carefully prepared bath of solid ice and liquid water As another check the melting point ofa pure substance such as silver could be used Or easier the steam point Accuracy requires a calibration to assess both random and systematic errors If in the preceding test the temperatures of the two constant temperature environments were known the above procedure could serve to establish the systematic error as well as random error of the instrument To do this The difference between the average of the readings obtained at some known temperature and the known temperature would provide an estimate of the systematic error 4411 page 60f9 ENGR 201 Homework 1 Text Problem 14 FIND Identify measurement stages for each device SOLUTION a thermostat Sensortransducer bimetallic thermometer Output displacement of thermometer tip Controller mercury contact switch openfurnace off closedfurnace on b speedometer Method 1 Sensor usually a mechanically coupled cable Transducer typically a dc generator that is turned by the cable producing an electrical signal Output typically a pointer scale note often a galvanometer is used to convert the electrical signal in a mechanical rotation of the pointer Method 2 Sensor A magnet attached to the rotating shaft Transducer A Hall Effect device that is stationary but detects each sensor passage by creating voltage pulse Signal Conditioning A pulse counting circuit maybe also digitalanalog converter if analog readout is used Output An analog or digital readout calibrated to convert pulses per minute to kph or mph 0 Portable CD Stereo Player Sensor laser with optical reader re ected light signal differentiates between a quotlquot and H Transducer digital register stores digital information for signal conditioning Signal conditioning digitaltoanalog converter and amplifier converts digital numbers to voltages and amplifies the voltage Output headsetspeaker note the headesetspeaker is a second transducer in this system converting an electrical signal back to a mechanical displacement 4411 page 7of9 ENGR 201 Homework 1 Text Problem 112 KNOWN Sequence calibration data set of Table 16 ri5mVro5mV ehmax SOLUTION By inspection of the data the maximum hysteresis occurs at X 30 For this case eh ehmax yup ydown 02 mV or ehmax 100 x 02 mVS mV 4 P112 5 4 9 3 E r 2 1 o o 2 4 XmV 4411 page8 of9 ENGR 201 Homework 1 Text Problem 119 KNOWN A bulb thermometer is used to measure outside temperature FIND Extraneous variables that might in uence thermometer output SOLUTION A thermometer s indicated temperature will be in uenced by the temperature of solid objects to which it is in contact and radiation exchange with bodies at different temperatures including the sky or sun buildings people and ground within its line of sight Hence location should be carefully selected and even randomized We know that a bulb thermometer does not respond quickly to temperature changes so that a sufficient period of time needs to be allowed for the instrument to adjust to new 1 By J quot quot of the effects due to instrument hysteresis and instrument and l 39 39 A t 39 quotquot can be 39 39 Because of limited resolution in such an 39 t different 1 t t t l observers might record different indicated temperatures even if the instrument output were xed Either observers should be randomized or if not the test replicated It is interesting to note that such a randomization will bring about a predictable scatter in recorded data of about 12 the resolution of 4411 page9 of9 ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 4 Uncertainty Analysis Part 1 T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University 19 April 2011 4182011 ENGR201 Lecture 4 Spring 2011 Announcements 1 Online Midterm will be given in Week 5 Coverage reading and notes from lecture 1 2 and 3 Homework problems for lectures 1 2 a d 3 Quizzes for lecture 1 2 amp 3 including this week s quiz Test will be TBD hour duration containing multiple choice truefalse vocabulary match and computational problems It is suggested that you have prepared a formula sheet and have a calculator or spread sheet program available There is NO MAKEUP Check Bb Vista later this week for additional details and for those with special arrangements 4182011 ENGR201 Lecture 4 Spring 2011 Announcements 2 Quiz this week on Finite Statistics Open Thursday 500 PM through Friday 500 PM Coverage Lecture 3 reading lecture notes and problem set 4182011 ENGR201 Lecture 4 Spring 2011 Homework and Readings for Week 4 Reading from text Section 51 52 53 54 and 55 make sure to pay attention to examples Problems from text 56 58 510 4182011 ENGR201 Lecture 4 Spring 2011 Definition Review Measurement Process of assigning a value to a physical quantity Error Difference between our assigned value and the true value Uncertainty Estimate of the probable error in our measurement Errors are a property of the measurement Errors come from instruments finite statistics procedures Uncertainty is a property of the result Using uncertainty analysis we will identify quantify and combine error estimates to estimate the error in our final result 4182011 ENG R201 Lecture 4 Spring 2011 This Lecture Purpose Estimating the iwhat as we design a measurement system Method Introduce uncertainty analysis Develop a methodical approach Make critical evaluations of potential elements of the apparatus Spend your money on components that will really make uncertainty a difference in reducing the 4182011 ENGR201 Lecture 4 Spring 2011 Repeated Measurements 3 Measured data True value Syste atic error plx lt Measured value x l 4 l Measurement number 4182011 ENGR201 Lecture 4 Spring 2011 7 Total Error 0 Here s what you will report as a result of your measurements x39 fiuxP 0 We are going to work on how to estimate uX when the experiment measurement system is being designed Uncertainty analysis is the method to quantify the u term 4182011 ENGR201 Lecture 4 Spring 2011 3 Assumptions for Uncertainty Analysis Test objectives are known amp process is clearly defined Knovvn corrections to systematic error have been applied Set the bathroom scale to zero before getting on Systematic errors are independent uncorrelated of each other Data are obtained under fixed operating conditions As the engineers you have some quotexperiencequot with the system components You can estimate systematic and random errors based on prior experience use speci cation sheets professional test codes or performance info in professional literature 4182011 ENGR201 Lecture 4 Spring 2011 9 Uncertainty Analysis Uncertainty analysis can be applied at several stages of measurement Design stage Advanced stage andor single measurement Multiple measurement We ll start with design stage the analysis performed before any measurement associated with the selection of equipment and techniques 4182011 ENGR201 Lecture 4 Spring 2011 10 Motivation DesignStage Uncertainty Analysis Before you perform measurements you want to make sure that the resulting data will have the accuracy and precision needed to help you make critical engineering decisions Select the proper sensors and instruments 39 Costbene t analysis why spend for an advanced instrument when the quantity it measures is not the one limiting uncertainty Select the proper measurement techniques Make an estimate of the resulting experimental uncertainty 39 Combine the uncertainties from several sources 4182011 ENGR201 Lecture 4 Spring 2011 11 DesignStage Analysis Zeroorder Uncertainty Even if all errors are zero there is still the problem of reading the instrument Zeroorder uncertainty of the instrument Uo 39 Do you remember the dif culty of reading the last digit ofthe vernier caliper and micrometer in lab 4182011 ENGR201 Lecture 4 Spring 2011 12 Uncertainty Analysis Zeroorder Uncertainty For zeroorder uncertainty the variation in measurement data is due to instrument resolution uo is the estimate of the random uncertainty caused by the data scatter due to reading the instrument rule 12 resolution and P 95 2 sigma u0 2 1resolution 95 2 Interpretation at 95 probability only 1 measurement in 20 would be outside this interval 09520 19 4182011 ENGR201 Lecture 4 Spring 2011 13 DesignStage Analysis Instrument Uncertainty Instrument uncertainty Uc Can be obtained from info published on datasheets web sites etc by the manufacturer Specifications are for specific environmental and operating conditions U0 is an estimate ofthe systematic error for the instrument MAY CONTAIN MULTIPLE TERMS 4182011 ENGR201 Lecture 4 Spring 2011 14 DesignStage Uncertainty Analysis Table 11 Manufacturer39s Speci cations Typical Pressure Transducer Operation input range 0 1000 cm H20 Excitation iii V dc Output range 0 5 V Performance Linearity error i05 FSO Hysteresis en39or Less than i015 FSO Sensitivity error i025 of reading Thermal sensitivity error i0i02 C of reading Thermal zero drift i0i02 C FSO Temperature range 0 50 C 4182011 ENGR201 Lecture 4 Spring 2011 15 RootSumSquares RSS for Combining Elemental Errors Measurement errorsfrom various sources combine to Increase the uncertainty Each error source iscalled an elemental error ie linearity and senSItIVIty errors prior table lfthere are K elemental errors we can find the total error using the rootsumsquares RSS method note uX for general formula uc seen later ux i11812ee 2 ux i 21521 P 4182011 ENGR201 Lecture 4 Spring 2011 16 RootSumSquares RSS for Combining Elemental Errors ux ix612 e e12 ux 32 e P Application ofthe method can be tricky because you must have all terms in the same units All errors should be estimated at the same probability level Generally in engineering this is the 95 P 95 or 2 standard deviation level Some may use the 68 OM standard deviation level The authors use 95 4182011 ENGR201 Lecture 4 Spring 2011 17 RootSumSquares RSS for Combining Elemental Errors The RSS method assumes that the likely variations in the value of an error taken over repeated measurements follow a normal distribution It produces a probable measure ofthe uncertainty due to these errors The alternative is to take the simple arithmetic sum of the elemental errors getting a larger uncertainty estimate This produces an uncertainty that assumes that all errors have the worst possible result values for each and every measurement It is not based on probability 4182011 ENGR201 Lecture 4 Spring 2011 13 DesignStage Combining Uncertainties uo uc Designrstage uncertainty l 2 2 uIf 7 uU up Interpolation error Instrument errors 0 quotc Designstage uncertainty Ud Combine the zeroorder uncertainty with the instrument uncertainty If other error sources are known at this point include them in the square root as well 4182011 ENGRZOt LectureASpring 20M 19 Ex 51 Find ud All Uncertainties have the same units Consider the force measuring instrument described by the following table Provide an estimate ofthe uncertainty attributable to this instrument and the instrument designstage uncertainty N N 020 N over 030 N over represent Process use RSS to find ucthen use RSS to find Ud 4182011 ENGRZOl Lecture 4 Spring 2011 20 Ex 51 Find ud All Uncertainties have the same units Find instrument uncertainty e1 is the linearity error at 020 N e2 is the hysteresis error at 030 N u0 1 020N2 030N2 036N 95 Instrument resolution is given as 025 N so 0 10125 N The designstage uncertainty becomes ud aug u 21036N2 0125N2 1038 N 95 4182011 ENGR201 Lecture 4 Spring 2011 21 Example 52 Find ud Uncertainties have different units A voltmeter is to be used to measure the output from a pressure transducer which is an electrical signal The nominal pressure is expected to be about 3 psi Estimate the designstage uncertainty for this combination Apparatus P Pressu re Chamber Sensor Voltmeter 4182011 ENGR201 Lecture 4 Spring 2011 22 Ex 52 Find ud Uncertainties have different units 0001 of VDC 1 V 25 over 2 over error 4182011 ENGR201 Lecture 4 Spring 2011 23 Ex 52 Find uOI Uncertainties have different units Apparatus P Pressu re Chamber Sensor Voltmeter Assumptions values represent the instrument at 95 probability Process nd the design stage uncertainty ua for each device ie udE and udP based on combining uo and uc then combine udE and udPto nd ua for the complete apparatus 4182011 ENGR201 Lecture 4 Spring 2011 24 Ex 52 Find udE for Voltmeter 00ft 100016 Using the resolution quot012 i5 V95 Specs give accuracy in terms ofthe meter reading 0001 ofreading Since we expect a nominal reading of 3 V ucE 3 Vx 000001 30 uv 95 Voltmeter design stage uncertainty udE 5x10 8 v2 30x10396 V2 304 v 95 4182011 ENGR201 Lecture 4 Spring 2011 25 Example 52 Find udP for Pressure Transducer Assume we are operating at the proper conditions The elemental errors are linearity and sensitivity using 3 psi operating condition we convert to mV ucF hef e 95 assumed J25mVpsi x 3psi2 20mVpsi x 3psi2 2961mV 95 Since the transducer has negligible resolution error we can ignore uo or uo 0 PT design stage uncertainty quotJP 1961mV 95 4182011 ENGR201 Lecture 4 Spring 2011 25 Example 52 Find uOI Apparatus Uncertainty 1 1 ud ud 0030 mV2 961 mV2 1961mV 95 39 Since the sensitivity is 1 Vpsi we can state the design stage uncertainty in psi Ud 100096 psi 95 Note that the uncertainty is completely dominated by the transducer look at the formula Spending more on a higher quality voltmeter will do nothing to lower the uncertainty Spend the money on the pressure transducer 4182011 ENGR201 Lecture 4 Spring 2011 27 Error Sources Calibration errors 39 Calibration reduces measurement errors but does not eliminate them Sources Systematic and random errors that are part of the calibration process The way the standard is applied to the measuring system Standards still have some element of uncertainty Lab standard values may change over time from the transfer standard they were compared to NIST National Institute of Standards and Technology 4182011 ENGR201 Lecture 4 Spring 2011 23 Error Sources 0 Data acquisition errors errors that occur during the act of measurement Process 4182011 ENGR201 Lecture 4 Spring 2011 29 Error Sources Data reduction errors Curve fit errors 0 Perhaps you have fit a calibration curve with a polynomial it may not fit everywhere Software resolution errors 0 Rounding errors 4182011 ENGR201 Lecture 4 Spring 2011 30 Systematic Error Systematic error is present on all repeated measurements Constant so statistics alone can not quantify its size May shift results either higher or lower than true value How do we know it is present How do we find its magnitude How do we minimize it Calibration is generally the best way to minimize systematic error 4182011 ENGR201 Lecture 4 Spring 2011 31 Systematic Error Analog Bathroom Scale Example Goal Calibrate your scale to be as accurate as possible The scale has output gradations every 2 pounds and a zeroadjustment knob You can get an idea of the random error by stepping on and off the scale several times and looking at the variation in readings The accuracy of the scale can not be better than this but it can be much worse if the systematic error is significant Follows Cornell Laboratory 0 ErrorAnalysis lrrn ii mi m ir rn Svstemic Errorshtml 4182011 ENGR201 Lecture 4 Spring 2011 32 Systematic Error Bathroom Scale Example Let s assume the scale is linear Construct a calibration curve measured weight vs actual weight Ifthe behavior is linear we only need two points to draw the curve 0 weight can be used but any two weights are ok The weights should be as far apart as possible for best results 4182011 ENGR201 Lecture 4 Spring 2011 33 Systematic Error Procedure With no weight applied rotate the zero adjustment knob so the scale reads zero Apply a known weight and read the scale How do get a known weight Weigh yourself on a scale known to be accurate doctor s office gym then weigh yourself on your scale 4182011 ENGR201 Lecture 4 Spring 2011 34 Systematic Error Sensitivity Apply 160 lbs and the scale reads scale 150 lbs Calibration oTrue value is 67 higher than the measured value 150 lbs 1067 160 lbs The 2point calibration curve shows that there is a sensitivity error Use the curve to reduce your systematic error 40L 1 Measured Weight lbs When you stand on the scale increase the weight by 67 If it reads 75 lbs it is really 80le 160 Applied Weight lbs 4182011 ENGR201 Lecture 4 Spring 2011 35 Reduction of Systematic Error You have two voltmeters A and B that you wish to calibrate for use at 100 V To do the calibration you have a standard voltage source available with negligible random error and an absolute accuracy of 05 V Measure the 100 V standard 10 times with each meter This example follows Chapter 1 Example Experimentation and Uncertainty Analysis for Engineers Coleman and Steele 4182011 ENGR201 Lecture 4 Spring 2011 Calibration Results 100 V Input I I0 0 I04 39 a s 39 39 9 98 395 92 E l gt0 I I I I I I I 86 39 Volumeter A so I Voltmezer B 0 2 4 6 8 IO Measurement Number 4182011 ENGR201 Lecture 4 Spring 2011 37 Reduction of Systematic Error Readings Averages Meter A 1006 V Meter B 900 V Systematic Errors Meter A 06 V Meter B 100 V Subtract 06 V from readings from meter A and add 100 V to readings from meter B for inputs near 100 V 4182011 ENGR201 Lecture 4 Spring 2011 33 Reduction of Systematic Error At this point can we say that we have removed all systematic error No since the standard itself had a 05 V error If you had both meters available to you which would you rather use 4182011 ENGR201 Lecture 4 Spring 2011 39 Systematic Error Can be estimated by Calibration as just shown Concomitant methodology Measure the same quantity with two different techniques Interlaboratory cooperation Different personnel equipment facilities doing the same measurement Experience 4182011 ENGR201 Lecture 4 Spring 2011 40 Review Topics Linear Interpolation Finite and Infinite Statistics Interpolation of Table Data Many times we need entriesthat are between tabulated values in tables called interpolation Consider that we require the 2 value for 025 probability not directly in Table 43 zi 006 007 008 06 02454 02486 02517 Obviously the answer is between 067 and 068 o A visual estimate is 0675 which is half the distance between the two entries 4182011 ENGR201 Lecture 4 Spring 2011 42 Interpolation of Table Data If we can assume a inear relationship between the two entries and find the equation of a line we can estimate values that are not as obVIous zl mpb 068 067 m 02517 02486 b 068 32258 gtlt 02517 01319 zl 32258gtltp 01319 32258 with p 025 zi 067455 0675 same as by eye Using the above formula we can also nd the value of zi corresponding to p 0251 zi 0678 4182011 ENGR201 Lecture 4 Spring 2011 43 Interpolation of Table Data There are many different interpolation techniques check out Wikipedia similar triangles is another form of linear interpolation The linear interpolation method is quick and sufficient for finding values within posted entries of a table 4182011 ENGR201 Lecture 4 Spring 2011 44 Using Statistics to make Engineering Decisions We ask the questions When do we use infinite statistics z When do we use finite statistics t We will explore two quality control problems to answer these questions 4182011 ENGR201 Lecture 4 Spring 2011 45 Infinite Statistics for Quality Control QC ln manufacturing a particular set of motor shafts only shaft diameters of between 3758 and 3810 mm are usable A manufacturing run produces a process mean diameter of 3784 mm and a standard deviation of 013 mm We want to know what percentage of the manufactured shafts are usable 4182011 ENGR201 Lecture 4 Spring 2011 46 Infinite Statistics for Quality Control QC Why are we using infinite statistics Process mean process standard deviation are interpreted as measures of the W Hence we may assume that the diameters are normall distributed Therefore this becomes a problem of finding the percentage of this population that lies between 3758 mm and 3810 mm 4182011 ENGR201 Lecture 4 Spring 2011 47 Infinite Statistics for Quality Control QC P3758 s x 3 3810 mm PZa s x s Zb za 3810 3784 mm013 mm 2 zb 3758 3784 mm013 mm 2 From table 43 noting symmetry Pza 04772 PZb 04772 PZa s x s Zb PZa PZb 09544 P3758 s x 3 3810 mm 954 3 sig fig 80 about 95 ofthe product is useable what do you do with the rest 4182011 ENGR201 Lecture 4 Spring 2011 48 Finite Statistics for Quality Control QC A manufacturer claims that motor shafts have an average density of 421 kgm3 We receive a large shipment of motor shafts and need to determine if the shipment meets the average density specification Do you accept the manufacturers claim 4182011 ENGR201 Lecture 4 Spring 2011 49 Finite Statistics for Quality Control QC We measure the density of 20 motor shafts selected at random from the large shipment obtaining Tc 2 475kgm3 and Sx 84kgm3 We have measured sample statistics the manufacturer is claiming a specific density for his product The question before us is whether his claim is supported by our data 4182011 ENGR201 Lecture 4 Spring 2011 50 Finite Statistics for Quality Control QC ASSUMPTION Sample is representative of the batch SOLUTION See if his claim for x is justified based on our sample measurements Calculate x x i tv95S 4182011 ENGR201 Lecture4 Spring 2011 51 Finite Statistics for Quality Control QC N 20 so 1 19 degrees of freedom At the 95 level tm95 2093 table 44 Std Dev ofthe Mean is S zsxJ So our estimate of x is x 475 i2093842012kgm3 475 i 393kgm3 The manufacturer s claim is 421 kgm3 So we reject the manufacturer s claim 4182011 ENGR201 Lecture 4 Spring 2011 52 References Experimentation and Uncertainty Analysis for Engineers 2nd ed HW Coleman and WG Steele VWeylnterscience 1999 Theory and Design for Mechanical Measurements 4th ed RS Figliola and DE Beasley John Wiley 2006 Lab 0 Minimizing Systematic Error Cornell University httQinstruct1citcornelleducoursesvirtual labL abZeroMinimizinq SVstemic Errorshtml read 10122009 4182011 ENGR201 Lecture 4 Spring 2011 53 Lecture 1 ENGR 201 Evaluation amp Presentation of Experimental Data T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University 29 March 2011 ENGR201 Lecture 1 Spring 2011 What is this sequence about All engineers regardless of area should have a common core of understanding related to measurements 39 Design ofa measurement strategy 39 Selection of sensors 39 Pursuit ofalternative solutions 39 Understanding ofthe limitations of measurement systems 39 Estimating the accuracy ofa measurement 39 Display of measurement results ENGR201 Lecture 1 Spring 2011 Our Sequence Concept Give you the basics in EPED l to understand measurement statistics several common sensors and computerbased data acquisition Apply these basics in EPED II where you will do a measurement system design as well as calibration testing and user interface development ENGR201 Lecture 1 Spring 2011 Course Goals 39 You are aware of sources of measurement errors and how they propagate and are capable of using this knowledge in calculations 39 You know the basic structure ofa LabVlEW program and recognize the importance of software of this type in engineering measurement 39 You know how to acquire measurements from a variety of engineering instruments and sensors 39 You understand the need to quantify accuracy and resolution in an engineering measurement plan and can put this into practice 39 You can prepare a high quality engineering report including presentation of goals background results analysis and conclusions ENGR201 Lecture 1 Spring 2011 How to Navigate Through This Course Check BbVista frequently for materials and announcements 39 Read the syllabus lab policies lab grading rubric and follow the instructions 39 Download and print lecture notes prior to class follow and annotate during lecture 39 Read and study the text do the homework check your solutions with those posted to prepare for weekly online quizzes 39 Read and prepare for labs prior to lab session Keep notebook up to date participate in the lab everybody needs hands on 39 The TA s are your rst line of defense contact them with questions about lab problems or material etc 39 lfyou have additional questions contact either ofthe instructors We are always available by emall See BbVista for office hours ENGR2 1 Lecture1 Spring 2011 5 The Textbook Theory and Design for Mechanical Measurements RS Figliola amp DE Beasley 5th Edition John Wiley MECHANICAL 2011 MEASUREMENTS 39 Hardcover and electronic versions available Glossary in back learn new terms No homework solutions in textbook Student Companion Site cut and paste httpbcswileycomhe bcsBooksactionindexampitemd04705474 13ampbcsd6234 ENGR201 Lecture 1 Spring 2011 6 EPED Labs Basic Mechanical Measurements LabVIEW Exercise Timing Measurement Reaction Time Measurement Voltage Measurement in LabVIEW Temperature Measurement Bicycle Cyclometer Project Note some labs take two sessions ENGR201 Lecture 1 Spring 2011 Staffing Faculty 39 Dr Thomas Chmielewski ECE Section B 39 Dr David Miller MEM Section A Teaching assistants Kenneth MalloryMEM kennethmallo drexeledu Nathaniel Taylor MEM nt65drexeledu Steven Leist MEM stevenkleistdrexeledu Marco Janco ECE ma3979drexeledu Jamie Kennedy ECE 39lk73drexeledu Zongquan Gu ECE zg33drexeledu Nan Xie ECE nanxiedrexeledu ENGR201 Lecture 1 Spring 2011 Week 1 Reading Figliola and Beasley 39 All of Chapter1 39 Graphing Standards pp 565567 Appendix A The Engineering Notebook Laboratory Items Folder 39 Best practices for recording data and procedures in your lab notebook Building Histograms Laboratory Items Folder 39 A introduction to building histograms from measurement data ENGR201 Lecture 1 Spring 2011 9 Week 1 Homework Problems Check Bb Vista not collected View LabVIEW Tutorial 1 Basic Introduction National Instruments httpwwwvoutubecomwatchvEm5R RM8E08 Gain some familiarity with the LV user interface front panel and block diagram run buttons See connection between user controls and block diagram icons See how to access the function and controls pallettes ENGR201 Lecture 1 Spring 2011 Lecture 1 Let s get started ENGR201 Lecture 1 Spring 2011 11 Engineering Measurement Applications Why do we do it As engineers we may need to do measurements to verify the capabilities or reliability of existing products materials or devices We may be developing measurement systems and processes to test new concepts research Our measurements may be needed to monitor or control an operating industrial process ENGR201 Lecture 1 Spring 2011 12 Engineering Measurement Applications How do we do it A test is designed to answer a question The measurements we make on a variable or collection of variables should help answer this question The test requires an integrated measurement system and a plan to analyze the data and present the results There is rarely a single solution to the measurement system design multiple ways to measure and collect and analyze ENGRZOW Lecture 1 Spring 2011 13 Generalized Measurement System Sensor uses physical phenomenon to sense variable being measured Calibration Transducer converts sensed info into a detectable signal Signal condittonmg stage Control signal Prunes Control stage Figure 11 Components of a gencm measurement system ENGRZOl Lecture 1 Spring 2011 Mercury Thermometer odiaatstaga Components of bulb a oaaiayscaie thermometer are equivalent to csensor liquid in bulb Gtransducer capillary senaaiaiaiaasage 39 OUtPUt Stage Sensor ENGRZOl Lecture 1 Spring 2011 Drexel SunDragon SunDragon v solar car Designed by students from many departments MEM ECE MATE primarily and built from carbon and kevlartiber Raced in the Sunrayce 3999 national competition ENGRZOl Lecture 1 Spring 2011 Drexel SunDragon mm Oumnl Timpmmru Tunp D cm Proqumay 9 Venus Sm Sm Sm Sm ENGRZEH Lemme 1 Sp ng ZEIM Wireless ECG Holter Monitor ENGRzm LEmurM Sprwg 2mm 18 Another Sensing System Measurand sound Sensor ear drum Transducer middle ear bones ossicles Signal Conditioning stapedius muscle reflex Output nerve impulses from the cochlea Feedback amp Control turn down your iPod ENGR201 Lecture 1 Spring 2011 19 Name Some Other Measurement Systems 39 What is the Measurand Sensor Stage Transducer Stage Signal Conditioning Stage Output Stage ENGR201 Lecture 1 Spring 2011 Experimental Test Plan We set about designing and performing an experiment to answer a specific question 39 Parameter Design Plan 39 What questions do I need to answer 39 What needs to be measured 39 What variables and parameters will effect my results 39 System and Tolerance Design Plan 39 How can I do this measurement and how good do the results need to be to answer my question 39 What technique what sensor what procedure 39 Data Reduction Design Plan 39 How to analyze present and use the collected data 39 How will I interpret the data How good is my answer Does it make sense ENGR201 Lecture 1 Spring 2011 2 1 Experimental Test Plan Successful engineers will review the measurement plan BEFORE Purchasing equipment Purchasing software Taking measurements ENGR201 Lecture 1 Spring 2011 Experimental Test Plan Decide on the question you want the test to answer Identify the relevant process parameters and variables Variables will influence the test We will measure the variables of interest but others may also affect the test ENGR201 Lecture 1 Spring 2011 Variables ndependent variable 39 Can be changed independently of other variables Dependent variable 39 Affected by the changes in one or more variables Extraneous variable 39 Variables that are not or can not be controlled during measurement but affect the value of the variable measured May cause differences in repeated measurements of the same variable Controlled variables 39 Variables that can be held at a constant value at least in a nominal sense ENGR201 Lecture1 Spring 2011 Results of a boiling point test for water 3 tests run on separate days Why do we get different results if the system accuracy Bngm 213 7 accounts for only 01 F Temperature F a Test1302inllg I Test2297lnHg o Test330 in Hg 1 2 3 4 5 6 7 Time min ENGR201 Lecture 1 Spring 2011 25 Noise and Interference Extraneous variables can impose noise and interference onto measured data Noise Random variation ofthe value of the measured signal caused by variation of extraneous variables 0 Increases scatter Can be reduced or removed using statistical techniques lnterference Effect that introduces a deterministic trend on the measured value le trend that is predictable in time or space such as a ramp or sine function ENGR201 Lecture 1 Spring 2011 26 Effects of Noise and Interference 5 Signal ye 2 sin21rt Signal interference at type 0f 4 Interference g is present Ta 3 EH Signal nmse 2 2 1 0 l l l l l 00 05 10 15 20 Time 5 ENGRZDl Lemurel Spring ZEIll Situation Dependent variable y Controlled by independent variables XE Xb Affected by extraneous variables zl wherej 1 2 3 How do we get the best estimate of y Fxa Xb zl Example The fuel usage rate y I should expect from my c r depends on fuel consumed x3 and miles or km travelled Xb Both of these are easily measured quantities Results are affected by driving style route chosen weather etc solicit some examples ENGRZEM Lemurel Spring IBM 28 Random Tests A random test strategy is used to try to remove the affects of the extraneous variables and find the influence of the independent variables xa xb on the dependent variable y Independent variables are changed in a controlled manner The influence of the 2 variables on y is not removed but the possibility ofthem introducing a false trend on y can be minimized A random test matrix sets up a random order to the change in one or more independent variables effective for extraneous variables that change in a continuous manner ENGRZOl Lecture 1 Spring 2011 Random Tests atmosphere Fl Voltrneter Cylinder What are the dependent independent and extraneous variables What strategy can be used to minimize the interference caused by the extraneous variables ENGRZOW Lecture 1 Spring 2011 Random Tests Extraneous variables P ower sou roe Voltmeter pen t atmosphere l gt V lL p V T Piston gt Pressure transducer Cylinder p f VTZlzzz3 Where V fxT Control variable x is changed Dependent variable p is measured If we are to do 6 measurements i16 each with the piston at Xi we should randomize the order in which they are done V2 V5 V1 V4 V6 V3 is one of many possibilities ENGR2D1 Lecture 1 Spring 2011 Replication and Repetition We can get a better estimate of the value of a measured variable by doing more measurements Ball bearing diameter cooling bath temperature resistor value etc Repeated measurements we make during any single test run or on a single batch are called repetitions Quantify the variation of the measurand for a single run or batch while operating conditions are held constant as much as possible ENGR2D1 Lecture 1 Spring 2011 Replication and Repetition lf we want to follow long term trends in a variable we need to do duplicate test runs on different days 39 Bearing diameters produced on a particular machine or by a particular operator lndependent duplication of a set of measurements under similar operating conditions is called replication Replication can be used to randomize the interference effects of different machines or operators ENGR201 Lecture 1 Sprlng 2011 33 Example 15 Repetition Set the room thermostat to a given temperature Make repeated measurements of the room temp repetitions Now you can see the average value and temp variation at that one temp setting 0 Variation might be expressed as the standard deviation Results let you see how well you can control the variable 0 How tight is the distribution In ll 4quot A 180x130iu2 ENGR201 Lecture 1 Sprlng 2011 34 Example 15 Replication temp Repeatthe same set or mum temp measurements rephcatmn Tne average and drstnoutren rertne second set or measurements may be drrrerent tnan tne nrst Tne drrrerenee m tne ayerages gryes an rndreatrun or new WEH we can set and untrm tne temperature m tne ruum Repneatren permrts tne assessment or new WEH we can duphcate a set or eendrtrens ENGR2D1Ledme15wmva 35 Calibration quotn a panoratron procedure you appty a known mputva ue to your asurement system Tne rnput may be a standard traceabte to a nattona aboratory Ora 00a Standard a see section 15 in text orten standardrzed procedures are used ANSWSA 39 m a ab you WM often See cahbratton Smokers on the measurement rnstruments rndrcatrng wnen tne ast panoratron was done by Whom and pernaps tne date tne next cahbratton 5 due Tne output ottne rnstrument m response to tne known rnput rs recorded r r r v panoratron curve a not atways hnear ENGR2D1Leduve15vvmva 36 Calibration Static calibration Apply a standard input to measurement system System is allowed to stabilize Output is recorded Repeat for another input Use a random test matrix to remove the effects of continuous extraneous variables or if applicable use an industry standard test procedure Generate a calibration curve Do not want to extrapolate beyond inputs used ENGRZOl Lecture 1 Sprlrig 20M Example Weighing Scale Calibration This is a O 5 lb nominal spring weighing scale To calibrate apply accurate weights in 05 lb increments from O to 5 lbs note these may be applied in a certain order per some spec Wheeler A N N ENGRZOl Lecture 1 Sprlrig 20M Calibration Curve Best Ill IIHL n 15 l 15 2 25 3 35 4 45 5 1m weight my may and m m m Engrmg mmmm m in parswpmm Hal mm ENGRZEH Lecture 1 Spring ZEIll Calibration 39Static sensitivity Slope at a particular point Xi on the calibration curve Kauai n For our scale example the slope sensitivity is a constant straight line Figure 19 of text shows one whose slope varies with x 39Dynamic calibration Finds the relationship between a time or space varying input and the output of the measurement system The input may be sinusoidal or a step chan e We get information such as time response gain or attenuation phase shift ENGRZEM Lecture 1 Spring ZEIll 20 Range and Span K Ecsl in line i l l l l l I 05 l 15 I 7 1 35 4 45 39l 39ruo weight lb Range variation of applied inputs If operating range of input is Xmin to Xmax n Xmax Xmin 0 to 5 lbs for scale Output span full scale operating range FSO If output operating range is Ymin to ymax ro Ymax ymin 05 to 6 lbs for scale eyeballlng plot a so see pg 43 of notes ENGR201 Lecture 1 Spring 2011 41 Accuracy DeVIatIon Plot for Weighing Scale 06 1 l I I 04 e X x 35 g 2 7 X Accuracy limits x Of the X x if x ll X difference 5 0 X g lt i a 7 between the V X 2 7 best fit line and D 70 7 X x i 7 measured 39 x x gtlt g i i i X values 704 X Q g 706 V l l l l l i l 1 ll 05 l l v 7 i V 35 4 45 5 l 2 M Wheeler rue wclehl bj M A MA ENGR201 Lecture 1 Spring 2011 4 21 Accuracy Estimates 39We can draw horizontal lines on the weighing scale deviation plot that will contain all data In this case the limits are 045 lb and 040 lb 39From the calibration curve the span ofthe scale is from 0374 to 6076 lb or 6076 0374 645 lb Here we used the equation R 129W 0374 with values ofW 0 5 39Expressing the accuracy limits as a percent of output span yields 70 and 62 2 sig fig ENGR ZEH Lecturel SpringZDll 43 Measured value unns Measurement Errors 10 independent measurements are made Apparent measured average Scatter due a 1d mndum errdr Test systematic error True dr known value Measured data 1 2 3 4 5 a 7 s 9 m Measured reading number ENGR ZEH Lecturel SpringZDll 44 22 Measurement Errors Apparent measured average Scenerdue a random error Test systematic error True or known vatue Measured value units Random Error Reading Average of Readings Systematic Error Average of Readings True Value ENGRIH Lemuel S l g mm 45 Random Errors Caused by a lack of repeatability Generates a scatter of measurement results Some sources I Temperature drill I Electrical noise I Instrument use or technique Minimization abilize environment eliminate noise I Take enough measurements to use statistical methods ENGRIH Lemuel S l g mm 46 Systematic Errors Constant for repeated measurement Some sources 39 Instrument drift Operator technique How large Characterize by using a concomitant measurement ENGRZEM Lecturet Spring 2mm 47 Visualizing Repeatability vs Accuracy Systematic Error inw random error but no Vandaquot arm systemahc errors iead to near accuracy direct indrcaimn or accuracy ENGRZEH Lecturei Spring 2m 48 24 Oulnux vama ompm mun Common Elements of Instrument Error Hysteresis annsula Mural dlli IIE d Li nearity yum g Sensitivity mm o mum cum u Zero shift mumrm Invukmus 39 a Hystwesb um b1 Unediy nnwr Maximum 6w mm amine rmursmn z Nm nalcuwe 3 on m a lmlyphcaiwdlce E i I E IA i z Mummar a 1 Mmlmum m g z 5 Frame 25 I Ax 17 W9 mm 5m dzhsrmrhandun magma masumem um npulvllue Input mus mm was 41516an mm m 2m mm mm arm m Remusmluy anal ENGRZEH Lemma 1 Sprmg 2m 49 Error as of Full Scale Output FSO 120 Uncertainty 100 7 interval 80 Idea devic Output of device 0 of full scale 39E 0 20 40 60 80 100 Value ol39measurand 5 of full scale wmzmam m WmWWWxpmmm 2m u mumWm m 2004 NGRZEM Lemure 1 Sprmg IBM 50 Solution Technique for Problems Answer the following questions What am seeking What facts do I know How can these facts be related to what I am seeking What facts are irrelevant What facts are missing 39 Equationtheory graph table etc What is the best format to express my result 39 Graph number table 39 How many digits signi cant gures ENGR201 Lecture 1 Spring 2011 Example Maximum Error pg 1 of2 A mechanicalshaft angularvelocity measuring device tachometer can measure shaft speed in the range of 0 to 5000 rpm It has an accuracy of i5 of full scale output FSO You notice that when the shaft speed is zero the device has a reading of 200 rpm What is the maximum errorthat you might estimate in reading a shaft speed of 3500 rpm ENGR201 Lecture 1 Spring 2011 26 Example Maximum Error pg 2 of 2 39Seeking max error estimate for 3500 rpm shaft rotation 39Facts 1 Input runs 0 5000 rpm so FSO 5000 rpm 2 accuracy 5 of FSO 3 there is a 200 rpm offset at 0 rpm 39Relate facts to goal Accuracy specification defines uncertainty of 1005 x 5000 1250 rpm We will have this error on all measurementsreadings The zero offset of 200 rpm is an additional systematic error 39The error estimate shows that our reading might be as much as 450 rpm too high 200 rpm 250 rpm 450 rpm 39The error could be reduced by subtracting the zero offset from all measurements in this case max is 250 rpm too high ENGR201 Lecture 1 Spring 2011 Developing a Vocabulary see Glossary pg 578 39 Sensor stage 39 Replication 39 Transducer 39 Repet39t39on 39 Calibration 39 Signalcondltlonlng 39 RangeSpan 39 Independent variable Resolution 39 Dependent variable Accuracy 39 Extraneous variable 39 Error Noise 39 True value 39 Systematic errors 39 Interference Random errors Hysteresis Random test strategy You are responsible for these definitions ENGR201 Lecture 1 Spring 2011 27 Lab 1 Overview 39Lab 1 Basic Measurements After performing this experiment you should be able to Determine the accuracy and precision of instruments Measure length using a linearscale ruler avernier caliper and a micrometer Properly acquire and record data using these instruments Analyze data to identify andor minimize error Select an optimum method of measurement for a given length measurement application Construct a histogram see BBVista for more details ENGRZEM Lecture1 Spring ZEI11 Lab 1 Overview D 1 Outside Diameter Thickness 2 inside Diameter 3 Hole Depth ENGRZEM Lecture1 Spring ZEI11 28 Lab 1 Overview Measurements to be made this week ID inside diameter OD outside diameter 9mm 39 instrument M agur ni m Ruler ID OD thickness Washer Caliper ID OD thickness Miorometer Thickness Ruler Diameter Ball Bearing Caliper Diameter Miorometer Diameter ENGR201 Lecture1 Spring 2011 57 Caliper Measurements of Washer ID 9 L Mclaughlin R Smith C Snavely EPED I Fall 200910 Frequency 7 Equally Axis Title 5 with Units N ENGR201 Lecture 1 Spring 2011 29 Summary All measurements have errors They are systematic and random types Calibrations can be used to identify the size of the systematic error and estimate the range of random error The method used to calibrate should mimic the intended measurement as much as possible A test is designed to ask a question The test plan is designed to answer that question Test planning Many factors can influence a measurement The test plan randomization repetition etc should be designed to minimize these effects ENGR201 Lecture 1 Spring 2011 30 ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 2 Statistical Measurement Theory part 1 5 April 2011 T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University ENGR 2m Spring 2cm Lecture 2 Announcements Quiz 1 will be on line this Thursday at 500PM thru Friday 500 PM You will have 50 minutes to complete it The coverage is homework 1 Remember this is independent work Check BbVista for more details There is lab this week please review the lab instructions before attending your lab session Download EXCEL files for lab ENGR 2m Spring 2cm Lecture 2 Focus Chapter4 Probability and Statistics Sections to Cover over two weeks 41 Intro 42 Statistical Measurement Theory 43 Infinite Statistics 44 Finite Statistics 45 ChiSquared Distribution 46 Regression Analysis 47 Criterion for Rejecting Data Points 48 Number of Measurements Required ENGR 2m Spring 2cm Lecture 2 Homework and Readings for Week 2 39Read textbook sections 41 43 Work through all of the author s examples the lecture uses different examples to get you even more exposure to the topics 39Problems from text 43 45 46 49 39Run the Probabilitydensityvi and Runninghistogramvi from the textbook s Student Companion Site You won t turn this in but it is both to reinforce the chapter material and get you more experience using LabVIEW Lab 2 is heavily LabVIEWrelated 39Check BbVista for additional material and information ENGR 2m Spring 2cm Lecturez Intro to Statistics There is a randomness in experimental measurements that lends to Interpretation usmg statistics Errors Systematic these we can adjust to deterministic Random the same system measuring the same measurand Will not always produce the same result 39 Contributions Measurement System resolution and repeatability Measurement Procedure and Technique repeatability 39 Measured Variable temporal variation spatial variation ENGR 2m Spring 2cm Lecture 2 Intro to Statistics Chapter 4 introduces concepts from probability and statistics that will help you make good engineering decisions 39 Reduce raw data into useful results 39 Locate bad data points 39 Determine how many measurements are needed for a given preCIsion Some of the material covered should be familiar 39 Mean and standard deviation 39 Linear regression ENGR 2m Spring 2cm Lecture 2 Motivation In your first lab you will be measuring the diameter of a set of about 10 ball bearings You could imagine this as a handful of parts that you pulled out of a crate of thousands of bearings we 2m Springzmi Lecturez After we measure these 10 parts we will be able to quantify A single representative value tha best characterizes the average of the data set A representative value thatprovides a measu re 0 he variation in the mea ured Set However what we really want to knowis howwell the average diameter of our small sample represents the true average diameter ofthe entire crate We now begin to provide the tools to accomplish this k we 2m Springzmi Lemurez Defining the Problem 39 Let s say we are measuring our 10 ball bearings from the crate of thousands 39We characterize the quality of the bearing through its diameter 39 Those we measure are a sample from an entire population 39 Let x represent a diameter measurement on our sample 39 Let x represent the true value of the diameter for the population 39 The true value of x is the mean value of all possible values of x We don t have the time or money to measure all those bearings We want to estimate x using our 10 measurements 10 values of x ENGR 2m Spring 2cm Lecture 2 Defining the Problem If the number of diameter measurements N is small any one data point can distort our mean value As N gt the effect of any one point is minimized and our measured average approaches that of the entire crate Note only finite samples are practical ENGR 2m Spring 2cm Lecture 2 Estimating the True Value We estimate the true value as x39 x i ux P where x is the true value of population 39 7 is the mean ofthe sample measurements iuX is the uncertainty interval in the estimate at some probability level P P can combine the random and systematic error we consider random only in Chapt 4 ENG 2m Spring 2cm Lecture 2 More Definitions The variable we are measuring x will have a random scatter and we will call it the random variable ENGR 2m Spring 2cm Lecture 2 Example Measurement Variability Measure hot gas in a duct Equipment is well calibrated and operated properly Over an hour a variety of values are measured The gas temperature is the random dependent variable Tabulate the values measured and the number of occurrences of each value Then plot the number of time each value occurs versus its value Plot is called Histogram ENGR 2m Spring 2cm Lecture 2 Results of 60 Temperature Measurements in a Duct Number of Readings Temperature C 1 1089 1 1092 2 1094 4 1095 8 1098 9 1100 12 1104 NwJgtUIUIJgt H H H o inrrudumun in Engineering ExperimentationA J Wheeler and A R 32ml ENGR 2m Spring 2cm Lecture 2 Gas Temperature Measurement Results Number of Occurances O 10876109021092810954 1098 11005110321105811084 1111 111361116211188 Temperature 2 ENG 2m Spring 2cm Lecture 2 Histogram 39 Note that there are bins associated with the number of occurrences 39 The bins can be labeled at their center or on their edges reference text An estimate for the number of bins K required for viable statistical analySIs Is K 187N1 4 1 11 for N 60 Choose conveniently Want gt 5 count for at least one interval reference text We have t a Gaussian curve red to the bins ENGR 2m SpringQUM Lecturez Frequency Distribution Ifwe divide the number of occurrences by the total number of measurements we obtain a Frequency Distribution Plot 39A tabular representation is shown on the next slide This is important in the introduction of the probability distribution function ENGR 2m Spring 2cm Lecture 2 Tabular Frequency Distribution Bin n fl nrlN100 1 1 17 2 1 17 3 6 100 4 0 00 5 17 283 6 0 00 7 16 267 8 10 167 9 4 67 10 3 50 11 2 33 Totals 60 1000 ENGR 2m Spring 2cm Lecturez Histogram to Frequency Distribution to PDF 39The histogram gives a visual indication ofthe likelihood or probabilig of finding a temperature within a certain range 39The probabiliy densiy function PDF px defines the probability that a measured variable might assume a particular value upon any individual measurement 39The PDF comes from the frequency distribution in the limit as N a and 6x a 0 We measure many many items and the bin width gets very very narrow n p x 11m New xeo N ENGR 2m Spring 2cm Lecture 2 Probability Distributions The PDF also provides the central tendency ofthe variable being measured Measures of central tendency include mean variance mode and median The shape ofthe distribution depends on the variable being measured and the process that produces this variable There are several standard distribution shapes that fit engineering data The shape of a histogram of measured results would determine which of the standard distributions fit the data best see Table 42 The statistics for that distribution would then be used to interpret the data We are going to discuss only the Normal or Gaussian distribution ENGR 2m Spring 2cm Lecturez Example Central Tendencies Text M sgs Sent U m lt Data on the number of text messages sent by a student has been recorded Find the measures of central tendency 10 mean median mode 11 OOIO39UIJgtWNH UixleUIbeiwamoow Measures of Central Tendency Mean sumcount 6014 429 Median Value that divides the distribution in half 4 12233344556778 Mode Value that appears most often 3 12233344556778 ENGR 2m Spring 2cm Lecture 2 Central Tendencies Excel Text Msgssm Enter data into table 1a 13 mm ME 14 mumNmbHumbNmmm ExamplePC Make sure that the Analysis TooIBox AddIn is installed Choose ToolsgtData AnalysisgtDescriptive Statistics Data Analysis Expanential Smaathinq west Ywarsamvle rm Vaviantes mien am Fa nl mm smm um Example PC Gmuved Ev D Labews m rm mw 0mm Humans 3 9mm Ranqe Cw New kasheet EN 0 New mkhaak ummaw stausucs D Kth Lavqest Dtag denceleveHmMean 39 D Kth SmaHest mam 5pm an may 2 25 Example Output Descrl ptlve Statistl cs Cu umnl Mean 4185714 Standard Errur 1568675 Medwan 4 Made 3 Standard Dewalmn Z 117786 Samp eVarwance 4 517473 nusws VD 93106 skewness D 290881 Mwmmum 1 Maxwmum 8 Sum GD Cmml 14 Central Tendency Mean A variable that has a central tendency can be quantified though its mean value and variance True mean value central tendency ofa continuous random variable xt having a probability density function px 1 T x 11m J xtdt TamT 0 For any continuous random variable X m x 1 xp xdt For a discrete random variable the mean of X where i 1 2 N is 1 N x ljm x ENGR 2m Spring 2cm Lecture 2 Central Tendency Variance The variance is a measure of the width of the peak of the density function For a continuous random variable the true variance is 2 1 T 2 039 11m J x x dt Tam T 0 Or for a continuous random variable x 0 no oer x x39ipw For a discrete random variable the variance is 1 N 0 0392 11111 Zx x j NsmN 11 1 The standard deviation is the square root of the variance ENGR 2m Spring 2cm Lecturez Infinite Statistics nfinite statistics require us to have an infinite number of measurements 39Obviously not very practical We look at finite statistics where we deal with samples in the next lecture Let us now focus on the Gaussian or normal Distribution based on infinite statistics ENGR 2m SpringZDll Lecturez Normal Distribution Measured value units 39The normal distribution describes the results of many engineering measurements 39 Bell curve 39As we take many many measurements the results center symmetrically around the 0 Measured uara me a n 1 2 3 4 5 a 7 a 9 10 Measured reading number Apparent measured average a Scatter due T in random errur Test systematic enor True or known value ENGR 2m SpringZDll LectuVEZ Normal Distribution Graphical Form The shape of the normal distribution is determined by the true mean x and the variance 02 or standard deviation 0 Mean 2 The curve is symmetric about the mean 2 for this example Note the symmetry and the change in the i u i 7 J distribution shape as o Rzintlnm viirialslc i increases 4i ENGR 2m SpringZDll LemureZ Normal Distribution Analytical Form pg has 75 02 39 The probability has its peak when x equals the true mean x consider exp0 1 The most likely probable measurement result is x The value at the peak depends on a As 0 increases the peak gets flatter Because ofthe squared term in the exponential the distribution is symmetric about the mean ENGR 2m SpringZUll LemureZ Normal Distribution Examining Probabilities Using the normal distribution you can predict the probability that a measurement will fall within a certain range The probability PX that a variable X will fall within x 1 6X is P 39 6x S x S x39 6x Lifp xdx px is a complicated function and we can ease the integration by doing a change of variable ENGR 2m Spring 2cm Lecture 2 Normal Distribution Examining Probabilities Let3 xx o Normalizing the deviation for a given x to the standard deviation From this we get dX 0 dB Let Z1 X1 x 2 replaces x as the variable The probability over the interval is now ricer2w ENGR 2m Spring 2cm Lecturez P 21S Sz Normal Distribution Examining Probabilities 39Since pX for the normal distribution is symmetric about X we can simplify this relationship a little more 39The probability over the interval is now 1 2 2 1 z EJ e zd 2E390 g zd 39Value in brackets known as the normalized error function Remember that factor of 2 in front The normalized error function is available in tabular form and through spreadsheet functions ENGR 2m Spring 2cm Lecture 2 Example using Tables Probability reading is between 9 and 10 Problem Results of a test follow a normal distribution having a mean of 100 and a standard deviation of 1 Find the probability that a single reading is between 9 and 12 Method We will solve this by changing the variables to z to specify the interval for a normalized Gaussian pdf and the Normal Error Function Table 43 pg 127 in text ENGR 2m Spring 2cm Lecturez Example using Tables Probability reading is between 9 and 10 3 2 1 l 1 V We need to nd the area under the curve as shown Break into two parts from 1 to O and O to 2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa l l ENGR 2m Spring 2cm Lecture 2 37 Example using Tables Probability reading is between 9 and 10 Due to the symmetry around 2 0 the area under the curve quotm from 1 to 0 is the same as 7 7 x fromOto1 a g ngExperfmentatmmAl wmummm Garm Use a table to nd the area under the curve Table 43 Probability Values for Normal Error Function uuuuuuuuuuuuuuuuuuuuuuu 22 38 Example using Tables Probability reading is between 9 and 10 P 1 s z s 0 PO s z s 1 03413 39ncgar39ezgggggrgiznty PO s z s 2 04772 P 1 s z s 2 03413 04772 08185 or 8185 Note to get area from 1 to 2 we add the area from 0 to 1 to that of the area from 0 to 2 ENG 201 Spring 2011 Lecture 2 Example Using EXCEL Probability reading is between 9 and 10 Excel can help solve this problem without resorting to change of variable and use of a table NORMDISTxmeanstandarddevcumuative 39When cumulative TRUE the formula is the integral of the normal distribution from negative infinity to x 39NORMDIST12101TRUE NORMDIST9101TRUE 390819 39Note we subtracted the integral from negative infinity to 9 from the integral from negative infinity to 12 yielding the integral from 9 to 12 ENGR 201 Spring 2011 Lecturez Example Using Labview Probability reading is between 9 and 10 Integral Frum emf to 12 El mean H 39 397 our 1 Continuous Integtral from 9 to 12 CDFvi gt E Integral from emf ta 9 El LabVIEW can solve the same problem graphically See file posted in Week 2 ENG 2m Spring 2cm Lecture 2 41 std Continuous CDF VVI Probability of Deviation from the Mean and Standard Deviation Memorize std and area percentages 44 rl n3 x 73t1 x chr x rv x l v u7 x 2a x 3a z 683 probability that any measurement will fall within one standard deviation of the true mean in an environment where the deviation from the mean value is totally influenced by random variables ENGR 2m Spring 2cm Lecturez 42 Summary 39 When we take many in nite measurements the results gather around the central tendency We describe the central tendency through terms such as the mean mode and median 39 Probability and statistics can be used to predict the performance of measurements 39 The results of many engineering measurements follow a probability density function called the normal distribution 39 The position and shape ofthe normal distribution can be speci ed through the true mean and the standard deviation or variance ENGR 2m Spring 2cm Lecture 2 Appendix Central Tendency Computations with MAC ENGR 2m Spring 2cm Lecture 2 22 Central Tendencies Macintosh Analysis Toolbox AddIn not available in Mac Office 2008 therefore must use alternative Possible free alternative is StatPlusmac LE by AnalystSoft mentioned in the Microsoft help when you look for the Analysis Toolbox Must download and install application ENGR 2m Spring 2w Lecture 2 45 Descriptive Statistics StatPlusMac Create data table 3 3 We ll use Excel file from previous exam W l it Open it 3 Open StatPlusmac Under Statistics m i l 39 Select Basic Statistics a J Statistics 77 7 7 7 7 7 7 7 Z Labelsin rstmw 39 Camel r OK ENGR 2m Spring 2w Lecture 2 46 23 Descriptive Statistics StatplusMac Click on the box to 7 the right of madam i Ranges Wm til Go to Excel and select the data you wish to be analyzed Click on the Range box again to move am nascme mm m c Dimmum back to StatPIus mm i 39 Click OK ENGR 2m Smmg IBM Lectuvel 47 39 SaPlusMaResults4xlt StatPlus returns an w luvquot i it i 14 i1 a ir m39i Excel file 0 A i a c u E mmmwm at 2 SINCE Yul EBIhl O 39 7 m umissx NOTE In the ri u g 23 free verston 2234 KHZZMWWEW 332 2 3 MWWM was 00 fldence quot33 stitftitl tlii t w 3933 level is set at 32335quot 2 5 WSSHMGHI l Mm smash Mmmzmv 5mm 3 mn will 25y 4m 3 3115 WINMm 7502l 55 is p as 65175 MAD 5 a w 24 ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 6 Temperature Measurement T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University saZDM ENGRZEM Lecture E Spring 2W 1 Midterm Performance Grade Histogram ranns251 Isszusm Count 259 nnsnis7s 525mm Average 811 m an 2551 Med r mum 11050 3253 Minlmum 400 lisrsz39m Standard Deviation 1363 SznrsRaME I i FrEauaniV saZml ENGRZEH Lecture E Spring mm 2 Reading and Homework Reading all of Chapter 8 Temperature Measurement Review Examples 85 86 87 88 ice baths are not always available or practical 811 Pay special attention to Section 85 Thermoelectric Temperature Measurement Homework Problems 88 89 815 817 and 818 53201 1 ENGR201 Lecture 6 Spring 2011 53201 1 Additional Announcement Quiz this week on Uncertainty analysis and Propagation of Error Open Thursday 500 PM through Friday 500 PM Coverage Lecture 4 5 reading lecture notes and problem sets ENGR201 Lecture 6 Spring 2011 Measuring Temperature Very common parameter in engineering body temperature fever monitor chemicalbiological reaction semiconductor wafer processing air temp HVAC environmental curing temperature epoxies concrete many many more applications 532011 ENGR201 Lecture 6 Spring 2011 History of Temperature Measurement Based on zeroth law of thermodynamics two systems each in thermal equilibrium with a third are in thermal equilibrium with each other Long history associated with some great names Galileo 15651642 barothermoscope temp and pressure Newton 16421727 fixed points freezing water and human armpit Fahrenheit 16861736 180 units between frozen and boiling water at atmospheric pressure successful Hg thermometer Celsius 17011744 100 units between boilingfreezing water inverted scale ie boiling was 0 Linnaeus 17071778 brought Celsius scale to present form 532011 ENGR201 Lecture 6 Spring 2011 International Calibration Standards Table 81 Temperature Fixed Points Defined by ITS90 Figiola amp Beasley VWey amp Sons 2006 LVE liquid vapor equilibrium SLE solid liquid equilibrium 011 NGR201 Lecture 6 Spring 2011 7 General Measurement System Calibration Signal conditioning stage 5 rial Sensor stage g Transducer path stage Prusess Control stage 532011 ENGR201 Lecture 6 Spring 2011 8 Central signal Temperature Sensor Parameters 39 Temperature Range 0 Intrusive NonIntrusive 0 Stability and Corrosion 0 Method of ce Resistan easurement 39 Sensitivity Technique de ends on Accuracy the Sensing ement 0 Linearity of output 39 Response Time 39 Cost 39 Direct Indirect ESZEIII ENGRZDI Lecture 6 Spring ZEIM g Sensing Element what property responds to Temperature Mechanical Property Sensors Density thermal expansion FluidExpansion Temperature Measurement Devices Bimetallic Temperature Measurement Devices Changeof State Temperature Measurement Devices Electrical Sensors Thermocouple Temperature Measurement Sensors TC Resistance Temperature Devices RTD Precious Metal RTD Semiconductor Thermistor Diode Optical Sensors Photometer Infrared Temperature Measurement Devices IR ESZEIII ENGRZEII Lecture E Spring ZEIII IEI Mechanical Property Sensors MctaA Bimetallic Element s abc Fluid Expansion d Change of State e eg liquid crystal M EEEEH 7 quot a n d 532011 ENGR201 Lecture 6 Spring 2011 11 Bimetal Element PrInCIple ofoperation me Difference in rate ofthermal WSW 1 260 L a expansion between different metals Characteristics Portable Do not require a power supply Not as accurate as thermocouples or RTDs Metal 1 Indicator metal 2 Mechanical linkage to dial or relays do not readily lend themselves to temperature recording Heat Added 532011 ENGR201 Lecture 6 Spring 2011 12 Bimetal Element Some Typical Applications Airducts boilers engines enclosures fire boxes plumbing piping refrigerators and many others Accuracy i2 of full scale Response Time Approx 1 minute M etal 1 Example Hotwater heater Metal 2 has a bimetallic sensing element telling the deVIce to turn on and off Heat Adda ESZtlll ENGRZEH Lecture E Sprlng Zull l3 Fluid Expansion Principle of operation Expansion of a liquid or gas with temperature Mercury organicliquid Alcohol Does not require electric power Stable even after repeated cycHng Range 40 to 400 C Very accurate 1100 C under carefully controlled conditions ESZtlll ENGRZEIl Lemure E Sprlng Zull M 53an Fluid Expansion Data are hard to record Cannot make point measurements Mercury is considered an environmental hazard Example Standard glass thermometer Fluid within the reservoir expands with temperature pushing up the capillary tube ENGRZEM Lecture 5 Spring mm H 53an Changeof State Principle of operation hange ofstate of material t e E i E i is Forms labels pellets crayons lacquers or liquid crystals LC appearance changes once a certain temperature is reached 395 Low Accuracy u I56 Slow Response Low Cost lrreversible One Time Use ept LC ENGRZDl Lecture 6 Spying 2m 15 Changeof State Examples Temperature Sensitive quotHE E E i i 3 Palm quot quot Temperature Indicating Labels Liquid Crystal Based o NonReversible Labels 1 AL Device damage indicators 760 Crayonsi1 d chal mark when its rating is reached the crayon leaves a liquid sme r 53an ENGRZEM Leclure a Spring mm 17 Electrical Sensors Resistance Temperature Devices RTD Precious Metal RTD Semiconductor Thermistor Thermocouple Temperature Measurement Sensors TC 532Ull ENGRZDl Lecture 6 Spying 2m 18 Relative Sensitivity Range The39 39iis39tur v Lv Iv Voltage or resistance Tlrammcouple saZEIM ENGRZEH Lecture E Spring IBM 19 Resistance Temperature Detectors RTD Principle of operation the electrical V resistance of a material changes as its r temperature changes R p203 A 39 RTD Precious Metal Pt resistance rising more or less linearly with temperature Range 200 C to 600 C Good sensitivity Uses external voltageresistance sensor Elle saZEIM ENGRZEH Lecture E Spring IBM In Precious Metal RTDs Plus Veryaccurate ery linear response compared to TC range Minus Expensive fra ile Complicated circuitry slow response Polynomial expression RO reference resistance at To with coefficients based on material constants 2 R R0EaT T0 T T0 Over short temperature range may be truncated to R R01zxT7T0 In general RTDs are not linear even though Table 82 lists values ofa you must consider the range ENSee g 86 in t xt e ESZEIl GRZDl Lecture E Spring ZEIll Zl Example of circuits for RTD Lem A M u t la i 7 s W RH ll K39H W Rm u Lle a Lend ll ik39ihlll ml llvl 2 wire and 3wire Wheatstone Bridge circuits 2 wire adequate if lead resistance is low and great accuracy is not required 3 wire used when lead resistance cannot be neglected saZEIM ENGRZEIl Lecture E Spring IBM 22 Thermistors thermally sensitive resistors Ceramic semiconductor the resistance drops nonlinearly With temperature rise Plus Very sensitive inexpensive immediate time response relatively simple circuit may be encapsulated for corrosive or abrasive environments Minus Low linearity small temperature range response changes with age very difficult to match characteristics when replacing sensors 532011 ENGR201 Lecture 6 Spring 2011 23 Thermistors thermally sensitive resistors Two Models Used Exponential SteinhartHart Equation RRexpl 7ojl AB1nRC1nR3 R in ohms T in Kelvin A B and C are the Steinhart Hart coefficients and depend on the type and model of thermistor and the temperature range of interest R in ohms T in Kelvin 3 depends on material temperature construction 532011 ENGR201 Lecture 6 Spring 2011 24 39hermistors Voltage and resistance normalized by value at 25DC7 2t irciiii outputvohagc i39hci mismr rcSlslanCC 20 40 60 80 Tenlpunilure T b Even though a thermistor is highly nonlinear a fairly linear response can be achieved as In a voltage divider circuit as shown 532mi ENGRZEH Lecture 6 Spring 2cm Thermocouple TC Principle of operation Two strips or wires made of different metals are joined to form two junctions A change in the temperature between the twojunctions induces a change in electromotive force Voltage as measured by the volt meter Measured Voltage due to temperature difference 532mi ENGRZEH Lecture 6 Spring 2m 26 A little more complicated Voltage is result of several cross effects Linear Effects emf voltage difference produces current flow while AT induces heat transfer 1 AT produces AV Seebeck Effect 1821 4877f if open 2 AV produces heat transfer Peltier Effect 1834 Quantity of heat in addition to PR that must be removed from junction to keep it at constant temperature Q 7Z39AB 3 AV and AT produce extra heat transfer Thomson Effect 1851 QaUITI T2 For TC all 3 effects may be present and contributing to overall emf of circuit 532011 ENGR201 Lecture 6 Spring 2011 27 Thermocouple Attributes Advantages Wide temperature range with appropriate selection of metals Fast response time if you use thin wires Point measurement Standard junctions require little or no calibration Concerns Requires high input impedance of signal conditioning stages Low voltage output Reference junction wiring not as simple as one might hope Very nonlinear response 532011 ENGR201 Lecture 6 Spring 2011 28 Comments Envtrnnment Bare Wire IRON CONSTANTAN COPPER 39 Fe I magnetica NICKEL Cu Ni LonquotgmE CtaanUxtdtztn and 1nert NthEL ALUMINUM NtA BHROMEGA NICKEL CHROMIUM NiCr Lumted Use 1n acuum or nedumng wue Tentpem ura Range Must Panta Dmbrntiun CONSTANTAN COPPER COPPER Cu NICKEL GurNi 0 391 1 rt cocwgggggmt Lgm t u in mw 727013 43am CHFQMIUM NICKEL estemawtggest 745410 75373 N rCr mm Der Dame OMEGAP OMEGANquot 9 74 45 NIQROSTL NISIL Mm 3 1017513 Nzms NiSiMg 532011 ENGR201 Lecture 6 Spring 2011 29 Typical TC Output amp Range 61 Note S type TC Has large range But low output swing Output mV 4 C 0 500 1001 1500 2001 39l cmpcratut39c CC 532011 ENGR201 Lecture 6 Spring 2011 30 Thermocouple Calibration a hulCtlUH Cutal c mts ul39 the calibration equation for lumpemm quot J llwrmmunpleu TJ L1 HEW 13153 megj 7 ul39 Eh l3 for l U in IUV T in C 4 1395an En g IImV llm39 39 En 39 12910111V L ZlII C 150 M031 3 urC m I ll r1 1052 m s gt m 19 izs v 1039 A m 71225 L115 3 2um 204 gtlt1J39l E 7393 7 tirn 217 H 1 0150000 A mquot2 3 q z 5 rm 303 3 w In E 71725 L371 3 gt1 lll l 75131513 7 1H4 75344 M x 104 33 n x in 321 1Wquot 5099 7390 x 10 Temperature C U 5320 1 l ENGRZO Lecture 6 Spring 20 l 3 Fundamental Thermocouple Laws pg 334 Law of Homogeneous Materials must have a junction of dissimilar materials to generate thermoelectric current Law of Intermediate Materials if any two junctions of the same two materials are at the same temperature no overall thermoelectric current will be produced The algebraic sum of the emf in a circuit composed of any number of dissimilar materials is zero if all of the circuit is at a uniform temperature MaterialA Material B 0 0 Material B Junction 3 Junction 4 T3 T Measuring device A Law of Successive or Intermediate Iemperatures IT a thermocouple pair produce emf12 when connected between T1 and T2 and emf23 when connected between T2 and T3 it will produce emf13 emf12 emf23 when connected between T1 and T3 532011 ENGRZO l Lecture 6 Spring 2011 32 Thermocouple TC Summary We are going to measure a TC voltage and relate this backto temperature Metal A Copper wire wire gt DVM DVM digital J volt meter Metal B W DVM terminal 532011 ENGR201 Lecture 6 Spring 2011 33 TC Junction Issues Metal A Copper wire wire g g 2 39 quot S a DVM Junction Metal B a Win DVM Cl39llttlldl This setup has 3 junctions so the voltage reading Is a function of 3 temperatures 532011 ENGR201 Lecture 6 Spring 2011 34 TC Junction Issues solution to multiple junctions Milll lL ll l W tappu I 39l l smug ll Junction 0 j M l x l l l I xx lvlul ll M lul lt39ml H 2 quotquot 139quot ercn39m39r isml mmlmm N erciunuc junction Il lullrem minimum The voltages generated at the Cu quot quotquotquotk39 to Metal A and Me a Ato Cu This circuit is ele dtrically equivalent junctions cancel each other if at To the one on left see Fig 817 in text same temperature same box 532011 ENGR201 Lecture 6 Spring 2011 35 Example 96 WheelerGanji A set of two type K thermocouples measures a voltage difference of 301 mV The coolerjunction is known to have a temperature of 300 C Find the temperature difference between the two junctions Chrome T1 T2 300 C 301th Alums Alumcl 532011 ENGR201 Lecture 6 Spring 2011 36 First Cut Incorrect Analysis Method Use a TC table of Temp vs Voltage or calibration for Type K to find AV 30100 mV AT722 C This would imply that the hotterjunction is 1022 C 722 C300 C This is not quite correct since 1 the table is based on two junctions one at 0 C ice bath and 2 the table is non linear note there is no reference junction seen in circuit 532011 ENGR201 Lecture 6 Spring 2011 37 Doing the computation correctly 0 TC calibrations are VERY nonlinear and standardized with respect to 0 C reference 0 Must analyze as two separate circuits as shown below insert two reference junctions one for each TC with each contributing the same voltage with opposite sign Alums Chrome hromcl T T3 300 C Alumc Alums 532011 ENGR201 Lecture 6 Spring 2011 38 Correct Analysis Technique Alumel Thromcl T T3 2 300 C 0 301 mV Alumcl Alumcl With two reference junctions the 30100 mV measured at the open circuit is the difference between the voltage produced by the 0 T1 C temperature difference x mV on the left and the 0 300 C temperature difference 12207 mV on right Hence x 12207mV 30100 mV or x 42307 mV Therefore T1 1027 C and AT 727 C 1027 C 300 C 532011 ENGR201 Lecture 6 Spring 2011 39 Electronic Cold Junction Rather than using an ice bath an electronic circuit can be used to mimic the reference junction Some instrumentation will have this built in so you can directly connect a TC Cold Metal A Wire Junction DMM Metal B Wire 10 TV per C Copper Wire 532011 ENGR201 Lecture 6 Spring 2011 40 Digital Temperature Measurement A variety of sensors are available that can be addressed digitally by a microprocessor Maxim D818B20 diode sensor Unique 1V re Interface Requires Only One Port Pin for Communication Can Be Powered from Data Line Measures Temperatures from 55 C to 125 C 67 F to 257 F Cost per unit 4 l05 C Accuracy from 10 C to 85 C Sensors Applications HVAC environmental controls temperature monitoring systems inside buildings equipment or machinery and 1Wire interface process monitoring and control systems pp 532011 ENGR201 Lecture 6 Spring 2011 41 Actual Measurement Scenario Hm Radiation r to S no T cooler duct wall 39l smpci atui39c sensor D gt 1 Conduction 10 cooler wall Ts To Even though the sensor is self contained we need to understand what it will measure when it is placed in the system The next slide shows additional detail of what we must consider 532011 ENGR201 Lecture 6 Spring 2011 42 Convection I to sensor Cool duct wall Complications Energy Losses SurfaceTemperature Conduction wire conduction qcond kA dT dx Loss to environment gas phase convection qm hAiTw Tp Radiation losses to walls qmd KgUSB T T3 Recovery losses high speed flows V2 Ake 7 0C AT md 532011 ENGR201 Lecture 6 Spring 2011 43 Optical Sensors do not perturb system Non contact do not change system Pyrometer Infrared Temperature Measurement Devices lR Thermal radiation from an object is related to its temperature and has wavelengths from approx 10397 to 10393 meters 532011 ENGR201 Lecture 6 Spring 2011 44 Blackbody Monochromatic Emissive Power Wavelength distribution for various temperatures mm mm mun Total Power Radiated w T25imlt Eb oT4 I 3 mun Stefan j Boltzmann I Law M l in mi leenf li mun Blackbody surface absorbs all incident radiation and as a result emits radiation in an ideal manner 532011 ENGR201 Lecture 6 Spring 2011 45 Relating to Frequency Monochromatic WM Emissive Power visim light 1 7 139 2qu K lllllll Nut 01 5 expC21Te1 W 7 Ebi lllllll E AlkWmzm m power emitted in a slice of wavelength AA l mu Wuwlcngni mini 532011 ENGR201 Lecture 6 Spring 2011 46 Pyrometer filament heat source Measurement of temp By color saZml ENGRZEH Lecture E Spring IBM 47 IR Devices Noncontacting devices 4 4 Surfaces emit in the IR spectrum qr 0 EHA7 Ta Infer temperature by measuring the thermal radiation emitted by a material Surface emissivity dependent Range 20 C to 870 C Accuracy 2 C a Medical ear thermometer with small range has higher accuracy saZull ENGRZEM Lecture E Spring IBM 48 53201 1 Lab This Week This week you will learn how to make a measurement using LabVIEW After the experiment you will be able to set up the analog to digital device to take a voltage measurement take measurements over a period of time plot data on the front panel write the results to a file to be manipulated later Report on the behavior of a power supply and DA during measurement We will make use ofthese Vl s in the following lab Measuring Temperature and in the speedometer project ENGR201 Lecture 6 Spring 2011 49 53201 1 Exercise I need to monitor the temperature of a silicon wafer oxidation process In this process Si is converted to SiO2 at high temperature 1100 C in the presence of oxygen and some moisture Give at least one candidate ANSI Code for a TC to make this measurement ENGR201 Lecture 6 Spring 2011 Si Oxidation 1100 C in moist oxygen Environment Wire A IRON CONSTANT3N COPPER 4309510 quot magnetic 69553 ALUMEGA lillCKEL CHROMIUM ALUMNUM 7645810 NiCr NH 54886 CONSTANTAN COPPER COPPERV Cu NICKEL Cu4li CHROMEGA NICKEL o l gl l CHRQMIUM MCKEL NiC Cu Ni Alteinative OMEGAPquot OMEGAN t 4345 NichosiL NlSlL to 17513 NiClSI NiSiMg at High Temps 532011 ENGR201 Lecture 6 Spring 2011 51 Si Oxidation 1100 C in moist oxygen 2 Comments ANSI Code fggzc mbmaffnad Environment Mimi BareWire p 9 Reducing Vacuum Inert Limited use in oxidizing at a J Iron Constantan high temperatures Not mom 1200 DC 346 to 2193 F recommended for low tem eratures Limited use in vacum or 270 to 1372 C K Chromel Alumel reducing Vlnde temperature 454 to 2501 F range Most popular Mild oXIgizmg39 r39 ducing vacuum or Inert Good where a T Copper Constantan moisture is present Low 270 to 400 DC 454 to 752 F temperature amp cryogenic pplication Oxidizing or inert Limited use in vacuum or reducing 270 to 1000 C E N39Chmme 0 Stanta Highest EMF change per 454 to 1832 F degree N Nicrosil NISIL Alternative to Type K More 270 to 1300 C NiCrSi NiSiMg stable at high temps 450 to 2372 F 532011 ENGR201 Lecture 6 Spring 2011 52 ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 7 Calculations for Lab 5 and Measurement of Angular Velocity T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University 5102011 ENGR201 Lecture 7 Spring 2011 1 Announcements On Line Quiz this week same process as before Final Exam Information Date June 72011 Time 13001500 Location unspecified Ifthere is interest Dr Chmielewski will hold a problem review session before the final 5102011 ENGR201 Lecture 7 Spring 2011 2 2 Reading and Homework 1 Reading Section 122 pp 522 525 in text Go to Wikipedia and look up shaft encoder then read about incremental rotary encoders In anticipation of your cyclometer project read Slosntext Help for DAme Base Create Virtual Channel Cl Cnt Edges Vl Counter Input Count Edges function Cl Period Vl Counter Input Period function 5102011 ENGR201 Lecture 7 Spring 2011 3 Reading and Homework 2 Problem 1213 from the text A magnetic pickup is used to detect the shaft speed of an experimental high speed compressor A gear with 12 teeth is used to excite the sensor lfthe shaft is turning at 18000 rpm how many positive pulses will be created per second By what number must the number of cgunts in a second be multiplied to produce a result in rpm A strobe based tachometer is shining at a disk rotating at 2400 rpm The disk has one circumferential marks for speed measurements a How many marks will you see ifthe strobe flashing rate is 1200 per minute b Repeat part a for flashing rates of 800 2400 4800 and 7200 flashesmin 5102011 ENGR201 Lecture 7 Spring 2011 4 Format for this lecture Background Theory to perform Time Constant computations for Lab 5 Measuring Angular Velocity Preparation for final project cyclometer 5102011 ENGR201 Lecture 7 Spring 2011 5 Lab 5 Capturing Temperature Measurements from a Thermocouple Lab Goals Build a subVl for voltage to temperature conversion Measure plot and record temperature measurements from a Type K thermocouple Correct the measured voltages with a calibration curve polynomial Find the time constants of the thermocouple cooling curves 5102011 ENGR201 Lecture 7 Spring 2011 6 Capturing Temperature Measurements from a Thermocouple The measurement Insert the TC into a power resistor Apply voltage to cause the resistor to heat up When at steadystate temperature remove the TC and allow it to cool in free space Record temperature data with time while the TC is cooling the result is an exponential decay similarto the graph on next slide 5102011 ENGR201Lecture7 Spring 2011 7 Sensor Cooling Curve mmuen ln Temperma a temperature 1mm 5102011 ENGR201Lecture7 Spring 2011 8 Determining a Time Constant In your lab this week you are asked to find the time constant of a thermocouple cooling cune The time constant is the time when the argument ofthe exponential is equal to 1 At one time constant the exponential is e391 and has a value of 0368 3 sig figs In order to find the time constant we must first understand the differential equation and its solution for the thermocouple s cooling curve 5102011 ENGR201 Lecture 7 Spring 2011 9 Some background on Differential Equations from ENGR 232 Given a differential equation ofthe form V ay 19 y0 yo The solution is b b at men lg Cl Cl As will be seen in later courses there are many ways to obtain this solution Note follow sign convention to apply the above formula 5102011 ENGR201 Lecture 7 Spring 2011 10 Differential equation describing Thermocouple cool down From the lab write up we nd that equating the expressions for the rate ofenergy change Eq 2 and 3 produces a linear first order differential equation This equation de nes a relationship between temperature and time dT mxcprths xT Too dT h x A S x T Too dt m x CI Note tau is called the dT 1 time constant gtltT Too T00 is roomtemp dt 139 To is the initial temp 5102011 ENGR201 Lecture 7 Spring 2011 11 Solving the differential equation We identify the terms a b and y d T J x T TOO dt 139 Hence the solution is t i T00 is room temp TtTcOTOTooe Tolstheinitialtemp The cool down curve is the form of above equation This is the model we expect our data to fit Will it fit exactly Explain 5102011 ENGR201 Lecture 7 Spring 2011 12 Some methods to determine time constant from experimental data We could find where am e391 0368 occurs We could do an exponential curve fit using Matlab or Labview or Excel We could convert the exponential curve to a linear curve and then find the slope Tmnpcrml c 5102011 ENGR201 Lecture 7 Spring 2011 13 Finding a Time Constant by conversion to linear curve We will focus on the latter approach Since we know something about analyzing linear cunes We can use a least squares fit to determine the parameters of a line that fit best Least squares is very robust in noisy environments or when some data is in error The slope will be the negative reciprocal of tau 5102011 ENGR201 Lecture 7 Spring 2011 14 Finding a Time Constant convert to linear curve Rearrange the solution of our differential t equation Ta TOO TO ToO 67 Tt Tw j T0 Too Take the natural log of both sides Inf Tow rTO Tw 1 5102011 ENGR201 Lecture 7 Spring 2011 15 Finding a Time Constant convert to linear curve We note that III 04001 kT0Too 1quot Has the formy mx b here b 0 y is the left side expression and m 1tau x t So plotting y vs x will yield a straight line with slope m which can be solved for tau 5102011 ENGR201 Lecture 7 Spring 2011 16 The process for EXCEL Start with a table of time t and temperature Tt data Create another column yt which is IllTurn TO Tw T is room temp TJ isthe initial temp These should be constants that are easily changed Plot yt vs t which should result in a straight line You will see some variation due to real data Find the slope ofthe line and then determine the time constant time constant 1slope To find the slope of the line is to use the EXCEL function LINEST this uses a least squares method good for data that may have noise or errors Using LINEST is not as straightfonNard as it may appear 5102011 ENGR201 Lecture 7 Spring 2011 17 Graphical Illustration see postgdwlllatlab file m Plot of experimental data vvhichjust happens to be yt21002exp12 t 7 Plot of nyt298 vs t e Slope of hence Tc 2 l l X1 237 y11185 39 X2 4793 y2 2396 39 l 2 3 4 5 d 5 7 3 s3 1 5102011 ENGR201 Lecture 7 Spring 2011 18 Additional comments A good way to validate your result is to use the computed time constant to plot the exponential solution on top of your measured data For the future when linearizing an exponential of the form y mx b we may need to include the b Considerthe solution and its n t Vc t Voe r 1 Mn 0 1nVo 139 5102011 ENGR201 Lecture 7 Spring 2011 19 Measuring Angular Velocity Techniques for measuring angular velocity Introduction to the ENGR 201 Project 5102011 ENGR201 Lecture 7 Spring 2011 20 Measuring Angular Velocity Rotation Why is this ofinterest In many mechanical systems the angular velocity of a rotating shaft is of great importance Safety AntiLock Braking Systems Electronic Stability Control V nd Turbines Process equipment printing press rolling mill for powder processing photoresist spinning for semiconductor processing centrifuges Robotics movement ofan arm or distance travelled as wheel rotates 5102011 ENGR201 Lecture 7 Spring 2011 21 Measuring Angular Velocity Rotation Some Methods Electric Generator Tachometers Magnetic Pickup Stroboscopic Tachometer Reed Switch Photoelectric optical interrupter optical encoder 5102011 ENGR201 Lecture 7 Spring 2011 22 Motivation V nd Turbine Safety Controller Starts the machine at Wind speeds of 8 to 16 mph Shuts off machine about 55 mph Turbines do not operate at higher Wind speeds because they might be damaged vvvvvv1 Eere Energy guvWindandhydruWindihuvv html 5102011 ENGR201Lecture7 Spring 2011 Electric Generator Tachometers A small electric generator can be attached to a shaft and the voltage output monitored Certain types of DC motors can provide an output voltage proportional to shaft speed Generator Direct connection or VOItmeter reduction gear or DAQ 5102011 ENGR201LectUi e7 Spring 2011 24 Electric Generator Tachometers The generator can be either direct current dc or alternating current ac For do you would normally be measuring the voltage and calibrating voltage to angular velocity For ac you would measure the frequency ofthe voltage produced and calibrate this to angular velocity 5102011 ENGR201 Lecture 7 Spring 2011 25 Tach Concerns Ripple Variations on a expected constant value must be small Linearity We must see a linear change in output with rotation speed Stability Tach output must be predictable overtime Temperature Sensitivity The operation environment must not cause a major change in performance o Tachometer gradient Measured in units such as volts1000 RPM need uncertainty Inertia and Friction Adding a tachometer can change the dynamics of the system it is measuring 5102011 ENGR201 Lecture 7 Spring 2011 26 SERVOTEK Tachometer BSeries Tech Generators Model Number man RPM max Face annn Flange Face Flange 5 u m 205 ElEIZEIM ENGRZEH Leg gmgmgwmmmpmgn HT 27 GE Tach Generator Particularly suited for heavy duty applications such as steel paper and textile drives and other industrial machinery utilizing adjustable speed drives Features and Bene ts C face or foot mounting Ball bearings Exceptional line in direct proporti Single or double shalt extensions Operational in either rotation or on a reversing device r voltage outputs on to speed an E RODUcTsauu innan ENGRZEH Lecture 7 Spring 2cm Magnetic Pickup Simple inexpensive Magnet wrapped with fine insulated wire Used in close proximity to teeth of a rotating ferrous gear Sensor requires no external power Non contact measurement as 39g compared to tachometer in j alpha ElEIZEIM ENGRZEH Lecture 7 Spring 2W Magnetic Pickups one cycle E CBN wsinNrwt E output voltage CEprop01tionajty constant Ni number ofteeth a angular velocity ofwheel Each gear tooth distorts the cores magnetic field as it approaches and departs The change in field induces a signal in the coil which can be measured externally Peaktopeak voltage can easily be tens of volts ElEIZEIM ENGRZEH Lecture 7 Spring ZEIM Magnetic Pickups Pulses from the sensor can be sent to a counter that produces a digital result or to a frequency to voltage converter that can use an analog or digital indicator It may be necessary to perform some signal processing to modify the Magnetic Sensers Cnrpnratin Shape of the pulse WWW 51 EIZEIM ENGRZDl Lecture 7 Spring 2mm 31 Magnetic Pickups Automobile Applications nusm wanting light In able fm auln hnns H mm nusmu nllllnl nnnnle Snark reiam engine nnmne an whanl 2 1m nme same speed sensn b quotIran Dnslllnn sensn mnnula r 3 1 Realhlake rm allnms nusum hu laullt mn ulalm nsn numn nmne lung an aim lines nmme A n r reiner Tnmme Dnslllnn sensm WWW aalcarcnmlibraQabsl mm 51 EIZEIM ENGRZDl Lecture 7 Spring 2mm 32 Magnetic Pickups Wheel Speed Sensor Signal Win wnul Speed s r San wwwaa1 car comlibraryabs1hlm 5102011 ENGR201 Lecture 7 Spring 2011 Magnetic Pickups 2 J commonswildmediaorgwikilmageABS www o ularmechanics comhow to centralam o mo Ive2265091hlm 5102011 ENGR201 Lecture 7 Spring 2011 Magnetic Pickup Computational Example Each time a tooth of a toothed wheel passes a magnetic pickup a single positive pulse will be generated If an electronic counter counts and displays the number of pulses that occur during a 1 s interval how many teeth must the toothed wheel have to give a readout in revolutions per minute 5102011 ENGR201 Lecture 7 Spring 2011 35 Example let n teeth on gear let N revolutions per minute of gear We want to have N counts in one second counts counts rev 1min N n x N x sec rev mm 60sec n 60 teeth This system counts the pulses that occur every second and displays rpm based on value of n 5102011 ENGR201 Lecture 7 Spring 2011 36 Stroboscopic Tachometer Mark on I pm by Flashing ligl Y RPM display Rotation 8340 Light flashing frequency adjustment Pullcy 5102011 ENGR201 Lecture 7 Spring 2011 37 Stroboscopic Tachometer Frequency of flashing light is adjusted until mark appears stationary Problem in that solutions can be found at integer multiples and 1 quotquot Vl11ll I I f I li wlllll mu submultiples of the rotation frequency Operators must follow proper procedures see text win it wlnm woman Jrm mm Nonco ntact advantage 5102011 ENGR201 Lecture 7 Spring 2011 Stroboscopic Tachometer FlashSpeed Rate 100 to 10000 FPMRPM 0 Accuracy 1005 of reading 1 digit Duty Cycle 5 to 30 min Sampling Time 1 s Size 83 x 48 x 48quot Weight22 lbs 1 kg Exiech 4612igiigi losiroboTach 4 LED display imn WWW ins iruman cumPruducl asuxPPruHudiDZZEESU 51EI2EI11 ENGRZEH Lecture 7 Spring ZEI11 39 Stroboscopic Errors 35 rpm 3500 mm 3600 mm aaan rpm 3600 rpm i800 Inm 360mm 101300 1pm fpm mm lmvv alum mm you w mm mm 3600n inm mu mm mm mm mm Submultiples divrde into top number 1 w M ivli i 51EIZEI11 ENGRZEH Lecture 7 Spring ZEI11 4D Photoelectric Tachometer Transmission slot or re ection Detector produces pulses as disk rotates O 0 a g Read by counter over C xed period of time LightgtD O D whom q l Produces a digital value for angular velocity 5102011 ENGR201 Lecture 7 Spring 2011 41 Incremental Encoder greatly simpli ed Two channels on disk in a quadrature relationship may also include a single reference channel Dude disk liacks shalt rd ll lil aledifh q IFl H r Z 77 17 7 Wm a cw rotation ifA leads B photollansisloli CCW rotation if B leads A 91 W39cha39eme Can count edges to increase resolution 2X or 4x multiplier 5102011 ENGR201 Lecture 7 Spring 2011 42 ENGR 201 Project Objective Design and build a measurement system to determine the angular velocity of a wheel Essentially a bicycle speedometer Sensor reed switch Adopt and reproduce the functionality of a commercial bike cyclometer as closely as possible EiElZEIM ENGRZEM Lecture7 Spring 2cm Reed Switch Two or more ferrous metal reeds within a glass tube Reeds make contact in the presence of a magnetic eld GLtssSEu GUSSWEE Nmmmwm warmarm lNOl 51 ElZEIM W 44 Simple inexpensive can carry significant currents Most common bike speed an measuremen Fragile due to glass Magnet must be very near to close switch Cadence 5102011 Reed Switch sensor for d cadence t httproc digikeycomlWebLibCoto Technolo IWeb DataRlOB Series df angular velocity of pedal cranks ENGR201 Lecture 7 Spring 2011 5102011 Bike Application VIagnet Sensor Side 1pdf ENGR201 Lecture 7 Spring 2011 Acquiring Data Sensor produces pulses How can you capture them Pulse Counting See USB6009 counter Frequency to Voltage Conversion VWI this work at very low frequency A limited number of frequency to voltage circuits will be available in the lab More information next week in lab write up 5102011 ENGR201 Lecture 7 Spring 2011 47 Computational Example 1 If a bicycle with 27 diameter wheels is moving at 15 mph how many sensor pulses are generated per second Assume that one pulse is read for each wheel rotation 5102011 ENGR201 Lecture 7 Spring 2011 48 Numerical Solution Ex 1 How many wheel rotations will we have per mile use dimensional analysis 1 rev 7Z39D inches 1rev DXL D in inches 63360 1n 1rev134x10 3 mi 5102011 ENGR201 Lecture 7 Spring 2011 49 Numerical Solution Ex 1 How many wheel rotations will we have per second at 15 MPH miX 1 rev X 1 hr hr 134 X10 3mi 3600 sec 311 revsec So we could expect 311 reed switch closures per second at 15 MPH 15 5102011 ENGR201 Lecture 7 Spring 2011 50 Computational Exercise 2 A magnetic pickup is excited by the rotation of a gear with 12 teeth on the shaft The shaft rotates at 3000 rpm How many cycles per second do you expect to be displayed by an oscilloscope that is used to view the signal A 3600 B 600 C 50 D 3000 5102011 ENGR201 Lecture 7 Spring 2011 51 Solution Computational Example 2 What do we know 12 teeth on gear each produces a pulse cycle when it passes the sensor 3000 RPM of shaft 12 cyclesrev 3000 revmin 36000 cyclesmin 36000 cyclesmin 60 secmin 600 cyclessec Solution is B 5102011 ENGR201 Lecture 7 Spring 2011 52 Questions regarding your design in lab Which detection technique is better pulse counting vs frequency to voltage Which can produce the best speed resolution How much resolution is needed you should be able to use what you learned in class to address this What about cost size weight etc 5102011 ENGR201 Lecture 7 Spring 2011 53 Caution In your software and reports remember that speed scalar and velocity vector are not the same thing 5102011 ENGR201 Lecture 7 Spring 2011 54 ENGR 201 Evaluation amp Presentation of Experimental Data Lecture 3 Statistical Measurement Theory part 2 T Chmielewski Electrical amp Computer Engineering D Miller Mechanical Engineering amp Mechanics Drexel University 12 Apnl 20M ENGR201 Lecture 3 Sprmg 20M Quiz 1 Performance Grade Hlstogram ID OU SYN CDWquot 25quot mm maul Average 56 a run You E m 6 l5 391 39 am m mu u m m1 mmm an ll ii h l Standard Dewaer 1902 55015100 SKurE Karlie l A 772733 7676777i s In u39iz 31517571137 in39i n 217221334 23 Tni mausmy 12 Apnl 20M ENGR201 Lecture 3 Sprmg 20M Announcements Quiz 2 will be online Thursday 414 at 500 PM thru Friday 415 at 500 PM Check BbVista for additional details Midterm is TBD Closed book and notes Calculators are permitted Scantron and Computational parts Covers Chapter 1 and 4 labs 1 and 2 12 April 2011 ENGR201 Lecture 3 Spring 2011 Homework and Reading Week 3 8 o Read textbook sections 44 45 474 make sure to pay attention to examples 44 45 46 412 413 and Table 47 on page 152 Problems from text 411 only analyze column 1 412 only analyze column 1 426 434 4 42 Additional information to support lecture concepts Video Statistics Standard Deviation cut and paste link ht tn39lwww vnutuhe cnmwatrhvH F Inh nnirquot 39 Compare Student s tdistribution to normal distribution httpvxwwstatStanfordedunaras39sm39l39DensityTDensityhtml Effect of sample size on finite statistics Finitepopulationvi in student area of textbook web site 12 April 2011 ENGR201 Lecture 3 Spring 2011 Recap of Lecture 2 Gaussian or Normal Distribution In nite statistics De ned by mean X and variance oz Probability Px that a variable x will fall within X 1 x is the area under curve PGLJX934504233176 Interval with probability PX o X S x o 6827 PX 20 X S x 20 9545 Px 3os x s x 30 9973 39 1 1 x 3N 2 xgtn a 2 13 x lZApanDll ENGRZEM LectureS Spring 2W 5 Back to original question from Lecture 2 We measure the diameter of 10 parts sample from a population in a crate and find the average a We want to know how well the average diameter of our small sample represents average diameter x of the entire crate Estimate truevalue x as x39 x i ux 13 tux is the uncertainty interval in the estimate at some probability level lZApanDll ENGRZEM LectureS Spring 2W 5 Today s Topics will answer that question and more Since you can rarely measure every item in a population Ifyou measure just part ofthe population a sample what can we say about the population as a whole Central Tendency mean and standard deviation Con dence Interval Type of distribution At the conclusion of this lecture we will have the tools to estimate the population mean and population standard deviation albeit with a confidence interval lZApanElll ENGRZEM LectureS Spring 2W Vocabulary Population and Sample Statistics Characteristics Paraeters Statistics Mean x x Standard S Deviation a X We say a population has parameters Sample has statistics Whereas infinite statistics describe the true behavior of the population of a variable finite statistics describe only the behavior of the sampled data set lZApanDll ENGRZEM LectureS Spring 2W Characteristics of a Sample It is easy to evaluate the statistics ofa sample Sample mean f Sample standard deviation SX These sample statistics are estimates of the population characteristics and are called statistical parameters How do we use these sample statistics to predict the behaviorproperties of the population we still need more tools 12 April 2011 ENGR201 Lecture 3 Spring 2011 9 Characteristics of a Sample complications There is no reason to believe that a random sample is going to have the same mean and standard deviation as the population Even worse different samples from the same population are going to give us different results Hence we need to de ne confidence intervals forthe estimates ie from sample 12 April 2011 ENGR201 Lecture 3 Spring 2011 10 Statistical Parameters of the Sample N Sample mean f 2 N T1 N 2 Sample variance Sx N1Z1xi xZ Sample standard deviation Sx S Note use of N1 for sample statistics Recall population used lim as N gtltgto 12 April 2011 ENGR201 Lecture 3 Spring 2011 11 ConfidencePrecision Intervals for infinite sample We d like to use the ideas from population analysis aka in nite statistics x395x P x 5x S x Sx395x J V 5 pxLv Or with a Change of variable 2 Z1 x1 x 039 P z1 s 3 3 21 673226 12 April 2011 ENGR201 Lecture 3 Spring 2011 12 Uncertainty in estimate Good News we can use the same forms Bad News normal distribution results are not valid if sample size is small lt30 Fix use the Student s t distribution that does work and replaces the normal distribution xi a i INSX 13 xi interval of values in which P of values lie tvp Sx precisionconfidence interval about sample mean at probability P note t depends on two values 12 April 2011 ENGR201 Lecture 3 Spring 2011 13 Student s tdistribution Student s t parameter twp xi yny P Probability Level v Degrees of Freedom Greek letter nu In general the number of independent measurements minus the minimum number of measurements that are theoretically necessary to estimate a statistical parameter For the t parameter I need the mean which I can estimate with one measurement lfl make measurements to get a more accurate result then v1019 12 April 2011 ENGR201 Lecture 3 Spring 2011 14 Student s tdistribution cont The Student s tdistribution is bellshaped and symmetric about zero The shape of the curve depends on the degrees of freedom v o The area under the curve 1 same as Gaussian hence we can compute probabilities The shape of the distribution is flatter than the normal distribution As v increases the curves look similar For v gt 30 the curves are nearly identical 12 April 2011 ENGRZOW Lecture 3 Spring 2011 Probability density function using the Student s tdistribution I39ll a 30 you can not h between the t n and the normal stribution 72 7 U I I I 12 April 2011 ENGRZOW Lecture 3 Spring 2011 Using Student s t distribution To find the probability of measuring a specified value from the population we would use the t distribution as we did the normal distribution except we need to know the appropriate v After determining v we would find the value of the tdistribution in a table Table 44 pg 131 or by using a computer statistical package just as with the normal distribution 12Ale ZEI11 ENGRZEM Lecture 3 Spring ZEI11 17 Section of TDistribution Table ith 9 l EM i 1 1000 6314 12706 63657 0816 292 4303 9925 2 3 0765 2353 3182 5841 Define a 1 P a is the area under the pdf where you will not expect to see a value Half ofthis area will appear on each side of the curve 12Ale ZEI11 ENGRZEH Lecture 3 Spring ZEI11 18 Area all Area 12 12Apnl2 11 ENGRZEH Lecture spring 2mm 19 Example 1 Interval with confidence 95 Problem Statement The manufacturer of MP3 players wants to estimate the mean failure time of a model A2 with 95 confidence Available Data A sample of 6 systems are tested to time of failure With these results 1250 1320 1542 14641275 and 1383 hours Estimate the interval of values over which 95 of the measurements ofthe failure time should be expected to lie 12Apnl2 11 ENGRZEH LecturEBSpring NM 20