Instrumentation, Measurements, and Statistics
Instrumentation, Measurements, and Statistics M E 345
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This 0 page Class Notes was uploaded by Chester Goldner III on Sunday November 1, 2015. The Class Notes belongs to M E 345 at Pennsylvania State University taught by John Cimbala in Fall. Since its upload, it has received 14 views. For similar materials see /class/233070/m-e-345-pennsylvania-state-university in Mechanical Engineering at Pennsylvania State University.
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Date Created: 11/01/15
Outline and Equation Sheet for M E 345 Author John M Cimbala Penn State University Latest revision 27 March 2009 Introduction Prima dimension 7 mass length time temperature current amount of light and amount of matter Signi cant digit 7 the rules for multiplication and division and for addition and subtraction Rounding off 7 round up if the least signi cant digit is odd and truncate if the least signi cant digit is even Dimensional Analysis Law of dimensional homogeneitv 7 Every additive term in an equation must have the same dimensions The method of repeating variables 7 there are 6 steps 1 2 3 9 95 List the parameters and count them n List the primary dimensions of each parameter Set the reduction j as the number of primary dimensions in the problem Then Reduce j by one if necessary Choose j repeating variables Construct the k Us and manipulate as necessary Write out the nal dimensionless functional relationship I and check your algebra Dimensional analysis is often extremely useful in setting up and designing experiments Errors and Calibration I Systematic errors bias errors 1 7 consistent repeatable errors Random errors precision errors 1 7 scatter in data a lack of repeatability unrepeatable inconsistent errors Accuracy 7 accuracy error is the measured value minus the true va ue Precision 7precision error is the reading minus the average ofreadings 0 Basic Statistics Mean bias error 7 de ned as lVIBE 39 De nitions 7 sample mean ther errors 7 zero linearity sensitivity resolution hysteresis instrument reapeatability drift Calibration 7 static time not relevant vs d 39 39 namrc time is relevant calibration sample variance S2 sample median half lower half higher sample mode most probable value 7 one that occurs most frequently population all values vs sample a selected portion of the total population Excel 7 learn how to use Excel s bui tin statistics functions Root mean square erro 7 de ned as Histograms and Probability Density Functions Normalized PDF 7 transform from x to Z using Z The Gaussian or normal PDF 7 how to predict 1 error function A z erf Histogram plots 7 frequency bins or classes bin width or class width Sturgis and Rice rules Normalized histogram 7 how to normalize the vertical scale and the horizontal scale DF 7 how to create a probability density function from a histogram Expected value 7 same as population mean Standard deviation 7 same as population standard deviation I I wu for systems with purely random errors using the erf giiiexpt iiyf See table ofAZ Z F where I Other PDFs 7 lognorrnal chisquared distribution student s 1 degrees of freedom con dence level and significance level I How to estimate the population mean u from a sample using the student s 1 PDF yd con dence level 1 7 or use table of critical values for the student s 1 PDF I How to estimate the o ulation standard deviation afrom a sample using the 7 PDF Correlation and Regression i Y X yx y I Linear correlation coef cient 7 11y 51 By de nition rm must always lie between Zw 7x y 7y 17139 11 I Regression analysis 7 leastsquares ts linear polynomial and multiple variables standard error of estimate Equations are provided in the lecture notes but we icall use Excel to do the nalysis Outlier Points I Outliers in a set of data point 7 use the modi ed Thompson tau technigue compare maximum absolute value of the deviation with Thompson Ttimes sample standard deviation TS to determine outliers I Outliers in a set of data pairs 7 for l 39 39 39 t of order rn S y 1 Examine the standardized residual e SW to determine outliers There are two criteria for a data pair to be called an outlier o leiSN gt2 0 The standardized residual is inconsistent with its neighbors as judged by a plot of e SW vs x Experimental Uncertainty Analysis I Maximum uncertaintv 7 assumes all errors have the same sign and add up unlikely Normally we do not use the maximum uncertainty Typically we use instead the RSS uncertainty as described below I RSS uncerta39 ted ert 39 t 7 assumes some errors are positive and some are neative more likely Use nmhinino elemental 39 quot using RSS uncertaint analysis 7 use it where K is the number of elemental uncertainties and u are the individual elemental uncertainties Experimental Design Hypoth I Choosing a test matrix 7 one parameter at a time full factorial vs Taguchi s experimental design arrays fractional factorial test matrices N L17 L levels P parameters for a full factorial experiment Level average 7 E g average over all the runs where parameter a is at level 1 Optimum Tag11chi desigp array 7 Meets two criteria 0 Each level of each parameter appears the same number of times in the array 0 Repetitions of parameterlevel combinations are minimized as much as possible Taguchi orthogonal array 7 know how to spot a poorly designed experimental design array Response surface methodology RS 7 goal is to efficiently hunt for the optimum values of parameters a b c etc such that response y is maximized or minimized 7 amin with similar equations for the other variables This is necessary to Coded variables 7 transform the physical variables into coded variables 61 2 a amidval a range amge a max properly use the RSM technique Direction of steepest ascent 7 vector v determined by regression analysis on coded variables 6amp9 39 a 7 For known M1 M2 and ax 39 0x ge 2 an Marching 7 Transform back to physical variables Aa then march in the direction of steepest ascent until y is no longer improving esis Testing Null hypothesis 7 a theory that is being considered or tested Typically the null hypothesis represents nothing is happening There are two parts to the null hypothesis The critical value extreme value 0 o The sides or Tails of the null hypothesis 7 select the least likely scenario 01 0 u lt 0 or u gt 0 Alternative hypothesi 7 also called the research hypothesi 7 the complement of the null h othesis O tstatistic 7 calculate for the extreme value critical value of the null hypothesis t p value 7 calculate based on the tstatistic pvalue is the probability of wrongly rejecting the null hypothesis if it is in fact true In Excel use TDISTt df tails pvalue is the area under the tails of the tPDF See also the tables of pvalues for a given value of df for the tdistribution one tail Note The values in the table are for onetail hypothesis tests multiply by 2 for two tails A small pvalue means that the null hypothesis is unlikely to be true 7 we reject the null hypothesis if the p value is less than some level typically 005 5 to standard engineering confidence level 39 h othesis testin when n is the same for the two samles 7 use 6 x3 7x4 as the variable no difference between the means of x3 and xA and set the value of Two sample 39 quot 39 testing when n is not the same for the two samples 7 set the null h pothesis as L v2 739 7 w n n set the cr1t1cal tstat1st1c to t39 xA x l39 and use dfN1NT 1 1 S z A 778 inAil quotA 39 quot3 1 quotB L Welch s equation as a kind of weighted value of the degrees of freedom Digital Data Acquisition I Binag to decimal and viceversa 7how to convert sum columns or successive division res ectively ADsystem Emmi mziyf Leakage 7 appears when the discrete data acquisition does not stop at exactly the same phase as it started Excel 7 how to create a frequency spectrum in Excel Windowing 7 reduces but does not totally eliminate leakage Analyzing the frequency content of a signal7 sample at higher and higher frequencies until no aliasing Antialiasing lter 7 use a lowpass lter to remove high frequencies to avoid aliasing Basic Electronics Review I Unity conversion factors Ohm s law and basic equations 7 41239 ewe V7 m I Impedance7 Filters I lstorder lowpass lter 7 I Closedloop quot with feedback loom 7 l Example circuits 7 buffer 7 I o Noninverting opamp ampli er E 353 1 ixsmnm qg 1r ij vLupumam qg i GBP supplied by manufacturer s Specs 0 Inverting opamp ampli er Lo ding7 input and output loading can cause voltage change due to internal resistances R1 input resistance or input impedance Stress Strain and Strain Gages I Axial stress and axial strain 7 for round rods Stress 39 39 quot Wire resistance 7 I 0 8 a D lt7 3 G D l E 5 H o g a D o I s Rum 153mm 55am aimed 39 Rm Dynamic System Response Quarter half and full bridge 7 o n l for a quarter bridge 0 n 2 for a half bridge 0 n 4 for a full bride Unbalanced bridge 7 Temperature Measurement Mechanical devices 7 liquidinglass thermometers bimetallic strips pressure thermometers Thermojunctive device 7 thermocouples sensin junctions reference junctions type K T etc Thermocouple law 7notation Wquot 0 Law of intermediate metals A third intermediate metal wire can be inserted in series with one of the wires without changing the voltage reading provided that the two new junctions are at the same temperature 0 Law of intermediate temperatures If identical thermocouples measure the temperature difference between T1 and T 2 and the temperature difference between T 2 and T 3 then the sum of the corresponding voltages V12 V23 must equal the voltage V13 enerated b an identical therrnocoule measurin the temperature difference between T1 and T 3 ie Lg 39 1 It 0 g Statement of the law of additive voltages For a given set of3 thermocouple wrres A B and C all measuring the same temperature difference T1 7 T 2 the voltage measured by wires A and C must equal the sum of the voltae measured by wires A and B and the voltage measured by wires B and C ie Thermopil 7 Several thermocouples in series Thermoresistive devices 7 RTDs metal R increases with increasing I thermistors semiconductor R decreases with increasing I Radiative devices 7 e g infrared pyrometer Tables 7 how to use thermocouple tables and RTD tables and interpolation Reference temperature is 0 C for all temperature tables Measurement ofMechanical Quantities Position 7 mechanical devices interferometer n otentiometer linear variable displacement transducer through transmission LVDT ultrasonic transducer pulseecho x Z G KEOAM so 8854 x 10 NC capacitance sensor Z laser displacement meter ra dian rotation quot 67039s s 2n radian min 39 rotation if Angular velocity and rpm 7 Nwm contacting tachometer rmn noncontacting tachometer V For stroboscopic tachometers watch out for aliasing 2 Torgue and powe 7 Mm wT 6ZprmT dynamometers prony brake cradled DC motor eddy current Pressure Measurement Types of pressure 7 absolute gage Pgage Pm 7 Pmm Liguid manometers 7 Pbdow Pabm pglAzl Mechanical pressure gage 7 Bourdon tube deadweight tester Flectronic pressure 39 139 l 39 type with quot l 39 measured by 0 Strain gage 7 a strain gage is mounted on the diaphragm itself sensing strain in the diaphragm o Capacitance 7 the diaphragm is mounted close to a xed parallel plate capacitance is measured 0 LVDT 7 diaphragm is attached to the core of a linear variable displacement transducer 0 Optical 7various optical techniques used to measure the degree of diaphragm deformation Pie oelectric pressure 39 7 l 39 causes a voltage that can be calibrated with pressure vacuum a McCleod gage used to measure very low pressures Velocity Measurement Linear velocity transducer QLVTQ 7 Similar in principle to LVDT but measure velocity not displacement os 6 A noise is ampli ed by dijj erentiation Doppler radar velocimeter 7 Doppler frequency shift A Vt dx t aft at dt noise is attenuated by integration Veloci of uid 7 Lagrangian methods follow uid particle moving with the ow Vt 61260 dt Eulerian methods measure velocity eld with a probe sitting in the ow Displacement sensor 7 Acceleration sensors 7 Laser Doppler velocimeter 1LDV1 7 particles in fringe pattern of focal volume V fs 251nltr 2 Hotwire and hot lm 7 wire heated to constant temperature King s law E Z a39 b V 2 P 7 P Pitotstatic probe 7 Pitot formula V 0 Other 7 rotating velocimeter cup anemometer tur Jine or vane anemometer electromagnetic velocimeter Volume Flow Rate Measurement Mass ow rate vs volume ow rate 7 V Notation V volume V velocity V volume ow rate V V At vs inline measurement 7 various types Obstruction 7 ori ce ow nozzle and Venturi Positive displacement 7 measure how many Wrapped or captured volumes pass through per unit time Turbine and paddlewheel 7 spin a turbine or paddlewheel connected to a shaft measure rpm and calibrate or f39 or variablearea 39 7 oat hovers when forces balance Miscellaneous owmeters 7 ultrasonic transit time and Doppler shift electromagnetic vortex others Endline measurement 7 Cd discharge coeff