AAE 520 Experimental Aerodynamics Measurement Uncertainty References Coleman

AAE 520 Experimental Aerodynamics Measurement Uncertainty References Coleman, H. W and Steele, W. G. Experimentation and Uncertainty Analysis for Engineers, John Wiley & Sons, 1989 Matlab Signal Processing Toolbox Manual Lab. View Manual Software Matlab Lab. View Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Generic Transducer System Areas of Concern • . Accuracy • . Static Sensitivity • . Frequency Response • . Loading (Impedance Matching) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics “degree of goodness” Anyone comparing results of a mathematical model with experimental data (and perhaps also with the results of other mathematical models) should certainly consider the "degree of goodness" of the data when drawing conclusions based on the comparisons. In Figure a the results of two different mathematical models are compared with each other and with a set of experimental data. In Figure b the same information is presented, but a range representing the likely amount of error in the experimental data has been plotted for each data point. It should be immediately clear that once the "degree of goodness" of the data is taken into consideration, it is fruitless to argue for the validity of one model over another based only on how well the results match the data. Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Experimental Approach - Questions 1. What question are we trying to answer? (What is the problem? ) 2. How accurately do we need to know the answer? (How is the answer to be used? ) 3. What physical principles are involved? (What physical laws govern the situation? ) 4. What experiment or set of experiments might provide the answer? 5. What variables must be controlled? How well? 6. What quantities must be measured? How accurately? 7. What instrumentation is to be used? 8. How are the data to be acquired, conditioned, and stored? 9. How many data points must be taken? In what order? 10. Can the requirements be satisfied within the budget and time constraints? 11. What techniques of data analysis should be used? 12. What is the most effective and revealing way to present the data? 13. What unanticipated questions are raised by the data? 14. In what manner should the data and results be reported? (Ref. Coleman and Steel) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Need for Uncertainty Analysis Is uncertainty analysis always necessary ? Four types of Experiments (a) quick-sort Flow visualization is used to establish the global nature of a flowfield. The visual results are then used to determine where to place probes. In development work, one needs to know whether something will work. A fast low cost yes or no answer is needed. In such a cases uncertainty analysis is not essential. (b) report of field test, development test, or acceptance test (c) report of research (d) calibration test No alternative to Uncertainty Analysis in (b), (c) and (d) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Replication Levels Three different replication levels are Zeroth Order is described by the following conditions time is frozen; the display of each instrument is considering to be invariant A single sample is taken. The values of uncertainty at this level are often assigned "one-half the smallest scale division" or some similar rule of thumb. First Order: At this order, time is the only variable; with the experiment running, the display for each instrument is assumed to vary stochastically about a stationary mean. The first order uncertainty interval includes the timewise variation of the display and its interpolation uncertainty. Valid estimates of the mean and standard deviation are obtained. N'th Order: At this order, time and the instrument identities are considered to be variables. For each conceptual replication, each instrument is considered to have been replaced by another of the same type. An example is the wind tunnel measurements of aircraft model drag in various facilities around the world using the same model. Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Measurement Error True Value NBS Standard Bias Error Systematic Error Remains Constant During Test Estimated Based On Calibration or judgement True Average m xk Measured Value Bias Error b Precision ( Random Error ) Index - Estimate of Standard Deviation Total Error dk dk = b + e k Purdue University - School of Aeronautics and Astronautics Random Error ek

AAE 520 Experimental Aerodynamics True Value Unbiased, Precise, Accurate True Value Biased, Precise, Inaccurate Unbiased, Imprecise, Accurate if N>>1 Biased, Imprecise, Inaccurate Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics • Accuracy Errors – Measure of how close the result of the experiment comes to the “true” value • Precision – Measure of how exactly the result is determined without reference to the “true” value • Bias Error – Reproducible inaccuracy introduced by calibration or technique. Sometimes this error is correctable • Random Error – Indefiniteness of result due to finite precision of experiment. Measure of fluctuation in result after repeated experimentation Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Normal Distribution ( Gaussian or Bell Curve ) The normal distribution is a two parameter family of curves. The first parameter, m , is the mean. The second, s, is the standard deviation. The usual justification for using the normal distribution for modeling is the Central Limit Theorem which states (roughly) that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity. The normal pdf ( probability density function) is: normalized so that the area under the curve = 1. 0 Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Parameter Estimation. A desirable criterion in a statistical estimator is unbiasedness. A statistic is unbiased if the expected value of the statistic is equal to the parameter being estimated. Unbiased estimators of the parameters, m, the mean, and s, the standard deviation are: estimate of the mean, m [ mean(data) ] estimate of the standard deviation, s [ std(data) ] Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Data Sample Signal from Hot Wire in a Turbulent Boundary Layer Output from an A/D Converter (in counts) at Equal Time Intervals Long Time Record Short Time Record 1000 980 960 Amplitude 940 920 900 880 860 840 820 800 0 20 40 60 Time Purdue University - School of Aeronautics and Astronautics 80 100

AAE 520 Experimental Aerodynamics Estimate of the Probability Density Function [ hist(data, # of bins) ] 800 700 600 500 400 300 200 100 0 850 900 950 1000 1050 1100 Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Errors in the Estimate of the Mean • For a normal Distribution m - true mean x - estimate of the mean s- estimate of the standard deviation N - number of samples t 95 - 95% confidence interval from Students t distribution t 95 = ~2 for N>20 The notation Means we are 95% confident that the mean lies between 12. 0 and 13. 0 Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Standard Deviation 95% Confidence Interval Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Propagation of Errors General Uncertainty Analysis Suppose the quantity of interest Q is to be calculated from the measured quantities by the equation If there are variations in the readings, at any instant it is and not that is measured. This consequently causes a variation in Q. If the are small then by the Taylor's series expansion the calculated Q is given as or Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Consider that many measurements have been made and that is the deviation for each reading set. It's standard deviation over many readings is given as Substituting for Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics If the measurements are independent , then there errors will be uncorrelated and the cross term will be zero. That is in the sums , etc. , any term is as likely to be positive as negative (assuming independent variations) then these sums tend to zero for large N. Hence But. is the standard deviation in the measurement Therefore, the standard deviation in the calculated result is Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics COMMON SENSE ERROR ANALYSIS. Examine the data for consistent. No matter how hard one tries, there will always be some data points that appear to be grossly in error. The data should follow commonsense consistency, and points that do not appear "proper" should be eliminated. If very many data points fa 11 in the category of "inconsistent, " perhaps the entire experimental procedure should be investigated for gross mistakes or miscalculations. . Perform a statistical analysis of data where appropriate. A statistical analysis is only appropriate when measurements are repeated several times. If this is the case, make estimates of such parameters as standard deviation, etc. . Estimate the uncertainties in the results. We have discussed uncertainties at length. Hopefully, these calculations will have been performed in advance, and the investigator will already know the influence of different variables by the time the final results are obtained. . Anticipate the results from theory. Before trying to obtain correlations of the experimental data, the investigator should carefully review theory appropriate to the subject and try to glean some information that wi 11 indicate the trends the results may take. Important dimensionless groups, pertinent functional relations, and other information may lead to a fruitful interpretation of the data. . Correlate the data. The word "correlate" is subject to misinterpretation. In the context here we mean that the experimental investigator shou 1 d make sense of the data in terms of physical theories or on the basis of previous experimental work in the field. Certainly, the results of the (Ref. they Holman, J. P. , ”Experimental Methods Engineers") experiments should be analyzed to show conform to or differ fromforprevious investigations or. Purdue standards that may be employed for suchand measurements. University - School of Aeronautics Astronautics

AAE 520 Experimental Aerodynamics Design of experiments Using Uncertainty Analysis • The choice of test and data reduction procedures can have an important impact on the accuracy of the results • It is important to specify the level of replication - 0 th, 1 st or Nth order • Reliable means for cross-checking and/or externally validating the results are necessary • In experiments where data are reduced by computer, uncertainty analysis can be done by sequential perturbation using the data reduction program Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Other Issues • • Threshold Resolution Linearity Hystersis Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Applications of Curve Fitting • Generate a Calibration Curve • Removal of measurement noise. • Filling in missing data points (for example, if one or more measurements were missed or improperly recorded). • Interpolation (estimation of data between data points; for example, if the time between measurements is not small enough). • Extrapolation (estimation of data beyond data points; for example, if you are looking for data values before or after the measurements were taken). • Differentiation of digital data. (For example, if you need to find the derivative of the data points. The discrete data can be modeled by a polynomial, and the resulting polynomial equation can be differentiated. ) • Integration of digital data (for example, to find the area under a curve when you have only the discrete points of the curve). • To obtain the trajectory of an object based on discrete measurements of its velocity (first derivative) or acceleration (second derivative). Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Method of Least Squares Straight Line Fit (Regression Analysis) to Set of Data Points (N points) Problem - Find m and b Vertical Deviation Minimize the Sum of the Squares of the Deviation Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Uncertainty Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics General Curve Fit Curve fitting analysis is a technique for extracting a set of curve parameters or coefficients from the data set to obtain a functional description of the data set. The algorithm that fits a curve to a particular data set is known as the Least Squares Method. The error is defined as where e(a) is the error, y(x) is the observed data set, f(x, a) is the functional description of the data set, and a is the set of curve coefficients which best describes the curve. To solve this system, you set up and solve the Jacobian system generated by expanding equation. After you solve the system for a, you can obtain an estimate of the observed data set for any value of x using the functional description f(x, a). Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Polynomial Fits to Hot Wire Data 4 th Order Linear 2 nd Order Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Oscilloscope The oscilloscope is basically a graph-displaying device -- it draws a graph of an electrical signal. In most applications the graph shows how signals change over time: the vertical (Y) axis represents voltage and the horizontal (X) axis represents time. The intensity or brightness of the display is sometimes called the Z axis. URL for a description of oscilloscope operation http: //www. tek. com/Measurement/App_Notes/XYZs/intro. html Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Analog Oscilloscope Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Digital Oscilloscope Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Data Acquisition (DAQ) Fundamentals The personal computer Transducers Signal conditioning DAQ hardware Software Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Computer Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics A/D PERFORMANCE SPECS Resolution describes the smallest standard incremental change in output voltage of a DAC or the amount of input voltage change required to increment the output of an ADC between one code change and the next adjacent code change. A converter with n switches can resolve 1 part in 2 n. The least significant increment is then 2 bn , or one least significant bit (LSB). In contrast, the most significant bit (MSB) carries a weight of 2 b 1. Resolution applies to DACs and ADCs, and may be expressed in percent of full scale or in binary bits. For example, an ADC with 12 -bit resolution could resolve 1 part in 2 12 (1 part in 4096) or 0. 0244% of full scale. A converter with 10 V full scale could resolve a 2. 44 m. V input change. Accuracy. An accuracy specification describes the worst case deviation of the DAC output voltage from a straight line drawn between zero and full scale; it includes all errors. Quantizing Error , Gain Error , Scale Error (full scale error), Hysteresis Error , Offset Error (zero error) and Linearity, Conversion Rate is the speed at which an ADC or DAC can make repetitive data conversions. Input Impedance of an ADC describes the load placed on the analog source. Number of Channels Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Analog to Digital Converter 12 Bit A/D - Resolution = one part in Binary Word - 101100111101 MSB Unipolar Mode (0 -10 Volts) Integer Output Voltage 4095. . . 1 0 9. 9975. . . 0024 0 LSB Volts=10/4096*Icount Scale +Full Scale - 1 LSB. . . Bipolar Mode 1 LSB -5. 0 to 5. 0 0 V=5/2048*(Icount-2048) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Dynamic Response of Measurement Systems Zero Order System Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics First Order System Impulse response – First Order System 1 1 0. 9 0. 8 0. 7 Amplitude Step response - First Order System 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 2 4 6 8 10 0 0 2 4 Purdue University - School of Aeronautics and Astronautics 6 8 10

AAE 520 Experimental Aerodynamics First Order System Sinusoidal Response amplitude Gain d. B Sinusoidal Response - First Order System 1 0. 8 Input 0. 6 0. 4 0. 2 Output 0 -0. 2 -0. 4 -0. 6 -0. 8 -10 0. 5 1 1. 5 2 2. 5 time 0 -10 -20 -30 -1 10 Phase deg Amplitude Decrease and Phase Shift 10 0 10 1 0 10 0 -30 -60 -90 -1 10 10 Purdue University - School of Aeronautics and Astronautics 1

AAE 520 Experimental Aerodynamics Second Order System Step response - Second Order System 2 1. 8 1. 6 Amplitude 1. 4 1. 2 1 0. 8 0. 6 0. 4 0. 2 0 0 5 10 Purdue University - School of Aeronautics and Astronautics 15 20

AAE 520 Experimental Aerodynamics Second Order System - Sinusoidal Response Amplitude (d. B) Phase (deg) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Second Order System - Impulse Response Impulse response - Second Order System 1 Logarithmic Decrement 0. 8 0. 6 Amplitude 0. 4 0. 2 0 -0. 2 -0. 4 -0. 6 -0. 8 0 20 40 60 80 100 Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Low Pass Filter Removes High Frequency Noise Band Pass High Pass Filter Removes DC and Low Frequency Noise (Such as 60, 120 Hz) Stop Band Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Filters and Transfer Functions In general, the z-transform Y(z) of a digital filter’s output y(n) is related to the z-transform X(z) of the input by: Filtering with the filter Function It is simple to work back to a difference equation from the z-transform relation shown earlier. Assume that a(1) = 1. Move the denominator to the left-hand side and take the inverse z-transform. In terms of current and past inputs, and past outputs, y(n) is: This is the standard time-domain representation of a digital filter, computed starting with y(1) and assuming zero initial conditions. This representation’s progression is Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Elliptic Filter [b, a]=ellip(5, 1, 20, . 5) Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Elliptic Filter Bessel Filter Chebyshev I Filter Butterworth Chebyshev II Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Effect of Filter Order Chebyshev II N= 5 N= 10 N= 2 Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Example Signal Fs = 100; t = 0: 1/Fs: 1; x =. 5+ sin(2*pi*t*5)+. 25*sin(2*pi*t*40); % DC plus 5 Hz signal and 40 Hz signal sampled at 100 Hz for 1 sec Total Signal DC Level Low Frequency Signal High Frequency Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics Cheby 2 Low Pass Recovers DC + 3 Hz Original Signal Filter Filtfilt Cheby 2 High Pass Cheby 2 Band Pass Cheby 2 Stop Band Recovers 40 Hz Recovers 3 Hz Recovers DC + 40 Hz Purdue University - School of Aeronautics and Astronautics

AAE 520 Experimental Aerodynamics 10 Point Averaging Filter b= 0. 1, 0. 1 a= 1. 0 Purdue University - School of Aeronautics and Astronautics