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Background
The Aerospace Engineering Department of Old Dominion University has partnered with the Department of Industrial Engineering at Florida State University to research methods, develop coursework, and educate students in a statistical based experimental methodology for wind tunnel testing. A Design of Experiments (DOE) approach to wind tunnel testing of aircraft and performance automobiles has been explored and developed at the Langley Full-Scale Tunnel.
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INTRODUCTION
The process of wind tunnel testing aircraft has the primary objective of characterizing and optimizing aerodynamic performance. Changes are made to independent variables such as vehicle attitude, configurations, and control surface deflections. The response variables include the three aerodynamic moments and forces. The traditional approach to testing is to vary one factor at a time while all other factors are held “constant”. Researchers will then pursue a course of experimentation aimed at improving one or more responses, such as minimizing drag while maintaining a specified level of lift. This approach requires that the entire measurement system, which consists of the wind tunnel facility, force measurement balance, and data acquisition system, be completely stable throughout the entire test entry. Any errors that result from systemic variation are confounded with precision errors and are fundamentally inseparable. In addition, if two or more inputs interact to affect a response, the one-factor-at-a-time experimentation approach will not detect these often important contributions to response predictions and system understanding.
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An alternate approach is to use formally designed experiments. This approach traces its origin to the seminal publication by Ronald Fisher, The Design of Experiments.1 Proponents of design of experiments have historically included engineers and scientists involved in manufacturing, the chemical process and semiconductor industries, and agriculture. In recent years, the aerospace community has begun formulating methods to exploit the benefits of DOE with regards to vehicle wind tunnel testing.2,3,4 DOE methods start by identifying all desired factors (independent variables) and all desired responses. Another key difference is the DOE approach to analyzing test results. A reduced size test matrix (run schedule) is formulated which, when executed, will provide data for developing statistically validated mathematical models of the responses in terms of the factors. The objective is to characterize the relationship between changes in system performance measures (responses) due to corresponding changes in system input factors. Bias errors due to uncontrolled or unknown systemic variations may be guarded against and uncertainty levels may be accurately estimated. Inherent to the DOE methodology is the development of empirical models detailing the response behavior being studied, capable of predicting performance measures over the factor design space studied.
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DOE Fundamentals
The run schedule of the designed experiment is formulated using three hallmarks of the DOE method: randomization, replication, and blocking.5 Randomization is the cornerstone of statistical analysis as applied to experimental design. The run order of the experiment design points must be randomly determined since statistical methods require that observations are normally, independently distributed random variables. Randomization also serves to average out the effects of unwanted extraneous factors that may be present. Replication refers to the act of duplicating measurements for the purpose of estimating experimental error and more precisely defining the effect of a factor in the experiment. A true replicate is a measurement that has been made after all the factors have been adjusted off-point and are then brought back on-point. For example, in the case of an aircraft test where the aileron position and yaw angle are factors, a sequential genuine replicate point would be obtained after both aileron position and yaw angle were adjusted above or below the desired value and then brought back to the design point. Blocking is a technique used to reduce the variability of nuisance factors and hence increase the precision of the measurements. For example, variations in wind tunnel measurements are often encountered in comparisons between overnight runs or shifts. Assigning blocks to shifts helps separate the shift-to-shift variability due to say environmental conditions or operators, from the force balance precision.
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Analysis of the experimental data is performed using statistical hypothesis testing and regression model building so that the response values can be accurately estimated or predicted using empirical models. These models are typically low order polynomial functions of the input variables. The model is also tested for adequacy relative to satisfying certain model assumptions, which are necessary for performing statistical tests. Replication of design points allows the modeler to determine an internal estimate of system noise or uncertainty (some refer to this informally as repeatability). Sufficient test runs will also provide the ability to test for potentially significant model terms of higher order, which may aid in prediction and estimation. Statistical tests are usually performed such that a p-value is reported for the modeler to assess significance. In many situations, p-values < 0.05 are indicative of statistical significance. The desired precision in a response is used to dictate the required number of data points for a given confidence level. For instance, if the desired precision in a drag coefficient is to be known with 95% confidence within one drag count (0.001) over a range of factors, then the number of samples required to meet this goal can be effectively determined.
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Interactions
One of the greatest benefits to pursuing DOE methods versus traditional OFAT methods is the inclusion of interactions in the analysis. Since OFAT methods require that only one variable be changed at a time, only the main effects will be directly discovered. By changing more than one factor at a time, DOE methods uncover interactions between variables. For instance, the change in drag on a race car due to varying the yaw angle may be found using OFAT methods - a main effect, while DOE methods can easily unearth the difference in the change in drag due to yaw angle for two different vehicle ride heights - an interaction effect. The relative magnitude of regression coefficients in the model give direct feedback as to the importance of the interaction effects to the overall response.
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Examples
The following case studies are provided to illustrate the variety of applications that can benefit from the DOE approach to experimentation.
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Aircraft Stability and Control Test
A general aviation aircraft was chosen as a test article for a 32-run basic stability and control test. The input factors for this initial investigation were angle-of-attack, sideslip, stabilizer, aileron, and rudder deflection. The responses were the six aerodynamic forces and moments. In the figure to the right we see a typical response surface result where pitching moment is plotted as a function of yaw and angle of attack. The underlying full mathematical model taken from the 32-run test schedule is shown below and is easily adapted for use with numerical simulations.
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Sedan Based Race Car Aerodynamic Characterization
Historically the OFAT test method has been used for virtually all automotive testing. For this study, a DOE approach was used to determine the aerodynamic sensitivities of a modern stock car, specifically a 1997 Chevrolet Monte Carlo Winston Cup car. In order to develop a complete evaluation of the methodology, a four-factor experiment was selected. The four factors were investigated at two levels each (high and low) using a factorial method. The factors were the front ride height, rear ride height, lower grille porosity and the yaw angle of the car. These choices represent the changes most often made by race teams to assess the aerodynamic performance of their particular racecar. As always a mathematical model for each response as a function of all the variables was created with a measure of uncertainty. The empirical model for CLF where the lower grille tape is removed in terms of the actual factor and only significant terms are retained is:
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CLF = - 0.0789 + 0.0161 Front Ride Height - 0.0163 Rear Ride Height
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At the bottom right hand corner is a cube plot showing the responses for the front lift coefficient from the 20-run experiment.
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Internal Strain Gage Balance Calibration
Strain gauge balances measure dynamic loads in six degrees of freedom. They include the normal force, axial force, pitch moment, roll moment, yaw moment and side force. To characterize the relationship between the six input load factors and the six output voltage responses of a strain gauge balance for balances rated at or above 3,000 lbs, the NASA Langley Research Center (LaRC) historically used a traditional approach consisting of 729 one-factor-at-a-time (OFAT) experimental runs that generally takes four to six weeks to complete. The OFAT experimentation invokes a strategy of varying one or two factors over a range using incremental changes, while holding the remaining factors constant at some nominal value. Although this approach can be used to characterize the underlying relationship, in this case between loads and voltages, the test matrices are often much larger than necessary. In an effort to reduce both the number of runs and the amount of time needed to characterize a strain gauge balance rated at or above 3,000 lbs, modern experimental design theory (DOE and response Surface Methods) was employed in a project that includes LaRC, ODU and FSU. The NTF-107 strain gauge balance, which had already been calibrated via manual and Single Vector System (SVS) methods, was used to evaluate and validate the proposed approach suggested in this study. The results of the study show that the new 65-run design provides a comparable calibration model to the OFAT method with a huge time savings and a statistically robust measure of uncertainty.
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Personnel
The key personnel involved in applying DOE to wind tunnel testing are Dr. Jim Simpson of the Department of Industrial Engineering, Florida State University (FSU) and Dr. Drew Landman of the Department of Aerospace Engineering at Old Dominion University (ODU). Dr. Landman is currently Assistant Manager of the Langley Full-Scale Tunnel where his duties include teaching experimental methods and automotive and aircraft aerodynamics as an associate professor in addition to supervising wind tunnel entries. Dr. Simpson received his PhD from Arizona State University under Douglas Montgomery in 1995, focusing on Statistical Process Control and Design of Experiments. Dr. Simpson served in the Air Force as an analyst where he applied DOE to flight test and weapons testing. He has been involved in many industrial experiment designs and currently performs research in the area of experiment design and analysis, including aerospace applications such as wind tunnel testing methods and balance calibration. Together Simpson and Landman have developed a four course sequence in modern experimental design with aerospace applications at the graduate engineering level. They are currently working on topics such as DOE with restrictions to randomization, efficient design space mapping and highly non-linear aerodynamic characterizations.
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