APPLICATION OF GLOBAL REGRESSION METHOD FOR CALIBRATION OF WIND TUNNEL BALANCES

Transcription

1 Symposium on Applied Aerodynamics and Design of Aerospace Vehicles (SAROD 2011) November 16-18, 2011, Bangalore, India APPLICATION OF GLOBAL REGRESSION METHOD FOR CALIBRATION OF WIND TUNNEL BALANCES Gireesh Yanamashetti* and H Sundara Mury** CSIR- National Aerospace Laboratories, Bangalore, India ABSTRACT Aerodynamic forces and moments on scaled models are measured in wind tunnels by multi-component strain gauge balances whose performance and accuracy are characterized by careful calibrations. It is well recognized at calibration loads at are representative of e combined loads experienced by e balance/model in wind tunnel test conditions lead to high accuracy of e measured loads. But, in e traditional meod e calibration loads are restricted to single-component and twocomponent loads at are unrepresentative of model loads. A recently developed meod called Global Regression Meod (GRM) places no such restrictions and erefore, permits application of multi-component calibration loads similar to model loads, leading to improved accuracy of measured loads. In addition, wi use of GRM it is possible to substantially reduce e number of calibration loads and hence, e calibration effort by optimization of load schedule. The GRM was recently implemented and a MATLAB software developed for balance calibration data analysis at NAL. In order to illustrate e application of GRM and use of e computer program calibrations of a 6-component internal balance were carried out wi different types and number of calibration loads at included combined loads similar to model loads and e data were analyzed using GRM. The balance accuracy was assessed for a set of check loads consisting primarily of combined loads. Results showed at e accuracy when e calibration loads were similar to model loads was significantly better an at obtained using single- and two component calibration loads. It was also found at improved accuracy could be obtained wi substantially reduced number of loadings. The paper presents an overview of e GRM and brief details of application of e GRM to calibration of multicomponent balances. A description of e above calibrations and discussions of e results are included. Key Words: Strain gauge balance, Calibration, Ma model, Regression analysis 1. INTRODUCTION Multi-component strain gauge balances are e primary instruments used for measurement of aerodynamic forces and moments on scaled models in wind tunnels. Careful calibration of a balance is necessary to characterize its performance and determine its accuracy. It is well known at apart from design and fabrication aspects, accuracy of a balance depends on e maematical model adopted and e type of loadings applied for balance calibration (Ref.1and 2). Accuracy of e measured aerodynamic loads can erefore be expected to be highest when e calibration loads are similar to ose experienced by e balance/model during wind tunnel tests, and an appropriate ma model is chosen. In e traditional meod e balance is calibrated by applying only single-component and two-component loads, and piece-wise curve fits to e output vs load data are obtained from each of e loading seuences. Slope of ese fits are e desired calibration coefficients (Ref.3). Such piece-wise curve fitting procedure, sometimes called as Cooke s meod, reuires at e load combinations be restricted to a maximum of two components (Ref.1). Conseuently, e calibration loadings in e traditional meod are also restricted to a maximum of two components, alough * Scientist, NTAF **Consultant, NTAF such loadings are generally unrepresentative of actual wind tunnel test conditions. Wi improvement in computational capability a new calibration data processing meod at does not place any restrictions on e type of calibration loadings has been developed (Ref. 1). The meod known as least suare Global Regression Meod (GRM) permits simultaneous loadings of any arbitrary combinations upto six components and it erefore, enables e calibration of a balance wi load combinations similar to ose experienced by e balance in actual wind tunnel test conditions. It is also noted at e Ground Testing Technical Committee (GTTC) of e American Institute of Aeronautics and Astronautics (AIAA) which recently reviewed various practices adopted at major facilities in USA and Canada on calibration and use of internal balances, recommended adoption of GRM for deriving e calibration matrix (Ref.1). Following is recommendation most major wind tunnel facilities have adopted e GRM for balance calibration data anlysis (e.g. Ref. 4 and 5). Calibration of half-model balances and oer external balances are freuently made by application of loads wi points of application at are laterally offset from e balance. Such loading conditions result in multi-component loading of e balance. For example, a normal force on a half-model results in combined

2 Gireesh Yanamashetti and H Sundara Mury loading of ree components viz., normal force, pitching moment and rolling moment on e balance. Use of piece-wise curve fitting of such combined load calibration data is not possible and it is necessary to utilize e global regression meod to determine e calibration coefficients of such balances. Balance calibration procedures at NAL have so far followed e traditional meod and efforts are now underway to bring ese procedures on par wi state-ofe-art and widely accepted practices in e wind tunnel community. As part of ese efforts, e least suares global regression meod for determination of balance calibration coefficients has recently been implemented at NAL and a computer program has been developed using MATLAB for is purpose. To illustrate e application of e GRM and demonstrate its advantages, calibrations of a sixcomponent internal balance were made wi combined loads at were representative of typical model loads and also wi traditional single-component and twocomponent loads. Data from ese calibrations which were carried out in an Automatic Balance Calibration System at NAL were analyzed using e GRM. The report presents details of e above work along wi a brief description of e GRM wi emphasis on its application to balance calibration data analysis. 2. CALIBRATION PROCESS OF A BALANCE Before describing e application of GRM to balance calibration data a summary of calibration process is included as a background. A balance is calibrated by applying known loads and electrical outputs from e strain gauge bridges on e balance elements are recorded. The outputs from e balance elements are related to e applied loads on e balance and is relationship is expressed rough a polynomial euation called as e ma model of e balance. The most general form of e euation features first order, second order and a limited number of ird order load terms (Ref.1). This model, called a ird order ma model also accounts for e dependency of balance outputs on e sign of loads rough e use of signed and absolute values of component loads. The ird order model expressing e output R e from e e balance element as a function of e applied loads ( F F j k ), can be written as (adapted from Ref.1): Re = a e + b1 e, j Fj + b2 e, j F j j=1 j=1 2 c1 e, j Fj c2 e, j F j F j j=1 j= c3e, j,k Fj Fk + c4 e, j,k Fj Fk j=1 k=j+1 j=1 k=j+1 + c5e, j,k F j F k + c6 e, j,k F j j=1 k=j+1 j=1 k=j d1 e, j Fj d2 e, j F j e=1 e=1 + + Fk... (1) The above ma model defines e balance behaviour for e following types of loads : linear + and loads load suared + and loads load cubed + and loads load cross product ++, +, + and second order load combinations This model features a total of 2(+2) calibration coefficients for each component of a - component balance. The ird order model for a 6- component balance will lead to a 6x96 calibration matrix. As noted earlier e traditional calibration procedure involves application of single-component loads followed by two-component loads which are applied by varying e load on one element while e load on e oer is held constant. The unknown calibration coefficients are determined by making separate or piece-wise curve fits to e output vs. load term for each of e load terms of e ma model (see Ref. 1-3 for more details). This procedure reuires at e calibration loads be restricted to a maximum of two components, which is infreuent in most wind tunnel test conditions. As noted earlier, e recently developed GRM does not place any such restrictions on e type of calibration loads and, in particular, is meod permits simultaneous loading of any combinations of components upto a maximum of six. The GRM erefore enables e calibration loading design to be representative of e model loads in a wind tunnel. A brief description of GRM follows. 3. A BRIEF REVIEW OF GRM Regression analysis is a widely used statistical techniue for investigating and modeling e relationship between variables in various fields including engineering and physical sciences. As noted earlier, regression analysis has recently been applied for balance calibration

3 Application of Global Regression Meod for Calibration of Wind Tunnel Balances data processing at several wind tunnel facilities. But details of application of e regression analysis to balance calibration are not available, at least in e open literature. It was erefore considered appropriate to include here a brief review of regression analysis wi emphasis on its application to determination of calibration coefficients of multi-component balances. The review is based primarily on e material presented in Ref. 6 and 7. Adopting notations commonly used in regression analysis e functional relation between e response y of a system to a set of independent variables x j, j = 1, 2 k is expressed by e following euation: y = β + β x + β x + + β x +ε o k k (2) where ε is e difference between e observed and fitted values of y, and x j s are called as predictors or regressors. The above euation is also called e regression model of e system under consideration. The model represented by en. (2) is called a multiple linear regression model wi k regressors and e coefficients β j, j = 0, 1, k are called e regression coefficients. The term linear is used because en. (2) is a linear function in unknown parameters β j, while e regressor x j can take any form (such as 3 x, sin x etc.). The term multiple refers to e fact at e euation involves more an one regressor. In order to find e unknown coefficients of en. (2) measurements of e response y are made for n different sets of predictors x j. When e number of measurements is greater an e number of unknown coefficients, i.e. n > k+1 e meod of least suares can be used to estimate e regression coefficients as noted below. The set of response values y for different sets of predictor values x j can be written in matrix notation as: y = X β +ε (3) where y is a n x 1 vector of response values, X is a n x p matrix of e levels of regressor variables, β is a p x 1 vector of unknown regression coefficients and ε is a n x 1 vector of random errors, where p= k+1. Using e least suares meod matrix of estimated regression coefficients can be obtained as (see Ref. 6 for derivations) : ( ) ˆ -1 β = X' X X' y (4) provided at e inverse matrix ( ) -1 X' X exists. The vector of fitted values of response corresponding to e measured values i.e. e fitted regression model is given by: ( ) -1 yˆ = Xβ ˆ = X X' X X' y (5) As part of regression analysis an assessment of e fitted regression model is made to ascertain e adeuacy and uality of fit rough statistical procedures called as hypoesis testing and tests for significance and using various uality metrics. Some of e commonly used uality metrics are : standard error of regression ˆ σ, coefficient of variation CV, coefficient of 2 multiple determination R, t-statistic and variance inflation factor VIF (Ref. 6 and 7). Estimates of standard error and confidence interval wids of e estimated regression coefficients are also used to assess e uality of fitted ma model. The above metrics are obtained using certain statistical uantities at are computed using e measured and fitted response data and e estimated regression coefficients. These statistical uantities are usually summarized in an analysis of variance (ANOVA) Table, a format of which is shown in Table.1 Table 1. Source of variation Regression Residual Total Analysis of variance (ANOVA) for significance of regression Sum of Suares SS R SS Res SS T Degrees of Freedom Mean Suare Euations for e various uantities noted in e ANOVA Table and e uality metrics are presented in Ref APPLICATION OF GRM TO BALANCE CALIBRATION DATA ANALYSIS Regression analysis meods described in e previous section are applied to determine e calibration coefficients of multi-component balances. For is purpose e polynomial euation representing e balance ma model (such as en (1)) is expressed in e format used in regression analysis, en (3). In order to do is, e various types of loads k n-k-1 n-1 MS R MS Res F o MS R MSRes

4 Gireesh Yanamashetti and H Sundara Mury 3 F j, F j, F 2 j, FjF k... F j in e polynomial euation are rewritten as load terms g j, j = 1, 2... k, and e coefficients a e, b1 e, b2 e,. d2 e are written as Ce, o, Ce, j, j = 1, 2... k respectively. Wi ese new symbols e balance ma model can be written as : k R e, i = C e, o+ C e, j g j,i j=1 i = 1, 2... n (6) where R e, i = output in e element due to i loading Li C calibration coefficient of e, j = e element associated wi j load term g j, i = value of load term j due to i loading L i L = i i loading In general, each applied load denoted by L i will cause loading of any one of e balance elements (single-component load) or simultaneous loading of two or more components (i.e. multi-component or combined loading). The load terms or regressors g j,i, j = 1, 2... k resulting from each such loading L i are computed from e functional relationships between g j and component loads F 1, F 2... F. These functional relationships are obtained from e chosen ma model. g j En. (6) represents e multiple linear regression model of e balance and e sets of measured values of balance outputs R e,i and e known values of load terms g j,i are utilized in a regression analysis to determine e unknown coefficients C e, j as noted below. En. (6) is written in matrix format as : R e = C E GN (1, n) (1, p) (p, n) (7) where R e is a e vector of outputs of e e element due to loadings L i, i = 1, 2... n, and is given by : R e = [R e,1 R e,2...r e,n ] C E is e extended vector of calibration coefficients of e e element and is given by : C E = [C e,0 C e,1 C e,2...c e,j...c e,k ] G N is e extended load matrix consisting of values of load terms g j for each of e loadings L i, i = 1, 2... n, and is written as : G N = g 1,1 g 1,2.. g 1,i. g 1, n g 2,1 g 2,2.. g 2,i. g 2, n g k,1 g k,2.. g k,i. g k, n The matrix euation (7) representing e ma model of e balance is in row format (which is customary in balance calibration data analysis), and is is converted to e column matrix format commonly adopted in regression analysis by taking e transpose of bo sides of e en. (7) and including e error vector e ma model of e balance is written as : ' ' ' e N E R = G C + ε (8) where indicates transposed uantities. Comparing en. (8) wi en. (3) e following euivalence of e terms used in balance calibration field wi ose used in regression analysis can be written: vector of response y vector of balance outputs matrix of predictors X matrix of load terms ' G N ' R e vector of estimated regression coefficients ˆβ vector of calibration coefficients ' C E. Using e above euivalence relations in en. (4) and after carrying out simplifications, e matrix of calibration coefficients is obtained as:

5 Application of Global Regression Meod for Calibration of Wind Tunnel Balances ( ) C ' ' -1 E = R e G N G N G N (9) The fitted regression model of e balance is given by: ˆ ' ' -1 R e = R e G N (G N G N ) G N (10) An assessment of e adeuacy and uality of e above fitted regression model of e balance can be made using various uality metrics noted in section 3. In order to implement e above GRM concepts for balance calibration, a MATLAB software was developed. The software computes e calibration matrix of a -component balance ( can be between 1 and 6) for e chosen ma model. In addition, e program computes e uality metrics used to assess e adeuacy and uality of e fitted regression model, noted in section 3. Computed uality metrics and oer data can also be utilized to carry out furer analysis to choose e most appropriate ma model or reduce e number of terms in e model by deleting insignificant terms. where F e, i is e residual error of i load in e e balance element and n is e total number of applied loads. The above standard deviation normalized wi respect to its rated load is regarded as a measure of e accuracy of e balance element, i.e., σe Accuracy of e balance element = x 100 % Fe Rated Balance element accuracies are computed separately for e calibration loads and e check loads. 6. CALIBRATIONS OF A 6-COMPONENT BALANCE To illustrate e application of e GRM and use of e computer program developed for is purpose and also to bring out e advantages of e GRM, calibrations of a 6-component internal balance Alough as per e current practice e choice is eier second order or ird order ma model wi or wiout considering e asymmetric behaviour of e balance, e program can also handle higher order ma models at include higher powers of component loads and load cross product terms wi more an two combinations of component loads. 5. ESTIMATION OF BALANCE ACCURACY Accuracy of a balance is estimated by comparing back-calculated loads wi actual applied loads on e balance. The back-calculated loads are e loads computed utilizing e calibration matrix generated from regression analysis and e recorded balance outputs for e applied loads. In general such back calculation of loads are made for all e calibration loads and also for a set of check loads at are not identical to e calibration loads. The difference F e between e back-calculated load and e actual applied load in a balance element e is e residual load error in e computed load F e. A standard deviation of e residual load errors in e e balance element is computed from e following euation : 1 1 n 2 2 σ e = Fe,i n 1 (11) i =1 Figure " diameter floating frame balance

6 Gireesh Yanamashetti and H Sundara Mury Cal 1 : a standard calibration featuring 935 loads consisting of 84 single-component loads and 851 two-component loads Cal 2 : e loads were similar to ose in (i), but, e total number of loads was 300 consisting of 43 single-component and 257 two-component loads, and Cal 3 : a combined load calibration featuring 300 loads consisting of 35 ree-component loads, 37 four-component loads, 38 six-component loads in addition to 30 single-component and 161 twocomponent loads. The magnitude of e reecomponent and six-component loads were similar to ose on a typical model at subsonic / transonic speeds in e NAL 1.2m tunnel. Calibration data from e above calibrations were analyzed using e GRM and ree different irdorder regression models and corresponding 6x96 calibration matrices were obtained. Assessment of e fitted regression model was also made using e various uality metrics noted in section 3 for e ree calibration models. R 2 value which is often used as a measure of e overall success of e regression model was > for all ree calibration models (R 2 = 1 implies a perfect fit). Overall uality of e ree fits was erefore judged excellent. A typical ANOVA table and values of some of e uality metrics are shown in Table 2. Figure 2. Automatic Balance Calibration System Table 2. ANOVA table for AF element Source Deg of Freedom Sum of Suares Mean Suare F were carried out and e calibration data were analyzed using e program. Details of ese calibrations and some results are presented below. The balance is of floating-frame type and strain gauged to measure 5 forces viz. N1, N2, S1, S2 and AF and one moment i.e. RM. Figure 1 shows e major dimensions, sign conventions of e load components and e rated loads of e balance. The balance was calibrated in an Automatic Balance Calibration System (ABCS) at NAL (Figure 2). The ABCS is of non-repositioning type featuring 6 servo-controlled hydraulic actuators for load application and e loads are measured by 0.02% accuracy load cells. Six high resolution optical sensors incorporated in e ABCS measure e balance deflections to an accuracy of 1 micron. More details of e ABCS are found in Ref. 8. The ree calibrations differed in e type and total number of calibration loads as noted below : (i) (ii) (iii) Regression Residual Total E E-6 5.8E+05 Element Std Error of Coeff of R^2 Statistic Regression Variation AF 1.78E For determining e balance accuracy for ese calibrations a set of 64 check loads consisting of 12 single-component loads, 28 four-component loads and 24 six-component loads were applied and balance outputs were recorded. The four-component and sixcomponent loads were generally similar to e loads AF % a) Cal Load index

7 Application of Global Regression Meod for Calibration of Wind Tunnel Balances AF % Cal 1 (935*) Cal 2 (300) Cal 3 (300) Single- & two-component loads Combined loads * indicates total number of calibration -0.4 b) Cal Load index Accuracy % FS AF % c) Cal Load index Figure 3. Comparison of typical residual load error distributions on a typical model during actual wind tunnel tests in e NAL 1.2m tunnel, and magnitude of e check loads were different from ose of e calibration loads. Balance outputs due to e check loads were processed using e computed calibration matrix for backcalculation of check loads for each of e ree calibrations. These back-calculated loads were compared wi known check loads, and residual load errors and balance accuracies were computed as described in section 5 for e ree calibrations. Figure 3 shows a comparison of a typical residual load error distribution for e ree calibrations. Magnitudes of e load errors are, in general, lowest for Cal 3 (which includes combined loadings similar to check loads) compared to ose for 0.00 N1 N2 S1 S2 AF RM Balance elements Figure 4. Comparison of balance element accuracies for calibrations wi different types and number of calibration loads Cal 1 and Cal 2 (which feature only single-component and two-component calibration loads). Figure 4 shows e accuracy of e six elements of e balance for e ree calibrations. It is seen at e balance element accuracies for Cal 3 were substantially better an ose for e oer two calibrations. As noted earlier, application of such combined loads during calibration is permissible only when e data analysis is made using GRM. It is also seen at ese accuracy improvements have been obtained wi a substantially reduced number of calibration loadings. The improvement in balance accuracy and e use of reduced number of calibration loads us bring out e advantages of e GRM for balance calibration. However, it is to be noted at application of combined loads similar to model loads on e balance is practical only wi automatic loading machines such as e ABCS. 7. CONCLUSIONS The traditional meod for calibration of multicomponent balances restricts e load combinations in e calibration load design to a maximum of two components, which are generally unrepresentative of actual wind tunnel test conditions. The Global Regression Meod does not place any such restrictions and permits multi-component calibration loads wi load combinations of more an two components at are representative of wind tunnel loads. It has been shown at calibration wi such multi-component loads along

8 Gireesh Yanamashetti and H Sundara Mury wi use of GRM leads to significant improvements in balance accuracy. It has also been shown at such improvements in accuracy can be obtained wi a substantially reduced number of calibration loads and a conseuent reduction of calibration effort. Use of GRM also enables adoption of higher order ma models (higher an e ird order used currently) including load cross product terms wi more an two load combinations for balance calibrations. Inclusion of load product terms featuring load combinations similar to ose on a model under wind tunnel test conditions may lead to furer improvements in accuracy of measured aerodynamic forces and moments. NAL Automatic Balance Calibration System (ABCS), NAL PD NT 0714, June 2007 ACKNOWLEDGEMENTS Auors ank Head and Dy Head of NTAF division for e encouragement given and Director NAL for kind permission to present is paper in e conference. REFERENCES 1. AIAA/GTTC Internal Balance Technology Working Group : Recommended Practice, Calibration and Use of Internal Strain-Gage Balances wi Application to Wind Tunnel Testing, AIAA R , Galway, R.D. : A Comparison of Meods for Calibration and Use of Multi-Component Strain Gauge Wind Tunnel Balances, NRC Aeronautical Report LR-600, March Cook, T.A.: A Note on e Calibration of Strain Gauge Balances for Wind Tunnel Models, RAE Technical Note AERO. 2631, December Philipsen, I: Improvements in Balance Calibration in DNW. Presented at Fif International Symposium on Strain Gauge Balances. Assoils, France, May Ulbrich, N. and Volden, T.: Predictive Capabilities of Regression Models used for Strain-Gage Balance Calibration Analysis. AIAA , Presented at e 26 AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Seattle, Washington, June Montgomery, D.C, Peck, E.A. and Vining, G.G.: Introduction to Linear Regression Analysis, Third Edition, John Wiely & Sons (Asia) Pte. Ltd, Singapore, Draper, N. and Smi, H. : Applied Regression Analysis, Third Edition, John Wiley & Sons, Inc., New York, Gireesh Yanamashetti, Jagadeeswarachar S.P. and Sundara Mury, H: A Description of Sub- Systems and Calibration Formulations of e