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Calibration uncertainty estimation of a strain-gage external balance
Measurement 46 (2013) 24–33
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Review
Calibration uncertainty estimation of a strain-gage external balance M.L.C.C. Reis a,⇑, R.M. Castro b, O.A.F. Mello c a b c
Instituto de Aeronáutica e Espaço, São José dos Campos, Brazil Instituto de Estudos Avançados, São José dos Campos, Brazil EMBRAER, São José dos Campos, Brazil
a r t i c l e
i n f o
Article history: Received 27 February 2012 Received in revised form 11 June 2012 Accepted 11 September 2012 Available online 12 October 2012 Keywords: Aerodynamic balance Calibration uncertainty Wind tunnel tests
a b s t r a c t Aerodynamic balances are employed in wind tunnels to estimate the forces and moments acting on the model under test. This paper proposes a method for the evaluation of uncertainty in the calibration of an external multi-component aerodynamic balance. In order to obtain a numerical relationship between aerodynamic loads and the balance sensor responses, a calibration is performed prior to the tests by applying known weights to the balance. A multivariate polynomial fitting using the least squares method is used to interpolate the calibration data points. The uncertainties of both the applied loads and the readings of the sensors are considered in the regression. The data reduction includes the estimation of the calibration coefficients, the fitted values of the aerodynamic load components and their corresponding uncertainties, as well as the goodness of fit. Ó 2012 Elsevier Ltd. All rights reserved.
Contents 1.
2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Multi-component aerodynamic balances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. The purpose of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Functional relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Loading combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Parameters estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The covariance matrix V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Matrix VW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Matrix VR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Matrix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. The fitted aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Matrix VW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Matrix VR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Matrix V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Fitted aerodynamic forces and moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
⇑ Corresponding author. E-mail addresses: [email protected] (M.L.C.C. Reis), [email protected] (R.M. Castro). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.09.016
25 25 25 25 27 27 28 28 28 28 29 29 29 29 30 30 31 31 31 32
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4.
3.6. Goodness of fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. The result of the calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The staff of the Aerodynamics Division of the Institute of Aeronautics and Space, Brazil, has been concerned with the quality of the data originating from aerodynamic tests. Following international recommendations based on the Metre Convention, several studies have been conducted in order to improve the metrological reliability of measurement procedures, tests and calibrations carried out in the Subsonic and Transonic Facilities, named TA-2 (wind tunnel number 2) and TTP (Pilot Transonic Wind Tunnel), respectively. One of these projects is the development of a methodology for the evaluation of uncertainty in measurement of the aerodynamic loads acting on the test article. The instrument used to measure the aerodynamic loads is the aerodynamic balance, whose calibration uncertainty contributes significantly to the overall uncertainty in wind tunnel testing. International organizations have been exchanging information and collaborating on the development of a balance calibration uncertainty methodology, but this has only been partially addressed [1]. 1.1. Aerodynamic loads Fig. 1 presents the terminology for designating the aerodynamic load components: drag force (D), side force (Y), lift force (L), rolling moment (l), pitching moment (m) and yawing moment (n). For simplicity, at TA-2 and TTP wind tunnels, they are named F1, F2, F3, F4, F5 and F6, respectively. 1.2. Multi-component aerodynamic balances Figs. 2 and 3 present the two balances employed at TTP and TA-2: internal and external, respectively. The former is
25
32 32 32 32 32
approximately 0.12 m long and is designed to fit within the test article (Fig. 2). The latter is 5.7 m high and carries the loads outside the tunnel before they are measured (Fig. 3). The balance measures the loads by using strain-gages arranged in a Wheatstone bridge. The results presented in this study originate from the calibration of an external balance. A six component external balance is used to measure the aerodynamic loads acting on the model during a wind tunnel test at the TA-2 aerodynamic facility. Fig. 4 presents a test article inside the test section. The schematic picture illustrating the connection between the model and the external balance is shown in Fig. 5. The positioning of the six load cells on the balance is also shown. A balance calibration is performed prior to the tests. The calibration is accomplished by applying loads to a calibration system constituted of a calibration cross, trays, cables and pulleys (Figs. 6 and 7). There are 14 trays for the application of weights. A set of approximately one hundred 10 kg weights is used. The load cells of the balance provide the readings Ri(i = 1, . . . , 6). The calibration cross is attached to the model support system, at the same location as a test article would be positioned. The applied calibration weights to the calibration system simulate the condition of wind tunnel airflow exerting forces and moments to the test article. Once the calibration is accomplished, the calibration cross is replaced by the model to be tested. 1.3. The purpose of the paper The calibration of wind tunnel balances requires the application of known weights to the calibration system and the reading of the electrical responses from the balance. In this paper, the weights are considered the output quantities and the readings are the input quantities of the calibration model.
Fig. 1. Aerodynamic forces and moments. a: angle of attack, b: sideslip angle.
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Fig. 2. Internal balance.
Fig. 3. External balance.
Fig. 4. Model under test at the TA-2 facility.
Wind tunnel facilities around the world are concerned about the mathematical modeling of the calibration [2], the design of balances [3], the development of balance calibration systems for minimizing sources of uncertainties [4,5], the optimization of software tools to process balance calibration data [6], the design of experiment of the calibration loading [7], and the analysis of uncertainty of the calibration [8–10]. Most research deals with more than one of these subjects and the interest is always to help experimentalists in the quantification and minimization of the balance
contribution to the uncertainty in the aerodynamic forces and moments measured during test campaigns. The evolution of the analysis of balance calibration uncertainty at TA-2 and TTP facilities can be seen in [11– 14]. These references present the data reduction employed to predict the aerodynamic forces and moments and associated uncertainties by applying least squares fitting to the calibration data [15,16]. Firstly, as the uncertainties in the calibration data were not known, the fitting was applied considering all the uncertainties in data points equal to 1 [11]. This was also done in Refs. [8,9]. In such an approach, the assigned uncertainties to the output quantities is performed considering information coming from the fitting only. No experimental investigation about the uncertainty in data to be fitted is done. In [12], the contribution of the load cells readings was considered applying the law of propagation of uncertainties as recommend by international standardization [17], but this information was only included after performing the fitting. The results obtained in [11] were validated using two methods: Monte Carlo [18] and Neural Networks. These results are presented in [13]. The failure of not performing the least squares fitting considering uncertainty in data points still remained and a study employing the internal balance to analyze the sources of uncertainty caused by the bridge readings was conducted. A particular loading, loading number 1, was replicated eight times during the calibration and a covariance matrix for the bridge readings was constructed [14], but the experimental evaluation of uncertainties in the applied weights was not accomplished. The purpose of the replicate measurements was to quantify the interdependence between the bridges of the balance. The method was improved and transferred to the calibration of the external balance, which is described in this paper. The enhancement over the previous procedure was to obtain the covariance matrix of the bridge readings from three different sets of balance calibration, instead of simply replicating a specific loading. Moreover, besides the uncertainties in the sensor readings, the uncertainties in the applied weights are also quantified experimentally and considered in the least squares regression. Such a procedure has not been found in aerodynamic balance calibration literature to date. The steps of the method proposed in this paper will be shown in Section 2: the loading plan, the calibration mathematical modeling and how the parameters of the mathematical modeling are estimated. The way the recognized sources of uncertainty were quantified and assigned to the calibration data points is also presented. A procedure on how to compose the covariance matrix of the bridge readings is explained. In Section 3, the method was verified for the fitted aerodynamic components by using data obtained from calibration and was analyzed in terms of quality of fitting. The data reduction employed in this study can be extended to the calibration of other multivariate measurement models which relate quantities of different kinds. Obviously, the investigation of sources of errors presented
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
27
Fig. 5. The interface model-external balance.
here must be adapted for the particularities of other measurement instruments. Metrological terms encountered in this paper comply with [19]. 2. Methodology
Fig. 6. Loading system for the balance calibration.
The calibration of an external balance involves the application of known weights to the balance and recording strain-gages readings at each force and moment combination. There are two ways of calibrating the balance: at sideslip angle beta equal to zero and at beta different from zero. At the subsonic wind tunnel TA-2, a calibration performed at beta equal to zero is called alpha calibration and beta calibration when otherwise. In this study only the alpha calibration will be considered and the results should be limited to tests conducted when the sideslip angle is set to zero. 2.1. Functional relationship The mathematical modeling of the calibration relates the aerodynamic components Fi to functions of the strain-gage bridges readings Ri (Eq. (1)). The system is multivariate and consists of a linear combination of 27 functions of R [15,16]. These functions are called basis functions and correspond to: R1 ; R2 ; R3 ; R4 ; R5 ; R6 ; R21 ; R1 R2 ; R1 R3 ; R1 R4 ; R1 R5 ; R1 R6 ; R22 ; R2 R3 ; . . . ; R26 . There are 27 adjustable parameters for each one of the six aerodynamic components. The model’s dependence on its parameters is linear.
Fig. 7. Calibration system, schematic. F: forces, M: moments.
28
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Table 1 Examples of nominal weights used in calibration for loadings number 1, 27, 46, 58 and 69. Unit: kgf for forces F1, F2, F3 and kgf m for moments F4, F5 and F6. Loading number
F1
F2
F3
F4
F5
F6
1 27 46 58 69
0 80 0 0 0
0 0 80 80 0
80 0 0 0 80
0 40 0 0 0
40 0 0 40 0
0 0 48 0 48
^ ¼ ðRT V 1 RÞ1 ðRT V 1 FÞ p
þ ai;7 R21 þ ai;8 R1 R2 þ ai;9 R1 R3 þ ai;10 R1 R4
V p^ ¼ ðRT V 1 RÞ1
þ ai;11 R1 R5 þ ai;12 R1 R6 þ ai;13 R22 þ ai;14 R2 R3 þ þ
ð2Þ
where RT is the transpose of the matrix R; V is the covariance matrix; and V1 is the matrix inverse of V. Matrix (RTV1R)1 contains information about uncertainties of the estimated parameters. Its diagonal elements correspond to the variances (squared uncertainties u2p^ ) and the off-diagonal elements are the covariances of the fitted parameters. The matrix can be denoted by V p^ . Therefore:
F i ¼ ai;1 R1 þ ai;2 R2 þ ai;3 R3 þ ai;4 R4 þ ai;5 R5 þ ai;6 R6
ai;27 R26 :
problem has a dimension of 73 27 and is constructed from the basis functions. Denoting the parameters of the ^ , the least squares results in: fitting by p
ð3Þ
ð1Þ
Cross-product terms involving the input quantities are included in the polynomial because it is not possible to eliminate the interactions between bridges completely. The task during the calibration is to apply known weights to the calibration system while recording the resulting bridge readings and to calculate the 27 unknown polynomial coefficients in the mathematical model. During the wind tunnel tests, using the bridge readings and the coefficients evaluated through the calibration, one can estimate the aerodynamic loads exerted by the airflow on the model under test by using Eq. (1).
2.4. The covariance matrix V The covariance matrix V is related to the uncertainties of the applied loads, uF. It is made up of the contributions of the sources of error due to the application of weights in the calibration system VW and the uncertainties in the readings of the bridges VR:
V ¼ V W þ CV R C T
ð4Þ
The terms in Eq. (4) are described in the following sections.
2.2. Loading combinations Seventy-three loading combinations are employed for alpha calibration. Table 1 presents some typical loading combinations which will be used in this paper to illustrate the results obtained when the method proposed here is employed. The values represent the weights applied to the balance originating from the application of weights on the trays T1, . . ., T14 of the calibration system (Table 2). The length of the lever arm of the cross is equal to 0.5 m for F4 and F5 and to 0.6 m for F6. 2.3. Parameters estimation According to the least squares method a calibration curve is fitted to each set of the N = 73 data points constituted by the bridge readings and the applied weights [15,16]. Each of the aerodynamic components is arranged in a vector F represented by the applied weights, whose dimension is 73 1. The design matrix R of the fitting
2.4.1. Matrix VW The matrix VW has a dimension of 73 73 and in this study it is considered diagonal. Its elements are based on the uncertainties of the weights employed in the loading process and also in an estimation of the uncertainties caused by the balance calibration system. The former contribution can be promptly evaluated by verifying the uncertainties of the weights applied to the calibration cross. The latter is based on the quantification of the sources of error that affect the resolution of the calibration system such as frictional forces and misalignments between cables and pulleys. The quantification of the calibration system resolution involves applying small weights to the trays and verifying when the load cells of the aerodynamic balance change their indications in a detectable way. Studies were carried out for all six aerodynamic components. The nominal values of the weights employed in this analysis are equal to 1, 2, 5 and 10 g.
Table 2 Nominal values of the weights applied to the trays T1–T14 for loadings number 1, 27, 46, 58 and 69. Unit: kg. Loading number
1 27 46 58 69
Weights (kg) T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
T13
T14
0 80 0 0 0
0 0 0 0 0
0 0 0 0 40
0 0 80 40 0
0 0 0 0 0
0 0 0 40 40
0 40 0 0 20
0 0 0 0 20
0 0 0 0 20
20 0 0 40 20
20 0 0 0 0
20 40 0 0 0
60 0 0 40 0
0 0 0 0 0
29
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2.4.2. Matrix VR Replicate measurements supply information for the construction of VR. Three calibrations under repeatability conditions were carried out over a short period of time. The diagonal elements of VR are represented by the variances and the off diagonal elements are the covariances between the load cell readings (Eq. (5)). The first 73 lines are elements related to R1, followed by the 73 lines related to R2 and so on, resulting in a 438 438 matrix.
2
uðR1;1 ; R1;1 Þ
uðR1;1 ; R1;2 Þ
uðR1;1 ; R1;73 Þ
The elements @F1,j/@Ri, j of matrix C are represented by Eq. (8). The parameters a of Eq. (8) are estimated by performing the least squares fitting to the calibration data, as shown in Section 2.3, with V equal to VW in Eq. (4), i.e., VR is not considered in the first regression. This means that the fitting is applied twice to the calibration data: the first time V = VW and the second time V = VW + CVRCT.
uðR1;1 ; R2;1 Þ
uðR1;1 ; R6;73 Þ
3
6 uðR ; R Þ uðR ; R Þ uðR ; R Þ uðR ; R Þ uðR ; R Þ 7 1;2 1;1 1;2 1;2 1;2 1;73 1;2 2;1 1;2 6;73 7 6 7 6 7 6 .. .. .. .. .. .. .. 7 6 . . . . . . . 7 6 7 6 V R ¼ 6 uðR1;73 ; R1;1 Þ uðR1;73 ; R1;2 Þ uðR1;73 ; R1;73 Þ uðR1;73 ; R2;1 Þ uðR1;73 ; R6;73 Þ 7 7 6 6 uðR2;1 ; R1;1 Þ uðR2;1 ; R1;2 Þ uðR2;1 ; R1;73 Þ uðR2;1 ; R2;1 Þ uðR2;1 ; R6;73 Þ 7 7 6 7 6 .. .. .. .. .. .. .. 7 6 5 4 . . . . . . .
ð5Þ
uðR6;73 ; R1;1 Þ uðR6;73 ; R1;2 Þ uðR6;73 ; R1;73 Þ uðR6;73 ; R2;1 Þ uðR6;73 ; R6;73 Þ
The first subscript in Eq. (5) identifies the load cells R1 to R6 and the second is related to each of the 73 loadings. 2.4.3. Matrix C Matrix C has a dimension of 73 438 and its elements are the sensitivity coefficients evaluated by taking the partial derivatives in Eq. (1). For example, for the aerodynamic force F1, matrix C is formed by:
2
@F 1;1
6 @R1;1 6 @F 1;2 6 @R1;1 6 6 . C¼6 6 .. 6 @F 6 1;72 6 @R1;1 4 @F 1;73 @R1;1
@F 1;1 @R1;2
@F 1;1 @R1;73
@F 1;1 @R2;1
@F 1;1 @R6;73
@F 1;2 @R1;2
@F 1;2 @R1;73
@F 1;2 @R2;1
@F 1;2 @R6;73
.. .
.. .
@F 1;72 @R1;73
@F 1;72 @R2;1
@F 1;73 @R1;73
@F 1;73 @R2;1
.. . @F 1;72 @R1;2 @F 1;73 @R1;2
..
.
3
7 7 7 7 .. .. 7 7 . . 7 7 @F 1;72 7 @R6;73 7 5 @F @R1;73 6;73
6 6 6 6 6 C¼6 6 6 6 4
0
0
@F 1;1 @R2;1
0
0
@F 1;2 @R1;2
0
0
0
.. .
.. .
..
.
.. .
.. .
..
.. .
0
0
0
0
0
@F 1;73 @R1;73
0 0
@R1;1
ð6Þ
.
0 @F 1;73 @R6;73
3 7 7 7 7 7 7 7 7 7 5
2.5. Goodness-of-fit The goodness-of-fit is evaluated through the chi-square quantity, v2 [15,16]. There is one v2 for each aerodynamic load. Taking into account the covariance matrix V:
v2 ¼ ðF Fb ÞT V 1 ðF Fb Þ
The first and second subscripts in Eq. (6) identify the aerodynamic component or load cell (i = 1, . . . , 6) and loading number (j = 1, . . . , 73), respectively. Note that the first block of 73 columns is related to R1, the second block is related to R2 and so on. In Eq. (6), only the sensitivity coefficients whose second subscripts correspond to the same loading have values different from zero. Therefore, matrix C is reduced to:
2 @F 1;1
Finally, matrix CVRCT corresponds to the contribution to the variances in F due to uncertainties in the bridge readings and has dimension of 73 73.
ð9Þ
where F is the vector of applied forces (or moments) to the b is the vector of fitted forces (or mocalibration cross; F ments), estimated by the least squares. A value for the chi-square which indicates a good fit is typically close to the number of degrees of freedom:
m ¼ N Np
ð10Þ
In Eq. (10) N = 73, the number of data points and Np = 27, the number of coefficients in Eq. (1) to be fitted by the least squares method. This leads to another quantity, the reduced chi-square:
v2red ¼
v2 m
ð11Þ
whose desired value is approximately equal to 1. 2.6. The fitted aerodynamic loads
ð7Þ After performing the least squares regression to esti^ of adjusted parameters, one can obtain mate the vector p b . The 73 fitted values of the fitted forces and moments F each aerodynamic component are calculated by multiply^: ing the matrix R by the corresponding vector p
30
@F 1;j @R1;j @F 1;j @R2;j @F 1;j @R3;j @F 1;j @R4;j @F 1;j @R5;j @F 1;j @R6;j
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¼ a1;1 þ 2a1;7 R1;j þ a1;8 R2;j þ a1;9 R3;j þ a1;10 R4;j þ a1;11 R5;j þ a1;12 R6;j ¼ a1;2 þ a1;8 R1;j þ 2a1;13 R2;j þ a1;14 R3;j þ a1;15 R4;j þ a1;16 R5;j þ a1;17 R6;j ¼ a1;3 þ a1;9 R1;j þ a1;14 R2;j þ 2a1;18 R3;j þ a1;19 R4;j þ a1;20 R5;j þ a1;21 R6;j ¼ a1;4 þ a1;10 R1;j þ a1;15 R2;j þ a1;19 R3;j þ 2a1;22 R4;j þ a1;23 R5;j þ a1;24 R6;j
ð8Þ
¼ a1;5 þ a1;11 R1;j þ a1;16 R2;j þ a1;20 R3;j þ a1;23 R4;j þ 2a1;25 R5;j þ a1;26 R6;j ¼ a1;6 þ a1;12 R1;j þ a1;17 R2;j þ a1;21 R3;j þ a1;24 R4;j þ a1;26 R5;j þ 2a1;27 R6;j
^ Fb ¼ Rp
ð12Þ
^ is 27 1. where R is 73 27 and p The associated uncertainty in the fitted aerodynamic component is the positive square root of the diagonal elements of:
V b ¼ RV p^ RT
ð13Þ
F
where V p^ is the matrix (RTV1R)1 as explained in Section 2.3. Eq. (13) is equivalent to applying the law of propagation of uncertainty in the polynomial relating the aerodynamic loads and load cell readings as represented in Eq. (1) [17]. Table 3 Values of the weights that must be applied to the calibration system in order to cause a perceptible change in the load cells readings (103 kg). Load cell
Applied weight
R1 R2 R3 R4 R5 R6
2 2 32 5 8 5
3. Results and discussion The least squares regression provides the fitted param^ , as well as its variances and covariances V p^ . eters vector, p The calibration data reduction also includes the fitted load b and their uncertainties V . The quality of the fit is values F bF quantified by the chi-square quantity, v2. Code in MATLABÒ and ExcelÒ worksheets were used to perform the data reduction. 3.1. Matrix VW The evaluation of uncertainties to compose matrix VW is of type B. The first contribution considered is that due to the errors of the weights employed in the loading process. It was found that the maximum error in the applied 10 kg weights is around 1 g in 10,000 g. Taking into account a rectangular distribution with limits equal to 1 103 kg, this leads to an uncertainty value of 5.8 104 kg. The resolution of the calibration system is the second contribution. It was estimated by applying incremental weights of the order of 103 kg to specific trays related to the aerodynamic component under investigation and by verifying the indication of the corresponding load cell. Different experiments were carried out to stimulate each one of the six sensors individually and the worst case found
Fig. 8. Study of the resolution of the calibration system related to R3. Each group of five readings corresponds to the same applied weight.
31
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33 Table 4 ^ for the force component F1. Second Estimated polynomial parameters p fitting. Parameter
^ p
up^
^ up^ =p
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27
3.525849 0.011648 0.008513 0.000051 0.001620 0.004485 0.000557 0.000234 0.001408 0.000242 0.000164 0.000087 0.000862 0.000599 0.000705 0.000283 0.000658 0.000080 0.000127 0.000369 0.000635 0.000197 0.000177 0.000583 0.000001 0.000250 0.000443
0.001298 0.001295 0.001440 0.000649 0.000534 0.000501 0.000058 0.000163 0.000185 0.000082 0.000067 0.000064 0.000364 0.000864 0.000383 0.000309 0.000290 0.000447 0.000430 0.000350 0.000328 0.000088 0.000155 0.000144 0.000057 0.000117 0.000058
2717.01 8.99 5.91 0.08 3.04 8.95 9.53 1.43 7.60 2.92 2.43 1.35 2.36 0.69 1.84 0.91 2.27 0.17 0.29 1.05 1.93 2.23 1.13 4.04 0.02 2.13 7.63
Table 6 Estimates of the uncertainty for the aerodynamic components. Units as in Table 5. Load number
Force
1 27 46 58 69
Moments
uF1
uF2
uF3
uF4
uF5
uF6
0.0125 0.0157 0.0126 0.0130 0.0127
0.0126 0.0148 0.0141 0.0128 0.0156
0.0132 0.0140 0.0124 0.0127 0.0137
0.0301 0.0137 0.0144 0.0354 0.0144
0.0147 0.0200 0.0132 0.0129 0.0128
0.0124 0.0138 0.0147 0.0129 0.0125
Table 7 Goodness of fit. Load cell
v2
v2red
F1 F2 F3 F4 F5 F6
232.33 271.65 453.91 67.69 165.99 529.83
5.05 5.91 9.87 1.47 3.61 11.52
the readings of the load cell R1 originating from calibration number 1, followed by the 73 elements of the readings of the load cell R2 and so on. The same procedure is repeated for columns 2 and 3, using the data set originated from the second and third calibrations, respectively. The matrix is then transposed.
was for R3 (Table 3 and Fig. 8). In Fig. 8, the first five readings correspond to the application of 2 g on trays 11, 12, 13 and 14, i.e., 8 g were applied to the system. In the sequence, from reading number 6 up to number 10, 16 g were applied, and so on. One can see that the behavior of the readings becomes stable for 32 g and the standard deviation of a rectangular distribution with lower and upper bounds equal to this limit was chosen as the resolution of the system, resulting in 1.8 102 kg. Table 3 presents the results of the type B evaluation of VW for the six load cells.
3.2. Matrix VR The covariance matrix VR is calculated through the MATLABÒ code, from a previous 438 3 matrix, named R, where the first 73 elements of column 1 correspond to
3.3. Matrix V After computing the terms VW and CVRCT of Eq. (4), ma^ of trix V can be used in Eq. (2) to estimate the parameters p the fitting. 3.4. Estimation of parameters Performing the least squares fitting for the second time to the experimental data, i.e., considering the uncertainties in the data points estimated through Eq. (4), results in the ^ presented in Table 4. Quantities a1 to a6 have units vector p of kgf mV1 and a7 to a27 have units of kgf mV2 when in the case of force components F1, F2 and F3. For moment components F4, F5 and F6, units are kgf m mV1 for a1 to a6 and kgf m mV2 for the remaining. One can see the important significance of a1 for F1, a2 for F2, a3 for F3, etc.
Table 5 Aerodynamic forces and moments resulting from the application of the least squares fitting. Units: kgf (force) and kgf m (moment). Load number
1 27 46 58 69
Force
Moments
F1
F2
F3
F4
F5
F6
0.0107 80.0792 0.0425 0.0313 0.0116
0.0164 0.0186 80.0198 79.9407 0.0949
79.9812 0.0141 0.0212 0.0415 79.9505
0.0437 39.9855 0.0017 0.0695 0.0358
40.0103 0.0394 0.0168 40.0592 0.0356
0.0443 0.0080 47.9342 0.0654 47.9397
32
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
Table 8 Values of aerodynamic forces and moments and corresponding uncertainty. Units: newton for force and newton metro for moment. Load
F1
uF1
F2
uF2
F3
uF3
F4
uF4
F5
uF5
F6
uF6
1 27 46 58 69
0.011 79.914 0.042 0.031 0.012
0.012 0.016 0.013 0.013 0.013
0.016 0.019 79.855 79.776 0.095
0.013 0.015 0.014 0.013 0.016
79.816 0.014 0.021 0.041 79.786
0.013 0.014 0.012 0.013 0.014
0.044 39.903 0.002 0.069 0.036
0.030 0.014 0.014 0.035 0.014
39.928 0.039 0.017 39.977 0.036
0.015 0.020 0.013 0.013 0.013
0.044 0.008 47.835 0.065 47.841
0.012 0.014 0.015 0.013 0.013
Nevertheless, some other parameters are not negligible, as a consequence of mechanical coupling between components of the aerodynamic balance. The uncertainties of the fitted parameters up^ (Table 4) are the positive square root of the diagonal elements of the matrix V p^ (Eq. (3)), as explained in Section 2.3. ^ =up^ between the fitted parameter and its The ratio p uncertainty is also shown in Table 4 and indicates the importance of each of the 27 fitted parameters to the polynomial modeling expressed by Eq. (1). Once again, values show that there is significant contribution besides those parameters a1, a2, a3, a4, a5, and a6 to the mathematical model, as expected. 3.5. Fitted aerodynamic forces and moments One can use the results of the external balance calibration to predict the aerodynamic loads which would act on a model under test in the wind tunnel. The values of the fitb (Eq. (12)), and the ted aerodynamic forces and moments, F associated uncertainties, V b (Eq. (13)), are presented in TaF bles 5 and 6, respectively, for the examples of calibration loadings expressed in Table 1. Each aerodynamic component was calculated by using its corresponding fitted parameters and by choosing the load cells values read at loadings number 1, 27, 46, 58 and 69. 3.6. Goodness of fit The chi-squared v2 (Eq. (9)) and reduced chi-square v2red (Eq. (11)) values of the fitting are presented in Table 7. Estimating the chi-square should result in a value around the number of degrees of freedom of the polynomial fit [15,16]. As there are 73 equations and 27 unknowns, the number of degrees of freedom is equal to 46 (Eq. (10)). The magnitudes presented in Table 7 are greater than expected, revealing that either the math model or the quantification of the uncertainties presented in the experiment, or even both, should be better investigated. 3.7. The result of the calibration The result of the calibration for loadings 1, 27, 46, 58 and 69 are shown in Table 8. The conversion to SI units was performed by multiplying the values in Tables 5 and 6 by the ratio between local g = 9.7864081 m/s2 and normal g = 9.80665 m/s2. 4. Conclusions A method for the estimation of uncertainty in external balance calibration was proposed. Results of the curve fit-
ting include the estimation of the values and uncertainties of the fitting parameters, as well as a statistical measure of goodness-of-fit. The law of propagation of uncertainty was employed to estimate the uncertainties in the aerodynamic components drag, side and lift forces and rolling, pitching and yawing moments. Least squares method was employed twice to fit the data points. The first step is necessary in order to estimate the contribution of the independent variable Ri to the overall uncertainty in Fi. The calibration model is multivariate and second order terms were considered in the mathematical modeling because the external balance is not able to separate the input quantities completely. Recognized uncertainty contributions to the calibration are due to the applied weights to the trays, resolution of the measurement system and replicated loadings. Studies were carried out to quantify the resolution of the system for each one of the six aerodynamic components, individually. Although different values were found, the main contribution observed for the reading of the lift force was ascribed to the resolution of the system because the calibration modeling is multivariate. However, in the future, experiments to investigate the contributions in a combined way must be considered. The values of the chi-square quantity greater than the degrees of freedom imply either that the adopted mathematical model is not suitable for the calibration or the uncertainties of the data points are larger than stated. Indeed, the identification of the sources of error caused by the calibration system and their consequent contributions to the uncertainty of the load components has still not been completely accomplished. Other studies considering different mathematical relationships between the aerodynamic loads and load cell readings are in progress.
Acknowledgements The authors would like to express their gratitude to FAPESP, the Foundation for Support of Research of the State of São Paulo, under Grant Nos. 2000/13769-0, 2002/01933-6, 2006/02628-3, and 2009/09370-0.
References [1] AIAA R-091-2003 Recommended Practice, ‘‘Calibration and Use of Internal Strain-Gage Balances with Application to Wind Tunnel Testing, American Institute of Aeronautics and Astronautics, VA, USA, 2003. [2] T.H. Johnson, P.A. Parker, D. Landman, Calibration modeling of nonmonolithic wind-funnel force balances, Journal of Aircraft 46 (6) (2010).
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33 [3] B.F.R. Ewald, Multi-component force balances for conventional and cryogenic wind tunnels, Measurement Science and Technology 11 (2000) R81–R94. [4] P.A. Parker, T. Liu, AIAA 2002-2792, uncertainty analysis of the single-vector force balance calibration system, in: 22nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, St. Louis, USA, 24–27 June, 2002. [5] R. Bergmann, I. Philipsen, AIAA 2010-4544, an experimental comparison of different load tables for balance calibration, in: 27th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Chicago, USA, 28 June–1 July, 2010. [6] N. Ulbrich, AIAA 2011-949, iterative strain-gabe balance calibration data analysis for extended independent variable sets, in: 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, USA, 4–7 June, 2011. [7] D. Landman, AIAA 2007-1604, a wind tunnel external balance calibration using design of experiments, in: U.S. Airforce T & E Days, Florida, USA, 13–15 February, 2007. [8] J. Tripp, P. Tcheng, AIAA 1994-2589, determination of measurement uncertainties of multi-component wind tunnel balances, in: 18th AIAA Aerospace Ground Testing Conference, Colorado Springs, USA, 20–23 June, 1994. [9] I. Philipsen, J. Zhai, AIAA 2007-143, comparative study of straingauge balance calibration procedures using the balance calibration machine, in: 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, USA, 8–11 January, 2007. [10] M.E. Kammeyer, M.L. Rueger, AIAA 2008-4029, estimation of the uncertainty in internal balance calibration through comprehensive error propagation, in: 26th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Seattle, USA, 23–26 June, 2008.
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[11] M.L.C.C. Reis, O.A.F. Mello, S. Uyeno, O.S. Sampaio, AIAA 2002-2793, a method for the estimation of the calibration uncertainty of an external six-component wind tunnel balance, in: 22nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, St. Louis, USA, 24–27 June, 2002. [12] M.L.C.C. Reis, O.A.F. Mello, S. Uyeno, AIAA 2003-3884, calibration uncertainty of an external six-component wind tunnel balance, in: 33rd AIAA Fluid Dynamics Conference and Exhibit, Orlando, USA, 23–26 June, 2003. [13] M.L.C.C. Reis, W.J. Vieira, I.M. Barbosa, O.A.F. Mello, L.A. Santos, AIAA 2004-2300, validation of an external six-component wind tunnel balance calibration, in: 24th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Portland, USA, 28 June–1 July, 2004. [14] M.L.C.C. Reis, R.M. Castro, J.B.P. Falcao Filho, O.A.F. Mello, calibration uncertainty estimation for internal aerodynamic balance, in: 12th IMEKO TC1 TC7 Joint Symposium on Man, Science Measurement, Annecy, France, 3–5 September, 2008. [15] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, second ed., Cambridge University Press, 1992. [16] P.R. Bevington, D.K. Robinson, Data Reduction and Error Analysis – For the Physical Sciences, third ed., McGraw-Hill, 2003. [17] JCGM 100: 2008 (F), Evaluation of Measurement Data – Guide to the Expression of Uncertainty in Measurement, Joint Committee for Guides in Metrology, 2008. [18] JCGM 101: 2008, Evaluation of Measurement Data – Supplement 1 to the Guide to the Expression of Uncertainty in Measurement – Propagation of Distributions Using Monte Carlo Method, 2008. [19] JCGM 200: 2012, International Vocabulary of Metrology – Basic and General Concepts and Associated terms (VIM), third ed., Joint Committee for Guides in Metrology, 2008.
Contents lists available at SciVerse ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Review
Calibration uncertainty estimation of a strain-gage external balance M.L.C.C. Reis a,⇑, R.M. Castro b, O.A.F. Mello c a b c
Instituto de Aeronáutica e Espaço, São José dos Campos, Brazil Instituto de Estudos Avançados, São José dos Campos, Brazil EMBRAER, São José dos Campos, Brazil
a r t i c l e
i n f o
Article history: Received 27 February 2012 Received in revised form 11 June 2012 Accepted 11 September 2012 Available online 12 October 2012 Keywords: Aerodynamic balance Calibration uncertainty Wind tunnel tests
a b s t r a c t Aerodynamic balances are employed in wind tunnels to estimate the forces and moments acting on the model under test. This paper proposes a method for the evaluation of uncertainty in the calibration of an external multi-component aerodynamic balance. In order to obtain a numerical relationship between aerodynamic loads and the balance sensor responses, a calibration is performed prior to the tests by applying known weights to the balance. A multivariate polynomial fitting using the least squares method is used to interpolate the calibration data points. The uncertainties of both the applied loads and the readings of the sensors are considered in the regression. The data reduction includes the estimation of the calibration coefficients, the fitted values of the aerodynamic load components and their corresponding uncertainties, as well as the goodness of fit. Ó 2012 Elsevier Ltd. All rights reserved.
Contents 1.
2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Multi-component aerodynamic balances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. The purpose of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Functional relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Loading combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Parameters estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The covariance matrix V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Matrix VW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Matrix VR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Matrix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. The fitted aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Matrix VW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Matrix VR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Matrix V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Fitted aerodynamic forces and moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
⇑ Corresponding author. E-mail addresses: [email protected] (M.L.C.C. Reis), [email protected] (R.M. Castro). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.09.016
25 25 25 25 27 27 28 28 28 28 29 29 29 29 30 30 31 31 31 32
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4.
3.6. Goodness of fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. The result of the calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The staff of the Aerodynamics Division of the Institute of Aeronautics and Space, Brazil, has been concerned with the quality of the data originating from aerodynamic tests. Following international recommendations based on the Metre Convention, several studies have been conducted in order to improve the metrological reliability of measurement procedures, tests and calibrations carried out in the Subsonic and Transonic Facilities, named TA-2 (wind tunnel number 2) and TTP (Pilot Transonic Wind Tunnel), respectively. One of these projects is the development of a methodology for the evaluation of uncertainty in measurement of the aerodynamic loads acting on the test article. The instrument used to measure the aerodynamic loads is the aerodynamic balance, whose calibration uncertainty contributes significantly to the overall uncertainty in wind tunnel testing. International organizations have been exchanging information and collaborating on the development of a balance calibration uncertainty methodology, but this has only been partially addressed [1]. 1.1. Aerodynamic loads Fig. 1 presents the terminology for designating the aerodynamic load components: drag force (D), side force (Y), lift force (L), rolling moment (l), pitching moment (m) and yawing moment (n). For simplicity, at TA-2 and TTP wind tunnels, they are named F1, F2, F3, F4, F5 and F6, respectively. 1.2. Multi-component aerodynamic balances Figs. 2 and 3 present the two balances employed at TTP and TA-2: internal and external, respectively. The former is
25
32 32 32 32 32
approximately 0.12 m long and is designed to fit within the test article (Fig. 2). The latter is 5.7 m high and carries the loads outside the tunnel before they are measured (Fig. 3). The balance measures the loads by using strain-gages arranged in a Wheatstone bridge. The results presented in this study originate from the calibration of an external balance. A six component external balance is used to measure the aerodynamic loads acting on the model during a wind tunnel test at the TA-2 aerodynamic facility. Fig. 4 presents a test article inside the test section. The schematic picture illustrating the connection between the model and the external balance is shown in Fig. 5. The positioning of the six load cells on the balance is also shown. A balance calibration is performed prior to the tests. The calibration is accomplished by applying loads to a calibration system constituted of a calibration cross, trays, cables and pulleys (Figs. 6 and 7). There are 14 trays for the application of weights. A set of approximately one hundred 10 kg weights is used. The load cells of the balance provide the readings Ri(i = 1, . . . , 6). The calibration cross is attached to the model support system, at the same location as a test article would be positioned. The applied calibration weights to the calibration system simulate the condition of wind tunnel airflow exerting forces and moments to the test article. Once the calibration is accomplished, the calibration cross is replaced by the model to be tested. 1.3. The purpose of the paper The calibration of wind tunnel balances requires the application of known weights to the calibration system and the reading of the electrical responses from the balance. In this paper, the weights are considered the output quantities and the readings are the input quantities of the calibration model.
Fig. 1. Aerodynamic forces and moments. a: angle of attack, b: sideslip angle.
26
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Fig. 2. Internal balance.
Fig. 3. External balance.
Fig. 4. Model under test at the TA-2 facility.
Wind tunnel facilities around the world are concerned about the mathematical modeling of the calibration [2], the design of balances [3], the development of balance calibration systems for minimizing sources of uncertainties [4,5], the optimization of software tools to process balance calibration data [6], the design of experiment of the calibration loading [7], and the analysis of uncertainty of the calibration [8–10]. Most research deals with more than one of these subjects and the interest is always to help experimentalists in the quantification and minimization of the balance
contribution to the uncertainty in the aerodynamic forces and moments measured during test campaigns. The evolution of the analysis of balance calibration uncertainty at TA-2 and TTP facilities can be seen in [11– 14]. These references present the data reduction employed to predict the aerodynamic forces and moments and associated uncertainties by applying least squares fitting to the calibration data [15,16]. Firstly, as the uncertainties in the calibration data were not known, the fitting was applied considering all the uncertainties in data points equal to 1 [11]. This was also done in Refs. [8,9]. In such an approach, the assigned uncertainties to the output quantities is performed considering information coming from the fitting only. No experimental investigation about the uncertainty in data to be fitted is done. In [12], the contribution of the load cells readings was considered applying the law of propagation of uncertainties as recommend by international standardization [17], but this information was only included after performing the fitting. The results obtained in [11] were validated using two methods: Monte Carlo [18] and Neural Networks. These results are presented in [13]. The failure of not performing the least squares fitting considering uncertainty in data points still remained and a study employing the internal balance to analyze the sources of uncertainty caused by the bridge readings was conducted. A particular loading, loading number 1, was replicated eight times during the calibration and a covariance matrix for the bridge readings was constructed [14], but the experimental evaluation of uncertainties in the applied weights was not accomplished. The purpose of the replicate measurements was to quantify the interdependence between the bridges of the balance. The method was improved and transferred to the calibration of the external balance, which is described in this paper. The enhancement over the previous procedure was to obtain the covariance matrix of the bridge readings from three different sets of balance calibration, instead of simply replicating a specific loading. Moreover, besides the uncertainties in the sensor readings, the uncertainties in the applied weights are also quantified experimentally and considered in the least squares regression. Such a procedure has not been found in aerodynamic balance calibration literature to date. The steps of the method proposed in this paper will be shown in Section 2: the loading plan, the calibration mathematical modeling and how the parameters of the mathematical modeling are estimated. The way the recognized sources of uncertainty were quantified and assigned to the calibration data points is also presented. A procedure on how to compose the covariance matrix of the bridge readings is explained. In Section 3, the method was verified for the fitted aerodynamic components by using data obtained from calibration and was analyzed in terms of quality of fitting. The data reduction employed in this study can be extended to the calibration of other multivariate measurement models which relate quantities of different kinds. Obviously, the investigation of sources of errors presented
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
27
Fig. 5. The interface model-external balance.
here must be adapted for the particularities of other measurement instruments. Metrological terms encountered in this paper comply with [19]. 2. Methodology
Fig. 6. Loading system for the balance calibration.
The calibration of an external balance involves the application of known weights to the balance and recording strain-gages readings at each force and moment combination. There are two ways of calibrating the balance: at sideslip angle beta equal to zero and at beta different from zero. At the subsonic wind tunnel TA-2, a calibration performed at beta equal to zero is called alpha calibration and beta calibration when otherwise. In this study only the alpha calibration will be considered and the results should be limited to tests conducted when the sideslip angle is set to zero. 2.1. Functional relationship The mathematical modeling of the calibration relates the aerodynamic components Fi to functions of the strain-gage bridges readings Ri (Eq. (1)). The system is multivariate and consists of a linear combination of 27 functions of R [15,16]. These functions are called basis functions and correspond to: R1 ; R2 ; R3 ; R4 ; R5 ; R6 ; R21 ; R1 R2 ; R1 R3 ; R1 R4 ; R1 R5 ; R1 R6 ; R22 ; R2 R3 ; . . . ; R26 . There are 27 adjustable parameters for each one of the six aerodynamic components. The model’s dependence on its parameters is linear.
Fig. 7. Calibration system, schematic. F: forces, M: moments.
28
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
Table 1 Examples of nominal weights used in calibration for loadings number 1, 27, 46, 58 and 69. Unit: kgf for forces F1, F2, F3 and kgf m for moments F4, F5 and F6. Loading number
F1
F2
F3
F4
F5
F6
1 27 46 58 69
0 80 0 0 0
0 0 80 80 0
80 0 0 0 80
0 40 0 0 0
40 0 0 40 0
0 0 48 0 48
^ ¼ ðRT V 1 RÞ1 ðRT V 1 FÞ p
þ ai;7 R21 þ ai;8 R1 R2 þ ai;9 R1 R3 þ ai;10 R1 R4
V p^ ¼ ðRT V 1 RÞ1
þ ai;11 R1 R5 þ ai;12 R1 R6 þ ai;13 R22 þ ai;14 R2 R3 þ þ
ð2Þ
where RT is the transpose of the matrix R; V is the covariance matrix; and V1 is the matrix inverse of V. Matrix (RTV1R)1 contains information about uncertainties of the estimated parameters. Its diagonal elements correspond to the variances (squared uncertainties u2p^ ) and the off-diagonal elements are the covariances of the fitted parameters. The matrix can be denoted by V p^ . Therefore:
F i ¼ ai;1 R1 þ ai;2 R2 þ ai;3 R3 þ ai;4 R4 þ ai;5 R5 þ ai;6 R6
ai;27 R26 :
problem has a dimension of 73 27 and is constructed from the basis functions. Denoting the parameters of the ^ , the least squares results in: fitting by p
ð3Þ
ð1Þ
Cross-product terms involving the input quantities are included in the polynomial because it is not possible to eliminate the interactions between bridges completely. The task during the calibration is to apply known weights to the calibration system while recording the resulting bridge readings and to calculate the 27 unknown polynomial coefficients in the mathematical model. During the wind tunnel tests, using the bridge readings and the coefficients evaluated through the calibration, one can estimate the aerodynamic loads exerted by the airflow on the model under test by using Eq. (1).
2.4. The covariance matrix V The covariance matrix V is related to the uncertainties of the applied loads, uF. It is made up of the contributions of the sources of error due to the application of weights in the calibration system VW and the uncertainties in the readings of the bridges VR:
V ¼ V W þ CV R C T
ð4Þ
The terms in Eq. (4) are described in the following sections.
2.2. Loading combinations Seventy-three loading combinations are employed for alpha calibration. Table 1 presents some typical loading combinations which will be used in this paper to illustrate the results obtained when the method proposed here is employed. The values represent the weights applied to the balance originating from the application of weights on the trays T1, . . ., T14 of the calibration system (Table 2). The length of the lever arm of the cross is equal to 0.5 m for F4 and F5 and to 0.6 m for F6. 2.3. Parameters estimation According to the least squares method a calibration curve is fitted to each set of the N = 73 data points constituted by the bridge readings and the applied weights [15,16]. Each of the aerodynamic components is arranged in a vector F represented by the applied weights, whose dimension is 73 1. The design matrix R of the fitting
2.4.1. Matrix VW The matrix VW has a dimension of 73 73 and in this study it is considered diagonal. Its elements are based on the uncertainties of the weights employed in the loading process and also in an estimation of the uncertainties caused by the balance calibration system. The former contribution can be promptly evaluated by verifying the uncertainties of the weights applied to the calibration cross. The latter is based on the quantification of the sources of error that affect the resolution of the calibration system such as frictional forces and misalignments between cables and pulleys. The quantification of the calibration system resolution involves applying small weights to the trays and verifying when the load cells of the aerodynamic balance change their indications in a detectable way. Studies were carried out for all six aerodynamic components. The nominal values of the weights employed in this analysis are equal to 1, 2, 5 and 10 g.
Table 2 Nominal values of the weights applied to the trays T1–T14 for loadings number 1, 27, 46, 58 and 69. Unit: kg. Loading number
1 27 46 58 69
Weights (kg) T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
T13
T14
0 80 0 0 0
0 0 0 0 0
0 0 0 0 40
0 0 80 40 0
0 0 0 0 0
0 0 0 40 40
0 40 0 0 20
0 0 0 0 20
0 0 0 0 20
20 0 0 40 20
20 0 0 0 0
20 40 0 0 0
60 0 0 40 0
0 0 0 0 0
29
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
2.4.2. Matrix VR Replicate measurements supply information for the construction of VR. Three calibrations under repeatability conditions were carried out over a short period of time. The diagonal elements of VR are represented by the variances and the off diagonal elements are the covariances between the load cell readings (Eq. (5)). The first 73 lines are elements related to R1, followed by the 73 lines related to R2 and so on, resulting in a 438 438 matrix.
2
uðR1;1 ; R1;1 Þ
uðR1;1 ; R1;2 Þ
uðR1;1 ; R1;73 Þ
The elements @F1,j/@Ri, j of matrix C are represented by Eq. (8). The parameters a of Eq. (8) are estimated by performing the least squares fitting to the calibration data, as shown in Section 2.3, with V equal to VW in Eq. (4), i.e., VR is not considered in the first regression. This means that the fitting is applied twice to the calibration data: the first time V = VW and the second time V = VW + CVRCT.
uðR1;1 ; R2;1 Þ
uðR1;1 ; R6;73 Þ
3
6 uðR ; R Þ uðR ; R Þ uðR ; R Þ uðR ; R Þ uðR ; R Þ 7 1;2 1;1 1;2 1;2 1;2 1;73 1;2 2;1 1;2 6;73 7 6 7 6 7 6 .. .. .. .. .. .. .. 7 6 . . . . . . . 7 6 7 6 V R ¼ 6 uðR1;73 ; R1;1 Þ uðR1;73 ; R1;2 Þ uðR1;73 ; R1;73 Þ uðR1;73 ; R2;1 Þ uðR1;73 ; R6;73 Þ 7 7 6 6 uðR2;1 ; R1;1 Þ uðR2;1 ; R1;2 Þ uðR2;1 ; R1;73 Þ uðR2;1 ; R2;1 Þ uðR2;1 ; R6;73 Þ 7 7 6 7 6 .. .. .. .. .. .. .. 7 6 5 4 . . . . . . .
ð5Þ
uðR6;73 ; R1;1 Þ uðR6;73 ; R1;2 Þ uðR6;73 ; R1;73 Þ uðR6;73 ; R2;1 Þ uðR6;73 ; R6;73 Þ
The first subscript in Eq. (5) identifies the load cells R1 to R6 and the second is related to each of the 73 loadings. 2.4.3. Matrix C Matrix C has a dimension of 73 438 and its elements are the sensitivity coefficients evaluated by taking the partial derivatives in Eq. (1). For example, for the aerodynamic force F1, matrix C is formed by:
2
@F 1;1
6 @R1;1 6 @F 1;2 6 @R1;1 6 6 . C¼6 6 .. 6 @F 6 1;72 6 @R1;1 4 @F 1;73 @R1;1
@F 1;1 @R1;2
@F 1;1 @R1;73
@F 1;1 @R2;1
@F 1;1 @R6;73
@F 1;2 @R1;2
@F 1;2 @R1;73
@F 1;2 @R2;1
@F 1;2 @R6;73
.. .
.. .
@F 1;72 @R1;73
@F 1;72 @R2;1
@F 1;73 @R1;73
@F 1;73 @R2;1
.. . @F 1;72 @R1;2 @F 1;73 @R1;2
..
.
3
7 7 7 7 .. .. 7 7 . . 7 7 @F 1;72 7 @R6;73 7 5 @F @R1;73 6;73
6 6 6 6 6 C¼6 6 6 6 4
0
0
@F 1;1 @R2;1
0
0
@F 1;2 @R1;2
0
0
0
.. .
.. .
..
.
.. .
.. .
..
.. .
0
0
0
0
0
@F 1;73 @R1;73
0 0
@R1;1
ð6Þ
.
0 @F 1;73 @R6;73
3 7 7 7 7 7 7 7 7 7 5
2.5. Goodness-of-fit The goodness-of-fit is evaluated through the chi-square quantity, v2 [15,16]. There is one v2 for each aerodynamic load. Taking into account the covariance matrix V:
v2 ¼ ðF Fb ÞT V 1 ðF Fb Þ
The first and second subscripts in Eq. (6) identify the aerodynamic component or load cell (i = 1, . . . , 6) and loading number (j = 1, . . . , 73), respectively. Note that the first block of 73 columns is related to R1, the second block is related to R2 and so on. In Eq. (6), only the sensitivity coefficients whose second subscripts correspond to the same loading have values different from zero. Therefore, matrix C is reduced to:
2 @F 1;1
Finally, matrix CVRCT corresponds to the contribution to the variances in F due to uncertainties in the bridge readings and has dimension of 73 73.
ð9Þ
where F is the vector of applied forces (or moments) to the b is the vector of fitted forces (or mocalibration cross; F ments), estimated by the least squares. A value for the chi-square which indicates a good fit is typically close to the number of degrees of freedom:
m ¼ N Np
ð10Þ
In Eq. (10) N = 73, the number of data points and Np = 27, the number of coefficients in Eq. (1) to be fitted by the least squares method. This leads to another quantity, the reduced chi-square:
v2red ¼
v2 m
ð11Þ
whose desired value is approximately equal to 1. 2.6. The fitted aerodynamic loads
ð7Þ After performing the least squares regression to esti^ of adjusted parameters, one can obtain mate the vector p b . The 73 fitted values of the fitted forces and moments F each aerodynamic component are calculated by multiply^: ing the matrix R by the corresponding vector p
30
@F 1;j @R1;j @F 1;j @R2;j @F 1;j @R3;j @F 1;j @R4;j @F 1;j @R5;j @F 1;j @R6;j
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
¼ a1;1 þ 2a1;7 R1;j þ a1;8 R2;j þ a1;9 R3;j þ a1;10 R4;j þ a1;11 R5;j þ a1;12 R6;j ¼ a1;2 þ a1;8 R1;j þ 2a1;13 R2;j þ a1;14 R3;j þ a1;15 R4;j þ a1;16 R5;j þ a1;17 R6;j ¼ a1;3 þ a1;9 R1;j þ a1;14 R2;j þ 2a1;18 R3;j þ a1;19 R4;j þ a1;20 R5;j þ a1;21 R6;j ¼ a1;4 þ a1;10 R1;j þ a1;15 R2;j þ a1;19 R3;j þ 2a1;22 R4;j þ a1;23 R5;j þ a1;24 R6;j
ð8Þ
¼ a1;5 þ a1;11 R1;j þ a1;16 R2;j þ a1;20 R3;j þ a1;23 R4;j þ 2a1;25 R5;j þ a1;26 R6;j ¼ a1;6 þ a1;12 R1;j þ a1;17 R2;j þ a1;21 R3;j þ a1;24 R4;j þ a1;26 R5;j þ 2a1;27 R6;j
^ Fb ¼ Rp
ð12Þ
^ is 27 1. where R is 73 27 and p The associated uncertainty in the fitted aerodynamic component is the positive square root of the diagonal elements of:
V b ¼ RV p^ RT
ð13Þ
F
where V p^ is the matrix (RTV1R)1 as explained in Section 2.3. Eq. (13) is equivalent to applying the law of propagation of uncertainty in the polynomial relating the aerodynamic loads and load cell readings as represented in Eq. (1) [17]. Table 3 Values of the weights that must be applied to the calibration system in order to cause a perceptible change in the load cells readings (103 kg). Load cell
Applied weight
R1 R2 R3 R4 R5 R6
2 2 32 5 8 5
3. Results and discussion The least squares regression provides the fitted param^ , as well as its variances and covariances V p^ . eters vector, p The calibration data reduction also includes the fitted load b and their uncertainties V . The quality of the fit is values F bF quantified by the chi-square quantity, v2. Code in MATLABÒ and ExcelÒ worksheets were used to perform the data reduction. 3.1. Matrix VW The evaluation of uncertainties to compose matrix VW is of type B. The first contribution considered is that due to the errors of the weights employed in the loading process. It was found that the maximum error in the applied 10 kg weights is around 1 g in 10,000 g. Taking into account a rectangular distribution with limits equal to 1 103 kg, this leads to an uncertainty value of 5.8 104 kg. The resolution of the calibration system is the second contribution. It was estimated by applying incremental weights of the order of 103 kg to specific trays related to the aerodynamic component under investigation and by verifying the indication of the corresponding load cell. Different experiments were carried out to stimulate each one of the six sensors individually and the worst case found
Fig. 8. Study of the resolution of the calibration system related to R3. Each group of five readings corresponds to the same applied weight.
31
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33 Table 4 ^ for the force component F1. Second Estimated polynomial parameters p fitting. Parameter
^ p
up^
^ up^ =p
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27
3.525849 0.011648 0.008513 0.000051 0.001620 0.004485 0.000557 0.000234 0.001408 0.000242 0.000164 0.000087 0.000862 0.000599 0.000705 0.000283 0.000658 0.000080 0.000127 0.000369 0.000635 0.000197 0.000177 0.000583 0.000001 0.000250 0.000443
0.001298 0.001295 0.001440 0.000649 0.000534 0.000501 0.000058 0.000163 0.000185 0.000082 0.000067 0.000064 0.000364 0.000864 0.000383 0.000309 0.000290 0.000447 0.000430 0.000350 0.000328 0.000088 0.000155 0.000144 0.000057 0.000117 0.000058
2717.01 8.99 5.91 0.08 3.04 8.95 9.53 1.43 7.60 2.92 2.43 1.35 2.36 0.69 1.84 0.91 2.27 0.17 0.29 1.05 1.93 2.23 1.13 4.04 0.02 2.13 7.63
Table 6 Estimates of the uncertainty for the aerodynamic components. Units as in Table 5. Load number
Force
1 27 46 58 69
Moments
uF1
uF2
uF3
uF4
uF5
uF6
0.0125 0.0157 0.0126 0.0130 0.0127
0.0126 0.0148 0.0141 0.0128 0.0156
0.0132 0.0140 0.0124 0.0127 0.0137
0.0301 0.0137 0.0144 0.0354 0.0144
0.0147 0.0200 0.0132 0.0129 0.0128
0.0124 0.0138 0.0147 0.0129 0.0125
Table 7 Goodness of fit. Load cell
v2
v2red
F1 F2 F3 F4 F5 F6
232.33 271.65 453.91 67.69 165.99 529.83
5.05 5.91 9.87 1.47 3.61 11.52
the readings of the load cell R1 originating from calibration number 1, followed by the 73 elements of the readings of the load cell R2 and so on. The same procedure is repeated for columns 2 and 3, using the data set originated from the second and third calibrations, respectively. The matrix is then transposed.
was for R3 (Table 3 and Fig. 8). In Fig. 8, the first five readings correspond to the application of 2 g on trays 11, 12, 13 and 14, i.e., 8 g were applied to the system. In the sequence, from reading number 6 up to number 10, 16 g were applied, and so on. One can see that the behavior of the readings becomes stable for 32 g and the standard deviation of a rectangular distribution with lower and upper bounds equal to this limit was chosen as the resolution of the system, resulting in 1.8 102 kg. Table 3 presents the results of the type B evaluation of VW for the six load cells.
3.2. Matrix VR The covariance matrix VR is calculated through the MATLABÒ code, from a previous 438 3 matrix, named R, where the first 73 elements of column 1 correspond to
3.3. Matrix V After computing the terms VW and CVRCT of Eq. (4), ma^ of trix V can be used in Eq. (2) to estimate the parameters p the fitting. 3.4. Estimation of parameters Performing the least squares fitting for the second time to the experimental data, i.e., considering the uncertainties in the data points estimated through Eq. (4), results in the ^ presented in Table 4. Quantities a1 to a6 have units vector p of kgf mV1 and a7 to a27 have units of kgf mV2 when in the case of force components F1, F2 and F3. For moment components F4, F5 and F6, units are kgf m mV1 for a1 to a6 and kgf m mV2 for the remaining. One can see the important significance of a1 for F1, a2 for F2, a3 for F3, etc.
Table 5 Aerodynamic forces and moments resulting from the application of the least squares fitting. Units: kgf (force) and kgf m (moment). Load number
1 27 46 58 69
Force
Moments
F1
F2
F3
F4
F5
F6
0.0107 80.0792 0.0425 0.0313 0.0116
0.0164 0.0186 80.0198 79.9407 0.0949
79.9812 0.0141 0.0212 0.0415 79.9505
0.0437 39.9855 0.0017 0.0695 0.0358
40.0103 0.0394 0.0168 40.0592 0.0356
0.0443 0.0080 47.9342 0.0654 47.9397
32
M.L.C.C. Reis et al. / Measurement 46 (2013) 24–33
Table 8 Values of aerodynamic forces and moments and corresponding uncertainty. Units: newton for force and newton metro for moment. Load
F1
uF1
F2
uF2
F3
uF3
F4
uF4
F5
uF5
F6
uF6
1 27 46 58 69
0.011 79.914 0.042 0.031 0.012
0.012 0.016 0.013 0.013 0.013
0.016 0.019 79.855 79.776 0.095
0.013 0.015 0.014 0.013 0.016
79.816 0.014 0.021 0.041 79.786
0.013 0.014 0.012 0.013 0.014
0.044 39.903 0.002 0.069 0.036
0.030 0.014 0.014 0.035 0.014
39.928 0.039 0.017 39.977 0.036
0.015 0.020 0.013 0.013 0.013
0.044 0.008 47.835 0.065 47.841
0.012 0.014 0.015 0.013 0.013
Nevertheless, some other parameters are not negligible, as a consequence of mechanical coupling between components of the aerodynamic balance. The uncertainties of the fitted parameters up^ (Table 4) are the positive square root of the diagonal elements of the matrix V p^ (Eq. (3)), as explained in Section 2.3. ^ =up^ between the fitted parameter and its The ratio p uncertainty is also shown in Table 4 and indicates the importance of each of the 27 fitted parameters to the polynomial modeling expressed by Eq. (1). Once again, values show that there is significant contribution besides those parameters a1, a2, a3, a4, a5, and a6 to the mathematical model, as expected. 3.5. Fitted aerodynamic forces and moments One can use the results of the external balance calibration to predict the aerodynamic loads which would act on a model under test in the wind tunnel. The values of the fitb (Eq. (12)), and the ted aerodynamic forces and moments, F associated uncertainties, V b (Eq. (13)), are presented in TaF bles 5 and 6, respectively, for the examples of calibration loadings expressed in Table 1. Each aerodynamic component was calculated by using its corresponding fitted parameters and by choosing the load cells values read at loadings number 1, 27, 46, 58 and 69. 3.6. Goodness of fit The chi-squared v2 (Eq. (9)) and reduced chi-square v2red (Eq. (11)) values of the fitting are presented in Table 7. Estimating the chi-square should result in a value around the number of degrees of freedom of the polynomial fit [15,16]. As there are 73 equations and 27 unknowns, the number of degrees of freedom is equal to 46 (Eq. (10)). The magnitudes presented in Table 7 are greater than expected, revealing that either the math model or the quantification of the uncertainties presented in the experiment, or even both, should be better investigated. 3.7. The result of the calibration The result of the calibration for loadings 1, 27, 46, 58 and 69 are shown in Table 8. The conversion to SI units was performed by multiplying the values in Tables 5 and 6 by the ratio between local g = 9.7864081 m/s2 and normal g = 9.80665 m/s2. 4. Conclusions A method for the estimation of uncertainty in external balance calibration was proposed. Results of the curve fit-
ting include the estimation of the values and uncertainties of the fitting parameters, as well as a statistical measure of goodness-of-fit. The law of propagation of uncertainty was employed to estimate the uncertainties in the aerodynamic components drag, side and lift forces and rolling, pitching and yawing moments. Least squares method was employed twice to fit the data points. The first step is necessary in order to estimate the contribution of the independent variable Ri to the overall uncertainty in Fi. The calibration model is multivariate and second order terms were considered in the mathematical modeling because the external balance is not able to separate the input quantities completely. Recognized uncertainty contributions to the calibration are due to the applied weights to the trays, resolution of the measurement system and replicated loadings. Studies were carried out to quantify the resolution of the system for each one of the six aerodynamic components, individually. Although different values were found, the main contribution observed for the reading of the lift force was ascribed to the resolution of the system because the calibration modeling is multivariate. However, in the future, experiments to investigate the contributions in a combined way must be considered. The values of the chi-square quantity greater than the degrees of freedom imply either that the adopted mathematical model is not suitable for the calibration or the uncertainties of the data points are larger than stated. Indeed, the identification of the sources of error caused by the calibration system and their consequent contributions to the uncertainty of the load components has still not been completely accomplished. Other studies considering different mathematical relationships between the aerodynamic loads and load cell readings are in progress.
Acknowledgements The authors would like to express their gratitude to FAPESP, the Foundation for Support of Research of the State of São Paulo, under Grant Nos. 2000/13769-0, 2002/01933-6, 2006/02628-3, and 2009/09370-0.
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