Calibration data of a wind tunnel sting balance was processed using a search algorithm that identifies an optimized regression model for the data analysis. The selected sting balance had two moment gages that were mounted forward and aft of the balance moment center. The difference and the sum of the two gage outputs were fitted in the least squares sense using the normal force and the pitching moment at the balance moment center as independent variables. The regression model search algorithm predicted that the difference of the gage outputs should be modeled using the intercept and the normal force. The sum of the two gage outputs, on the other hand, should be modeled using the intercept, the pitching moment, and the square of the pitching moment. Equations of the deflection of a cantilever beam are used to show that the search algorithm s two recommended math models can also be obtained after performing a rigorous theoretical analysis of the deflection of the sting balance under load. The analysis of the sting balance calibration data set is a rare example of a situation when regression models of balance calibration data can directly be derived from first principles of physics and engineering. In addition, it is interesting to see that the search algorithm recommended the same regression models for the data analysis using only a set of statistical quality metrics

Bader, Jon B.

Ulbrich, Norbert

NASA Technical Reports Server

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
	
AIAA 2009-XXXX
2-5 August 2009, Denver, Colorado
Analysis of Sting Balance Calibration Data
Using Optimized Regression Models
(Extended Abstract of Proposed Conference Paper)
N. Ulbrich *
Jacobs Technology Inc., Moffett Field, California 94035–1000
J. Bader**
NASA Ames Research Center, Moffett Field, California 94035–1000
Calibration data of a wind tunnel sting balance was processed using a search
algorithm that identifies an optimized regression model for the data analysis.
The selected sting balance had two moment gages that were mounted forward
and aft of the balance moment center. The difference and the sum of the two
gage outputs were fitted in the least squares sense using the normal force and
the pitching moment at the balance moment center as independent variables.
The regression model search algorithm predicted that the difference of the gage
outputs should be modeled using the intercept and the normal force. The sum of
the two gage outputs, on the other hand, should be modeled using the intercept,
the pitching moment, and the square of the pitching moment. Equations of the
deflection of a cantilever beam are used to show that the search algorithm’s two
recommended math models can also be obtained after performing a rigorous
theoretical analysis of the deflection of the sting balance under load. The analysis
of the sting balance calibration data set is a rare example of a situation when
regression models of balance calibration data can directly be derived from first
principles of physics and engineering. In addition, it is interesting to see that the
search algorithm recommended the same regression models for the data analysis
using only a set of statistical quality metrics.
Nomenclature
d = distance between forward and aft moment bridge
do = distance between model reference center and balance moment center
F = normal force at balance moment center, [ lbs]
M = pitching moment at balance moment center, [ in - lbs]
M1 = moment at forward moment bridge, [ in - lbs]
M2 = moment at aft moment bridge, [in - lbs]
R1 = electrical output at forward moment bridge, [μV/V]
R2 = electrical output at aft moment bridge, [μV/V]
α1 , α2 , • • • , α6 = regression coefficients
β1 ,β2 , • • • , β6 = regression coefficients
γ1 ,γ2 = regression coefficients
δ1 ,δ2 = regression coefficients
e1 , e2 = regression coefficients
η1 , η2 ,η3 = regression coefficients
* Aerodynamicist, Jacobs Technology Inc.
** Branch Chief, Code AOI, NASA Ames Research Center
American Institute of Aeronautics and Astronautics
https://ntrs.nasa.gov/search.jsp?R=20090033643 2019-08-30T07:49:56+00:00Z
I. Summary
During the past 4 years a software package was developed at the Ames balance calibration lab that
is used for the analysis of wind tunnel strain–gage balance calibration data. The software uses an innova-
tive candidate math model search algorithm in order to find an optimized regression model for the balance
calibration data analysis. This so–called recommended math model meets strict statistical quality require-
ments that the user specifies before the math model search is performed (see Ref. [1] and [2] for a detailed
description of the algorithm). Figure 1 shows key elements of the candidate math model search algorithm.
So far, the search algorithm was applied to a wide variety of strain–gage balance calibration data sets
(see, e.g., Refs. [3], [4], [5], and [6]). Recently, calibration data of a sting balance was processed using the
candidate math model search algorithm. The analysis of this data set will be discussed in great
detail in the proposed paper. The selected sting balance had two moment gages that were mounted
forward and aft of the balance moment center. Figure 2 shows the location of the two moment gages relative
to the balance moment center. The electrical outputs of the two balance gages are called R1 and R2 . These
gage outputs are a function of the normal force F and the pitching moment M that the balance experiences
at the balance moment center. The two loads are defined by the following equations
F = 
M2 − M1	 (1a)
d
M = 
M1 + M2	 (1b)
2
where M1 is the moment at the forward moment bridge, M2 is the moment at the aft moment bridge, and
d is the distance between the two balance gages (see Fig. 2).
During the analysis of the calibration data and during the regression model search (i) the difference and
(ii) the sum of the two gage outputs were fitted in the least squares sense using the normal force and the
pitching moment as independent variables. The regression model search algorithm needed an upper and a
lower bound of the regression models in order to define the regression model search space. It was decided to
use the following two regression models as upper bounds for the candidate math model search:
FIRST UPPER BOUND =⇒
 
R2 −R1 = α1 + α2 ·F + α3 ·M + α4 ·F2 + α5 ·M2 + α6 ·F ·M (2a )
SECOND UPPER BOUND =⇒
 
R1 + R2 = β1 + β2 · F + β3
 
·M + β4 · F2 + β5 · M2 + β6 ·F ·M (2b)
From Eq. (1a) we know that the normal force is approximately proportional to the difference of the
moments at the forward and aft moment gages. We also know from Eq. (1b) that the pitching moment is
approximately proportional to the sum of the moments at the forward and aft moment gages. Therefore,
the following two linear regression models were selected as lower bounds for the regression model search:
	
FIRST LOWER BOUND =⇒
 
R2 − R1 = γ1 + γ2 · F	 (3a )
	
SECOND LOWER BOUND =⇒
 
R1 + R2 = δ1 + δ2 · M	 (3b)
In the next step the candidate math model search algorithm was applied to the sting balance calibration
data set. A total number of 20 regression models were tested during the search. Afterwards, the algorithm
chose the following two recommended math models for the analysis of the data:
FIRST RECOMMENDED MATH MODEL =⇒
 
R2 − R1 = E1 + E2 · F	 (4a )
SECOND RECOMMENDED MATH MODEL =⇒
 
R1 + R2 = η1 + η2 · M + η3
 
· M2 (4b)
It is a surprising result of the regression model search that (i) the recommended math model of the gage
output difference (R2 − R1) is only a function of the normal force and that (ii) the regression model of the
sum of the gage outputs (R1 + R2 ) is only a function of the pitching moment and the square of the pitching
moment. An explanation of this search result had to be found.
2
American Institute of Aeronautics and Astronautics
Fortunately, the overall geometry and design of the sting balance is very simple. It may be approximated
as a cantilever beam. Then, it is possible to apply the equations of the deflection of a cantilever beam in
order to show that the two recommended math models, i.e., Eq. (4a) and (4b), can also be obtained after
performing a rigorous analysis of the deflection of the sting balance under load (see, e.g., Ref. [7] for the
description of the deflection of a cantilever beam). This assertion will systematically be proven in
the proposed conference paper.
In conclusion, the regression analysis of the given sting balance calibration data set is a rare example
of a situation when regression models of a balance calibration data set can directly be derived from first
principles of physics and engineering. It is also interesting to note that the candidate math model search
algorithm predicted the same regression models for the balance calibration data using only a set of statistical
quality metrics.
II. Acknowledgements
The authors would like to thank Thomas R. Wayman of The Gulfstream Aerospace Corporation for
providing the sting balance calibration data set for the present study. The work reported in this paper was
partially supported by NASA’s Aeronautic Test Program and the Wind Tunnel Operations Division at Ames
Research Center under contract NNA04BA85C.
III. References
1Ulbrich, N., “Regression Model Optimization for the Analysis of Experimental Data,” AIAA 2009–1344,
paper presented at the 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January 2009.
2 Ulbrich, N. and Volden, T., “Regression Analysis of Experimental Data Using an Improved Math
Model Search Algorithm,” AIAA 2008–0833, paper presented at the 46th AIAA Aerospace Sciences Meeting
and Exhibit, Reno, Nevada, January 2008.
3 Ulbrich, N. and Volden, T., “Application of a New Calibration Analysis Process to the MK–III–C
Balance,” AIAA 2006–0517, paper presented at the 44th AIAA Aerospace Sciences Meeting and Exhibit,
Reno, Nevada, January 2006.
4Ulbrich, N. and Volden, T., “Analysis of Floor Balance Calibration Data using Automatically Gen-
erated Math Models,” AIAA 2006–3437, paper presented at the 25th AIAA Aerodynamic Measurement
Technology and Ground Testing Conference, San Francisco, California, June 2006.
5 Ulbrich, N. and Volden, T., “Analysis of Balance Calibration Machine Data using Automatically Gen-
erated Math Models,” AIAA 2007–0145, paper presented at the 45th AIAA Aerospace Sciences Meeting and
Exhibit, Reno, Nevada, January 2007.
6 Ulbrich, N., Volden, T., and Booth, D., “Predictive Capabilities of Regression Models used for Strain–
Gage Balance Calibration Analysis,” AIAA 2008–4028, paper presented at the 26th AIAA Aerodynamic
Measurement Technology and Ground Testing Conference, Seattle, Washington, June 2008.
7Timoshenko, S. and MacCullough, G. H., “Elements of Strength of Materials,” D. Van Nostrand
Company, Inc., Toronto, New York, London, 2nd ed., 1940, p.161.
3
American Institute of Aeronautics and Astronautics
FUNCTION CLASS CONSTRAINT CONSTRAINT
COMBINATION THRESHOLD THRESHOLD
CHOICE CHOICE 1 CHOICE 2
CANDIDATE MATH MODEL
SEARCH ALGORITHM
ALL AVAILABLE “HIERARCHY” “P-VALUE” “VIP' “PRESS”	 I
MATH MODELS CLASS
SINGULAR VALUE ACCEPT ONLY PRIMARY SECONDARY
PRESS
	 I
OF-
DECOMPOSITION HIERARCHICAL SEARCH SEARCH RESIDUAL
REGRESSORS MATH MODELS CONSTRAINT CONSTRAINT MINIMIZATION
AND
RESPONSES
(REQUIRED) (OPTIONAL)
--	 -
--	 --
(REQUIRED)
----
-- - --
(REQUIRED)
-
-
(REQUIRED)
II	 MATH MODELS
I
I
MATH MODELS
WITH MATH MODELS NON- MATH MODELS MATH MODELS WITH	
I
UNWANTED WITH HIERARCHICAL WITH WITH INFERIOR	 I
FUNCTION CLASS LINEAR MATH MODELS INSIGNIFICANT NEAR-LINEAR PREDICTIVE
I COMBINATIONS
DEPENDENCIES TERMS DEPENDENCIES CAPABILITIES
II-----------------------------REJECTED MATH MODELS ------
Fig. 1 Key elements of candidate math model search algorithm.
WIND TUNNEL MODEL	 rh^ft?
` 
^r	
` i s 	` Ir
I	 ,	 I
1
1
1	 i	 1t 1 /Z	 1	 1	 /2	 11
1	 ,
1	 1
GAGE 1	 GAGE 2
(FORWARD)	 i	 (AFT)
1	 r	 11
1	 1
MODEL	 BALANCE
REFERENCE	 MOMENT
CENTER	 CENTER
Fig. 2 Forces and moments acting on a wind tunnel sting balance.
MATH MODEL
4
American Institute of Aeronautics and Astronautics


Analysis of Sting Balance Calibration Data Using Optimized Regression Models

Calibration data of a wind tunnel sting balance was processed using a candidate math model search algorithm that recommends an optimized regression model for the data analysis. During the calibration the normal force and the moment at the balance moment center were selected as independent calibration variables. The sting balance itself had two moment gages. Therefore, after analyzing the connection between calibration loads and gage outputs, it was decided to choose the difference and the sum of the gage outputs as the two responses that best describe the behavior of the balance. The math model search algorithm was applied to these two responses. An optimized regression model was obtained for each response. Classical strain gage balance load transformations and the equations of the deflection of a cantilever beam under load are used to show that the search algorithm s two optimized regression models are supported by a theoretical analysis of the relationship between the applied calibration loads and the measured gage outputs. The analysis of the sting balance calibration data set is a rare example of a situation when terms of a regression model of a balance can directly be derived from first principles of physics. In addition, it is interesting to note that the search algorithm recommended the correct regression model term combinations using only a set of statistical quality metrics that were applied to the experimental data during the algorithm s term selection process

Analysis of Sting Balance Calibration Data Using Optimized Regression Models

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