[american institute of aeronautics and astronautics 47th aiaa/asme/asce/ahs/asc structures,...

American Institute of Aeronautics and Astronautics1

A Statistics based Methodfor Mapping Flight Strains to Loads

Madhusudan A. Padmanabhan*

Aeronautical Development Agency, Bangalore, Karnataka, India, 560 017, India

Karthik Y. Nagesh†

Altair Engineering India Pvt. Ltd., Bangalore, Karnataka, 560 078, India

and

Hemalatha Elattuvalappil*

Aeronautical Development Agency, Bangalore, Karnataka, India, 560 017, India

Loads experienced by flight vehicles are conventionally obtained from strain gagecalibrations done on ground. Intensive effort is required to generate the calibration data andto fit them into equations of acceptable accuracy. An alternative load estimation method,which directly correlates flight strains to the ground data, is proposed here. A best-fit linearmodel of the structure, inherent in the combination of flight and ground data, isparameterized via least squares based regression analysis. The statistical concept ofinfluence is used in an elegant way to identify and exclude strains that deviate from themodel. The approach is in tune with the current trends of smaller test matrices, employmentof flight-like distributed loads, and maximum use of collected data. The method isdemonstrated for a fighter aircraft with delta wings of composites construction.

Nomenclatureg = acceleration due to gravityH, h = hat matrix, elementK = number of loads of interestM = number of ground testsN = number of strain gagesPflt = flight load vectorPgt = ground test load matrix, one column per testRst = externally studentized residuals = variance estimateS = subspace spanned by regressor vectorsR2 = coefficient of determinationw = weighting coefficientX = regressor matrixy = response vectorεflt = flight strain vectorεgt = ground test strain matrix, one column per testθ = angle made by response vector with regressor subspaceνσ = measurement noise with zero mean and σ2 variance

* Scientist, Loads & Aeroelasticity Group, Airframe Directorate, P.B.No.1718, Vimanapura Post, Bangalore,560017, India.† Software Engineer, E P M, 163 / B, 6th Main, 3rd Cross, J. P. Nagar Phase III, Bangalore, 560078, India.

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

AIAA 2006-2005

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


I. IntroductionHE design of present day aerospace structures is based on detailed multidisciplinary analyses withrepresentative theoretical models. The complexity of the structure, however, gives rise to a degree of

uncertainty, which is addressed by providing factors of safety at various levels of design. While safety of flight is ofparamount importance in any flight vehicle, be it commercial (passenger / cargo) or military, the cost factor in thecase of the former, and performance requirements in the case of the latter dictate a move towards tighter tolerances.It therefore becomes important to generate load models based on field data 1, as they are generally more accuratethan theoretical models. Reliable load models can provide benefits in terms of quicker envelope clearances, reducedconservativeness of fatigue load spectra and more efficient MDO based flight vehicle designs.

Load calibration, or the formation of strain-to-load equations from ground based static test data, has for severaldecades been the backbone of the conventional, load estimation process 2-4, and thereby of load modeling. Intensiveeffort is required to generate calibration data through ground-based tests and to fit them into (usually linear)equations of acceptable accuracy. Technology is currently evolving towards a smaller number of distributed loadtests in place of the costlier, traditional unit load test matrix, and the use of computationally intensive searchmethods to generate the equations.

Distributed loading, made more feasible over the years by developments in testing and test control methods, ispreferred for its relative closeness to the operational state. For instance, total load magnitudes in single point loadingare limited by local strength or stability considerations, and generally cannot reach levels that are representative ofreality.

In the equations, which are generated ahead of flight tests, the number of gages that figure for a specific load isusually limited to 5 or less. Whereas appropriate gage selections are often finalized based on exhaustive searchtechniques 4, alternative approaches such as Genetic Algorithms have been suggested to handle large datasets 5. Inany case, the accuracy of the load computed from a specific equation depends strongly on the integrity of theparticipating gages in the flying specimen.

In this paper, a novel, statistics based method of estimating flight loads is proposed. The method does not usecalibration equations. Flight strains are directly correlated with ground test strains via linear regression, and flightloads are obtained as appropriate combinations of ground test loads. Maximum use is made of the availablemeasurements, from flight and from ground tests, to obtain averaged loads. At each flight instant of interest, theinitial gage selection, consisting of all gages common to the ground test specimen and the flying vehicle, is refinedby removal of noisy or erratic gages. A quantitative measure of statistical influence is used in an iterative manner,leading to an acceptable final gage set and converging load estimates.

The paper is organized as follows. The formulation is covered in detail in Section II. Results of ground test datachecks that provide proof-of-concept, and load estimates for specific test maneuvers of a fighter aircraft, arepresented and discussed in Section III. The strengths of the method, limitations encountered during application, anddirections for future work are summarized in Section IV.

II. Formulation

A. Linear Regression ConceptsThe basic linear model relating a vector, y, of response measurements to a matrix of regressors, X, is given by

Eq. (1), where νσ is a random additive measurement noise with zero mean and variance σ2, and w is the weightingvector. It can be shown least squares based regression yields unbiased estimates of w and σ2.

{ } [ ] { } { }σν+⋅+= wXw0y (1)

The concept of a multidimensional leastsquares fit is illustrated in Fig. (1). Theresponse vector, y, in N-space is approximatedby its orthogonal projection, ŷ, onto thesubspace, S, spanned by M (≤N) regressorvectors. The parallelogram represents thesubspace and the dotted line is the residualerror that cannot be bridged by the model.

T

y

e

Fig 1. Multidimensional least squares fit

ŷθ


Examination of the characteristics of the error provides useful information on the quality of the fit and also thequality of the measured data. In Fig. (1), the angle, θ, contained between y and S is an obvious indicator of thecloseness of the fit. The coefficient of determination, R2, which equals the square of the cosine of θ, is oftenpresented as a measure of the degree of linearity and an inverse measure of the noise level of the data.

While there are several ways to measure the influence exerted by each data point on the slope component of theweighting vector, the externally studentized residual, Rst is recommended over others 6. In Eq. (2), Rst(i) is expressedas a combination of the ith residual error, ei, a σ estimate, s(i), that does not use the ith data point and a leveragemeasure based on the so called hat diagonals.

( ) ( ) ( )iiiist hseiR −= 1/ , i = 1 to N (2)

The diagonals of the hat matrix of Eq. (3) fall between 1/N, for points at the regressor mean, and unity, for pointsat infinity. Outliers at the regressor-mean, which influence the intercept, w0, but not the slope, w, have the lowestleverage values. Points with Rst values falling outside the interval [-3, 3] are generally considered to be outliers withexcessive influence, and are hence preferably omitted from the analysis.

( ) TT XXXXH1−= (3)

B. Load Estimation ProcedureA two-step, linear regression based procedure, proposed in Ref. 7, is used. In the first step, the instantaneous

flight strain profile is expressed as a linear combination of ground test strain profiles, and the weights are solved forin the least squares sense. This is shown in Eq. (4), where εflt is a vector of instantaneous flight strains, εgt is thematrix of ground test strains (one column per test), w is the set of weights, N is the number of strain gages and M isthe number of ground tests. The bias vector, ε0, is an averaged value representing the drifts or offsets present in thestrain readings. For ease of analysis, it may be conveniently absorbed into w by appending a column of ones to εgt.

{ } [ ] { } 101 ××××+= MMNgtNflt wεεε (4)

In the second step, flight loads are obtained as weighted linear combinations of ground test loads. This is shownin Eq. (5), where Pflt is the vector of instantaneous flight loads, Pgt is the matrix of ground test loads (one column pertest) and K is the number of loads of interest.

{ } [ ] { } 11 ××××= MMKgtKflt wPP (5)

In essence, the pair of ground test matrices, εgt and Pgt, constitutes the stiffness model that is needed to transformstrains into loads.

III. ResultsThe method is demonstrated for a typical fighter aircraft in the 10-ton class with aeroelastically tailored

composite wings that carry split trailing edge elevons and leading edge slats (Fig. (2)). The multi-spar constructionof the wing adds to the structural complexity by introducing redundant load paths.

The ground test database contained 68 strain gages at locations common to the gages on the flying wing, namelythe wing-fuselage and wing-elevon attachments. The loads of interest are the resultant shear force, bending momentand torsion at the wing-root. The data was sourced from strength assessment tests on a flight standard wing that wasmounted on a rig simulating fuselage flexibility. The tests involved application of 12 different flight-like loadprofiles, corresponding to extreme performance conditions for which the wing had been designed. For each loadprofile, the tests were conducted in ascending steps (followed by descending steps) of 10% of full load, with strainsbeing recorded at every step.

For the present work, the “regress” function of MATLAB 8 was used to generate least squares solutions, aftersuitable augmentation to compute externally studentized residuals as per Eq. (2).


A. Ground Test Data ChecksBefore applying the above method to flight data, the ground test database was studied extensively to confirm its

quality and utility. Each of the 12 test cases was considered as check case (akin to a realistic flight case) for whichthe loads were to be estimated, using the other tests. Sample results are plotted in Fig. (3), where the measuredstrains of test cases #1 and #12 are seen to be very close to the corresponding reconstructed values. For ease ofvisualization, the measured data are connected with continuous lines, and the reconstructed data with dashed lines.

Fig 2. Wing geometry & strain gage locations

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

0

G a g e in d e x

Mic

rost

rain

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

0

G a g e in d e x

Mic

rost

rain

M e a s u re dR e c o n s tru c ted

(a )

(b )

Figure 3. Measured versus reconstructed strains: (a) test #1 and (b) test #8

Wing-fuselage attachments

Win

g-el

evon

atta

chm

ents


Values of R2 are presented in Fig. (4), along with the angles made by the strain vector for each check case with thesubspace spanned by the remaining cases. The closeness of R2 to unity is indicative of the following facts, inaddition to acting as a proof of concept: (a) the measurement noise level is quite low and (b) the available gagelocations are adequate to observe the loads of interest. The close match between measured and reconstructed strainsalso implies that the load profiles are linearly dependent. The maneuver corresponding to the third case, whichshows a dip in the R2 curve and makes a relatively higher angle, was seen to contain a pitch acceleration componentthat is not present in the other maneuvers. Removal of a load case chosen at random from the total set of load casesdid not lead to any noticeable fall in the quality of these results.

The absence of any systematic pattern in the residual error, seen from plots such as that of Fig. (5), confirms theadequacy of the linear regression model.

The estimated loads are plotted in Fig. (6) and estimation errors in Fig. (7). The errors are generally well below20%, except for the torque of test #3 and the shear force and bending moment of tests #7 & #11 in Fig. (7a), whoseload magnitudes are quite low. Since errors taken relative to individual check case loads tend to get amplified whenload values are low, it seems more sensible to peg them to the maximum loads taken over all the tests (Fig. (7b)).

The above checks were repeated with available strain data corresponding to fractional loads, and it was foundthat the estimates degrade gradually and are quite inaccurate below 40% of full load.

1 2 3 4 5 6 7 8 9 1 0 1 1 1 20

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

G r o u n d t e s t i n d e x

R2

Figure 4. Quality of fit – ground data checks

0

0

R e c o n s t r u c t e d s t r a i n

Rec

on

stru

ctio

ner

ror

Figure 5. Scatter plot for test #1

6.285.0122.737.994.919.416.147.2210.008.2315.196.82

Angle (in degrees) between measured strain vector for a loadcase and the subspace spanned by the remaining load cases


1 2 3 4 5 6 7 8 9 10 11 12

0

Ground test index

Sh

ear

forc

e

1 2 3 4 5 6 7 8 9 10 11 12

0

Ben

din

gm

om

ent

1 2 3 4 5 6 7 8 9 10 11 12

0

To

rque

ActualEstimate

ActualEstimate

ActualEstimate

Figure 6. Wing root reactions

1 2 3 4 5 6 7 8 9 10 11 12-100

-80

-60

-40

-20

0

20

40

60

80

100

Ground test index

Per

cen

terr

or

1 2 3 4 5 6 7 8 9 10 11 12-100

-80

-60

-40

-20

0

20

40

60

80

100

Ground test index

Per

cen

terr

or

SHEAR FORCEBENDING MOMENTTORQUE

Figure 7. Relative error of load estimates pegged to(a) individual check case loads and (b) maximum of all check case load magnitudes

(a)

(b)


B. Flight Load EstimatesUnlike ground data that is acquired under well-controlled conditions, strains measured in flight are often noisy,

contain spikes or tend to drift. Sample strain histories, pertaining to a test sortie, are shown in Fig. (8). The data wasde-spiked with an interactive graphical tool and the drifts that occurred between initialization of the straininstrumentation and the takeoff phase were nullified by resetting the strains to zero just prior to takeoff. Since themaneuvers performed were relatively benign, the data falls in a band of 0-500 microstrains. Subsequent drifts thatoccurred during the course of flight were generally of the order of 200 microstrains or less.

The ground test database used for flight load estimation is the same as that mentioned earlier in Section III,namely, 68 strains and 3 loads for each of 12 tests.

Wing root shear force, bending moment and torsion were computed for two maneuvers, namely a stabilized 4gturn and a 133.5 deg/s pure roll.

The estimates were refined repeatedly by discarding the gage with the highest absolute value of Rst in each step.A sample plot of Rst, corresponding to the first iteration, is shown in Fig. (9). The two farthest points (with values of–11 and +4) correspond to the highly drifted histories marked in Fig. (8).

7800 8000 8200 8400 8600 8800 9000 9200 9400 9600 9800-6000

-4000

-2000

0

2000

4000

6000

Time, s

Mir

cro

stra

in

H ighly drifted strain histories

Figure 8. Flight strains for a sortie (cleaned & initialized at takeoff point)

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

G a g e i n d e x

Rst

Figure 9. Externally studentized residuals for the first iteration


The evolution of load estimates during the refinement process is seen in Fig. (10a) for the sustained turn and inFig. (10b) for the roll maneuver. The corresponding evolution of R2 values is shown in Fig. (11) and Fig. (12).

0 10 20 30 40 50 60 700

Iterationb

Sh

ear

forc

e

0 10 20 30 40 50 60 70

0

Iterationb

Sh

ear

forc

e

0 10 20 30 40 50 60 700

Ben

din

gm

om

ent

0 10 20 30 40 50 60 70

0

Ben

din

gm

om

ent

0 10 20 30 40 50 60 700

To

rqu

e

0 10 20 30 40 50 60 70

0

To

rqu

e

Figure 10. Evolution of estimates for (a) stabilized 4g turn (b) 133.5 deg/sec roll

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iteration index

R2

Figure 12. Evolution of R2 – 133.5 deg/sec roll

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iteration index

R2

Influence basedSequentialRandom

Figure 11. Evolution of R2 - stabilized 4g turn

(a) (b)


The estimates fall into three clear sequential segments, which can be characterized as variable, stable, and degraded.The variation is seen during the process of removal of strain readings containing large drifts. The degradation iscaused by indeterminacy of the weights, as the number of gages falls below the number of ground tests. Theaccompanying R2 values (connected by a solid curve) are very low in the first segment, from where they riseconsiderably and become comparable to those of the ground test data (Fig. (4)) in the stable segment. Figure (11)also depicts how R2 evolves when gages are removed from the selection in an ad hoc manner instead of through theRst criterion. The two broken lines correspond to elimination of gages in the order of their occurrence in the gagelist, and in a random sequence.

It may be seen in Fig. (10) that the evolution of the wing root torque estimates is more erratic than that of shearforce and bending moment. It has been shown by others 4 that adequate loading of leading & trailing edge surfacesand collection of strain data at or near these surfaces are strong requirements for accurate wing root torqueestimation. Hence, the poorer trend obtained here is attributed to the absence of strain information from the leadingedge slat region and from one of the wing-fuselage attachments.Match with finite element model predictions was seen to be somewhat poor. However, it must be noted that (a) theavailable strain data is restricted to (4 out of 5) wing-fuselage attachments and wing-elevon attachments (of onewing only), with none in the slat region or on the skins, (b) attachment free play causes some gages to registerunequal strains for equal tension / compression loads, (c) ground test strains are in the order of only a few hundredmicrostrains and (d) flight strains are even lower. It is expected that the nonlinear effects of attachment free play willbe minimized and greater accuracy of load estimates achieved when (a) surface strains are used and (b) extremes ofthe maneuver envelope are approached.

IV. ConclusionThe load estimation method proposed in this paper uses a two-step, linear least squares regression based

procedure to map strains measured on a flying vehicle to structural loads, with ground static test data providing thestiffness model. It operates in an iterative manner to refine the gage selection and provide averaged load estimates.Internal consistency checks on the ground test data, wherein each test was considered as a check case akin to arealistic flight situation for which loads were to be estimated using the remaining tests, have provided proof-of-concept. The sample results presented for flight loads show clearly explainable trends. Stable estimates are obtainedafter removal of gages whose readings contain large drifts. However, the flight load estimates are not close to modelpredictions, for which possible reasons have been discussed.

The proposed method, which incorporates several new features, is simple, intuitive and effective. It is in tunewith the emerging trends of fewer tests and more intensive data processing. In contrast to the calibration equationroute, here (a) all the available strain data are put to use simultaneously and (b) the number of ground tests isrequired to be lower than the number of gages. The emphasis is not on how small a gage set is sufficient to derive aload of interest, but on how many gage readings fit the linear stiffness model inherent in the ground test data. Thepower and elegance of the influence based refinement procedure is revealed by the much poorer performance of adhoc refinements wherein gages are removed from the selection in a sequential or random manner. The feasibility ofreal-time estimation by this method, however, depends on the availability of computing resources.

The present formulation places more emphasis on the statistical properties of the data than the structuralmechanics of it, and more work is needed to address its limitations. The expression of flight strains in terms ofground test strains does not fit into the conventional categories of observational study and designed experiment: theregressors are the ground test strains (not the applied loads) and may be subject to the same kinds of errors as theflight strains. The measure of influence used in this work, Rst, is centered at the regressor mean, where it has theleast leverage. An alternative measure that is centered elsewhere, such as at zero, may lead to more refined gageselections.

On a more general note, the authors feel that simultaneous processing of the complete set of flight measuredstrains has useful potential. For example, the problem of damage detection in structures with multiple load pathsmay be addressed by extraction of patterns that are lost when one is dealing with subsets of the available data.

AcknowledgmentsFacilities for this work were provided by the Aeronautical Development Agency, Bangalore, India. The ground

test database was sourced from the Aircraft Research & Design Center, Hindustan Aeronautics Limited, and theflight data from the National Flight Test Center. The authors thank J. V. Kamesh and A. Banerjee for the manydiscussions that helped in the development of the proposed load estimation method, and T. R. Rajanna for provision


of ground test data. The encouragement provided by Dr. R. M. O. Gemson in the course of the work and thepreparation of this paper is gratefully acknowledged.

References1Allen, M. J., and Dibley, R. P., “Modeling Aircraft Wing Loads from Flight Data using Neural Networks,” NASA/TM-

2003-212032, 2003.2Skopinksi, T. H., Aiken, W. S., Jr., and Huston, W. B., “Calibration of Strain-Gage Installations in Aircraft Structures for

the Measurement of Flight Loads,” NACA Report 1178, 1954.3Jenkins, J. M., and DeAngelis, V. M., “A Summary of Numerous Strain-Gage Load Calibrations on Aircraft Wings and

Tails in a Technology Format,” NASA TM-4804, 1997.4Lokos, W. A., and Stauf, “Strain Gage Loads Calibration Parametric Study,” NASA/TM-2004-212853, 2004.5Nelson II, S. A., ”Strain Gage Selection in Loads Equations using a Genetic Algorithm,” NASA CR-4597, 1994.6Vining, G. G., Statistical Methods for Engineers, Chap. 6, Duxbury Press, 1998.7Karthik, Y. N., “Estimation of Aircraft Loads from Flight Strain Data,” M.Tech. Dissertation, Aeronautical Engineering, M.

V. J. College of Engg., Bangalore, India, 2005.8MATLAB, Statistics Toolbox Users Guide, Ver. 4.0 (R13), Natick, MA, 2002.

[american institute of aeronautics and astronautics 47th aiaa/asme/asce/ahs/asc structures,...

Documents