An Inverse Interpolation Method Utilizing In-Flight Strain Measurements for Determining Loads and Structural Response of Aerospace Vehicles

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1 An Inverse Interpolation Method Utilizing In-Flight Strain Measureents for Deterining Loads and Structural Response of Aerospace Vehicles S. Shkarayev and R. Krashantisa University of Arizona, Tucson, AZ 8572 and A. Tessler Analytical and Coputational Methods Branch, M/S 240 NASA Langley Research Center, Hapton, VA ABSTRACT An iportant and challenging technology aied at the next generation of aerospace vehicles is that of structural health onitoring. The key proble is to deterine accurately, reliably, and in real tie the applied loads, stresses, and displaceents experienced in flight, with such data establishing an inforation database for structural health onitoring. The present effort is aied at developing a finite eleent-based ethodology involving an inverse forulation that eploys easured surface strains to recover the applied loads, stresses, and displaceents in an aerospace vehicle in real tie. The coputational procedure uses a standard finite eleent odel (i.e., direct analysis ) of a given airfrae, with the subsequent application of the inverse interpolation approach. The inverse interpolation forulation is based on a paraetric approxiation of the loading and is further constructed through a least-squares iniization of calculated and easured strains. This procedure results in the governing syste of linear algebraic equations, providing the unknown coefficients that accurately define the load approxiation. Nuerical siulations are carried out for probles involving various levels of structural approxiation. These include plate-loading exaples and an aircraft wing box. Accuracy and coputational efficiency of the proposed ethod are discussed in detail. The experiental validation of the ethodology by way of structural testing of an aircraft wing is also discussed. INTRODUCTION Advanced structural health onitoring is generally regarded as a vital technology for the next generation of aeronautical and space systes []. This technology is aied at preventing catastrophic structural failures and is coprised of three facets: (a) deterination of stresses and deforations of structural copo-

2 nents, (b) identification of external loads, and (c) detection of critical daage echaniss such as cracking, delaination, and corrosion. The developent of the health-onitoring technology involves ultidisciplinary research in the areas of coputational echanics, intelligent inforation systes, and sensor networks. Recent advances in the design of health-onitoring systes for aerospace applications are discussed in [, 2]. The present effort is aied at developing a finite eleent-based ethodology involving an inverse forulation that eploys easured surface strains to recover the applied loads, stresses, and displaceents in an aerospace vehicle in real tie. The deterination of loads, stresses, and displaceents using experientally easured structural response (strains) is defined as an inverse proble. This type of proble ay result in an ill-posed governing syste of equations and, therefore, requires a special approach to obtain an accurate and stable solution. The atheatical fundaentals of inverse probles ay be found in [3-5]. A review of recent literature indicates that inverse ethods are used quite extensively in echanics. Maniatty and co-workers proposed a finite eleent-based ethod for solving inverse elastic [6] and elastoviscoplastic [7] probles. Their ethod uses a regularization procedure involving two atrices that ipose soothness on the solution. Several one- and two-diensional exaples illustrate the application of the ethod. More recently, Maniatty [8] studied the regularization procedure utilizing a statistical approach by Tarantola [4]. This analysis provides an estiate of the errors in the solution of an inverse proble. The ain drawback of this ethod is that it requires iterations. For coplex three-diensional structures, it ay lead to convergence difficulties and high coputational costs. The application of artificial neural networks to load identification was proposed by Cao et al. [9]. In their approach, the load-strain relationships are established by using a learning algorith for a ulti-layer neural network. The convergence of this algorith, however, strongly depends on the set of chosen learning paraeters. The learning can even diverge if the cobination of paraeters is not appropriate. Martin et al. [0] discussed a non-iterative ethod for the reconstruction of surface tractions using a boundary eleent ethod. The restriction of the ethod is that displaceents and tractions have to be applied siultaneously on a portion of the surface structure. Okua and Oho [] studied the identification of dynaic characteristics using an inverse proble fraework. The set of spatial atrices is deterined by using experientally easured frequency response functions. This ethod, however, requires additional constraint equations related to the daping atrix. The present approach cobines a coputational echanics ethodology with experientally easured strain data to deterine the in-flight response characteristics in real tie. In-flight internal and external loads, unlike other flight paraeters, cannot be directly easured. Therefore, an inverse approach is required to construct the solution. A finite eleent-based ethod for experiental data analysis is developed and leads to the deterination of stresses, displaceents, and external loads. Nuerical siulations are perfored using a linearly elastic plate and wing structure to study the accuracy and coputational effectiveness of the ethod.

3 INVERSE INTERPOLATION METHOD Consider an airplane that perfors a aneuver or is subjected to atospheric turbulence. A change of the external and internal forces causes linear and angular displaceents, strains, and stresses at each point of the structure. The sensors are ebedded in the structure along specified patterns, allowing strain coponent easureents at certain locations. The strain sensors easure the changes in the strain coponents { ε } at n specified locations and appropriately store this inforation in an onboard coputer. These strain data are used as input to the inverse analysis of the coputational odel based on a finite eleent forulation. Since the current approach is based on a finite eleent ethod, a finite eleent odel of the airfrae is first developed. Consider external loads applied to the airplane increentally. Any change of forces acting on the aircraft is assued to be sall. At each increent or step of loading, the governing equations of the linear finite eleent approxiation have the for [ K] { U} = { P } () where [ K] is a stiffness atrix, { U} is a nodal displaceents vector, and { P} is a vector of equivalent nodal forces at the current load increent. The nodal strains can be expressed in ters of nodal displaceents using the strain-displaceent atrix [ B] as { ε} = [ B]{ U } (2) Given a surface load, ps, () nodal forces are readily expressed as T { } [ ] ( ) P = N psds (3) s where the integration is perfored over the surface of the structure s. The inverse proble is forulated as follows. First, one needs to establish a set of possible load cases,. Each load case is characterized by the specifics of the aerodynaic load distribution and inertia forces. For the i th load case, the load approxiation is expressed in paraetric for as l F() s a R () s = (4) i ij ij j= where Rij () s are chosen spatial distribution functions and aij are unknown approxiation paraeters. These unknown paraeters represent flight perforance characteristics such as load factor, angle of attack, and speed. There can also be certain inequity constraints iposed on these paraeters

4 ϕ ( a ij ) 0 (5) The approach proceeds with a direct finite eleent analysis perfored for each i th load case. The corresponding displaceents, { U ij}, and strains, { ε ij}, are deterined fro Eqs. () and (2) as { ij} [ ] [ ] T U = K N Rij ( s) ds (6) s { ij} [ ] [ ] [ ] T ε = B K N Rij () s ds (7) The stresses are then coputed fro Hooke s constitutive relations { ij} = [ ]{ ij} s σ D ε (8) where [ D] denotes the elasticity atrix. The displaceents, strains, and stresses corresponding to the i th load are coputed fro the relations { i} aij { ij} U = U (9) j= { i} aij { ij} ε = ε (0) j= { i} aij { ij} σ = σ () The paraeters of the load interpolation, Eq. (4), are coputed using a least squares procedure iniization, i.e., for the i th load case, we have j= T Si = { εi εi} { εi εi} = aij{ εij} εi aij{ ε ij} εi (2) j= j= The iniization of Eq. (2) with respect to the paraeters a ij, while accounting for the constraints (5), results in a governing syste of linear algebraic equations that is readily solved for the aij coefficients. In this anner, a set of the aij solutions for load cases can be obtained. It is clear that a suitable criterion ust be established in order to select the ost appropriate load case. One such criterion eploys a quality function, QS ( i, n, ),in which the calculated Si values are used. A particular for of this function is constructed based on coputer siulations and experiental statistics. For exaple, if Chebishev s polynoials are chosen to represent the Rij ( s) functions, the quality function can be represented as [5] T

5 (,, n) Q S i Si = n ( + ) ln lnh n + + (3) where H is the probability at which this estiate is valid. A iniu of the quality function QS ( i, n, ) corresponds to the k th load case, providing the best solution to the overall inverse proble. For the known akj ( j =, l) paraeters, the resulting forces acting on the structure are calculated fro Eq. (4), followed by the displaceent, strain, and stress coputations using Eqs. (6-). NUMERICAL SIMULATIONS The present approach is validated using a two-step procedure. A finite eleent odel is first developed for a given linearly elastic structure under the applied load ps. () Surface strains { ε } are then coputed at n specified locations, with these quantities representing the easured experiental strains. In the second step of the analysis, an inverse ethod is eployed that uses these easured strains, { ε }, and the paraetric approxiation of the load, Fi ( s ),asgivenineq.(4).thestrains are coputed fro the loads represented by the Rij ( s) functions. The paraetric approxiation of the load is obtained via a least-squares iniization of Eq. (2). Then, the applied loads and structural response are recovered. Two nuerical exaples are presented. The first is a cantilever plate under transverse loading, whereas the second is a siplified odel of an aircraft wing box. The aterial properties used are those for a typical aluinu alloy with Young s odulus E = 72 GPa and Poisson s ratio ν =0.3. The finite eleent analysis is perfored using a coercial finite eleent code, ANSYS [2], with both odels eploying SHELL 63 shell eleents. Plate Analysis Consider a cantilevered rectangular plate under transverse loading 2 2 p( x, y) = bx + b2y + b3y+ b4 applied noral to the top of the plate as shown in Figure. Two loading cases are considered: (a) a doinant bending load given by the paraeters b = 0.25, b 2 = 4, b 3 = 0, and b 4 =.5 ; and (b) a doinant torsion load with b = 0.25, b 2 = 4, b 3 = 5,and b 4 = Noral strains in the x-direction, { ε x }, and in the y-direction, { ε y}, siulating experiental easureents, are first coputed fro the finite eleent analysis (first step) at 7 strain-gage locations ( n = 34), uniforly spaced along the line y = 0.25 on the upper surface of the plate, z = 0. The plate is discretized using a esh of 4 6 shell eleents.

6 px,y ( ) y z x l=2 t=0.002 h=0.5 ε y ε x Strain gage rosette Figure. The cantilevered plate proble. 2 2 Assuing that the load approxiation is represented by F( x, y) = ax + a2y + ay 3 + a4, the present inverse approach gives rise to the values of the aj ( j =,4) coefficients suarized in Table I. Their favorable coparison with the corresponding values of the bj ( j =, 4) coefficients for the actual load case clearly deonstrates that the proposed ethod defines the load accurately. TABLE I. COEFFICIENTS OF LOAD APPROXIMATION a 4 Load Case a a2 3 a a b Aircraft Wing Box The second exaple concerns the wing box depicted in Figure 2. The upper and lower panels are stiffened with integral stringers. As shown in the figure, three ribs 2 2 are attached spanwise on the inside of the wing. The load pxy (, ) = b l x ( b = 0.5 ) is applied noral to the upper surface of the wing. There are straingage locations for { ε x } ( n = ), uniforly spaced along the line y = 2.4 on the upper surface of the wing. 2 2 Here, we select the load approxiation function as F( x, y) = a l x.in order to study the effect of easureent errors on the accuracy of the ethod, the easured strains are assued in the for { ε x }{ }. The coponents of the vector of relative errors { } are given as i = δ i,whereδi is a rando variable having a standard noral distribution. The recovered a coefficient is in the range of These values agree well with the actual load paraeter b = 0.5.

7 px,y ( ) Q x t=0. y z Rib l= Rib 2 Rib 3 h=0.3 Stringer δ=0.00 d=0.02 Wing cross-section Figure 2. Aircraft wing box and loading. FUTURE DEVELOPMENTS Additional nuerical studies based on different loadings will be perfored. In order to correlate and quantify the results of the inverse approach, an experiental validation of the ethodology is underway. Structural testing of an aircraft wing will be perfored in an experiental setup coprised of a wing box, threediensional frae, and loading and easureent systes. These tests will allow: (a) the deterination of the accuracy of the proposed ethod, (b) a study of the errors associated with the application of strain gages, and (c) developent of recoendations for the nuber and locations of strain gages. REFERENCES. Noor, A. K., S. L. Venneri, D. B. Paul, and M. A. Hopkins Structures Technology for Future Aerospace Systes, Coput. and Struct., 74: Sipson, J. O., S. A. Wise, R. G. Bryant, R. J. Cano, T. S. Gates, J. A. Hinkley, R. S. Rogowski, and K. S. Whitley Innovative Materials for Aircraft Morphing, presented at SPIE s 5 th Annual International Syposiu on Sart Structures and Materials, March -5, Tichonov, A. N. and V. Y. Arsenin Solution of Ill-Posed Probles. Washington, DC: V. H. Winston and Sons. 4. Tarantola, A Inverse Proble Theory: Methods for Data Fitting and Model Paraeters Estiation. New York: Elsevier. 5. Vapnik, V. N Algoriths and Progras for Dependencies Reconstruction. Moscow: Nauka in Russian.. 6. Maniatty, A., N. Zabaras, and K. Stelson Finite Eleent Analysis of Soe Inverse Elasticity Probles, J. Eng. Mech., 5(6): Maniatty, A. and N. Zabaras Method for Solving Inverse Elastoviscoplastic Probles, J. Eng. Mech., 5(0): Maniatty, A Regularization Paraeters and Error Estiating in Inverse Elasticity Probles, Int. J. Nuer. Methods Eng., 37: Cao, X., Y. Sugiyaa, and Y. Mitsui Application of Artificial Neural Networks to Load Identification, Coput. and Struct., 69:63-78.

8 0. Martin, T. J., J. D. Halderan, and G. S. Dulikravich An Inverse Method for Finding Unknown Surface Tractions and Deforations in Elastostatics, in Inverse Probles in Mechanics, S. Saigal and L. G. Olson, eds. New York, ASME, pp Okua, M. and T. Oho Experiental Spatial Matrix Identification as a Practical Inverse Proble in Mechanics, in Coputational Methods for Solution of Inverse Probles in Mechanics, L. G. Olson and S. Saigal, eds. New York: ASME, pp ANSYS User s Manual, Vol. I Canonsburg, PA: Swanson Analysis Systes, Inc.