Quantification of Uncertainty

For the quantification of uncertainty affecting the static aeroelastic response differ­ent structural and aerodynamic output parameters are considered. Evaluation cri­teria commonly used for characterization of the aeroelastic response are the lift and drag coefficients for a given angle of attack, natural frequencies, or flutter speed of

the investigated aircraft. In this study only static aeroelasticity is treated, therefore the study is concerned with the aerodynamic performance of an aircraft wing un­der cruise flight conditions (see table 2). For this purpose the converged angle of attack aEqSt is evaluated iteratively for a given lift. A coupled fluid structure ana­lysis is performed using derivates of the wing structural model presented in section

1.1 which is affected by different types of uncertainties. For each derivate relative deviation A aEqSt / aEqSt compared to the result obtained for a reference structure (without modifications) is calculated.

The global values for relative difference to the converged angle of attack A aEqSt / aEqSt presented in this work are influenced not only by the change of struc­tural parameters, but are dominated by the aerodynamic properties of the wing as well as by the given flow conditions and aerodynamic method used within the static aeroelastic analysis. For this reason, the results presented within this work should be considered as sample values to demonstrate the degree of deviation within aero­dynamic output parameters for a special test case.

To examine the change within the wing box stiffness the structural response (without aeroelastic coupling) is calculated for different derivates of the FE test model subjected to a reference load case. The reference load case is represented by a pressure distribution and inertia loads obtained for a reference structure un­der cruise flight conditions. Local values of bending angle w'(y) and twist 0(y) are computed along the structural wing span. These "beam-like" deformations are extracted from the nodal solution of the 3D finite-element model by means of the method presented by Malcolm and Laird [9]. The procedure employs a least squares fitting to extract three translational and three rotational section deformations from the nodal displacements in x-, y – and z-direction for each wing section. This process is applied to a series of sections along the wing span to calculate the bending and tor­sion. For the local values of bending and torsional angle the deviations Aw'(y) and A0(y) are calculated relative to the deformations obtained for the reference struc­tural model. The local deviations are related to the maximum reference values of the corresponding deformation, w'(y)max and 0(y)max respectively. This approach en­sures that no singularities can occur due to very small local values within the torsion deformation.

To estimate the effect of stiffness variation on the wing aerodynamics, a well – known concept for the elastic angle of attack ael is used. This kinematical term describes the local change of the geometric angle of attack ag in flight direction due to elastic deformation of the wing. It affects the load distribution caused by the flexible structure of lifting surface and thus the overall lift coefficient. Deviations in torsion and bending stiffness of the wing box cause a change of the lift distribution over the wing span compared to the reference structure. Under conditions of steady cruise flight this lift change must be corrected by adapting the angle of attack ag of the aircraft iteratively until target lift will be achieved and aEqSt = ag.

For swept wings the local angle ael (y) depends on the torsion deformation 0 (y) as well as on the bending angle W(y):

From the kinematical interrelationship in equation (1) follows that for a wing with positive angle of sweep back ф the torsion and bending contributions of the elastic angle of attack are directed mutually. For this reason, the change of bending angle due to reduction of bending and shear stiffness of the wing box structure can be compensated by the change of torsion deformation caused by the reduced torsion stiffness to a certain degree. For common transonic transport aircraft wing struc­tures the angle ael is dominated by the bending deformation and for this reason is negative.

To estimate the effect of the variation of torsion and bending distortions on the deviation of the elastic angle the propagation of uncertainty is applied on equation

(1)

. For a local relative deviation A ael (y)/a^ in elastic angle of attack a mathem­atical correlation (2) is the following:

The local values A0(y)/0max and Aw'(y)/w’max are relative deviations of tor­sion and bending distortions due to the local change of wing box stiffness caused by different degrees of modeling simplification. These values are structural parameters depending on the stiffness properties of the structure. The terms 0max/a^ cos ф and w’max/aeax sin ф in equation (2) are ratios of the local torsional and bending angles relative to the maximum value of elastic angle of attack. These values de­pend on the local sweep back angle ф(у) of the wing box reference axis, the load distribution in chord and span-wise directions (ratio of the distributed moment relat­ive to the distributed load) as well as on the ratio of the torsion stiffness GJ relative to the bending stiffness EI.

The local deviations of the elastic angle of attack A ael (y) / aeax are related to the maximum value obtained for the reference FE model in the same manner like deviations of structural deformations. Since the effect of the deviation of this para­meter on the geometric angle of attack ag and thus on the local lift distribution is depending on the magnitude of ael (y) this approach seems to be more suitable for the objective of the present study than relating this term to the reference local values as commonly done. The latter method would overestimate the influence of the deviation A ael (y) considering local variations of the elastic angle of attack near the root as well, which have no appreciable effect on the load distribution due to the very small values of ael (y) within this area.

The distribution of the local deviations A0(y)/0max, Aw'(y)/w’max and A ael (y) / a<miax varies along the wing structural axis, depending on the stiffness and load distribution of the present wing structure and aerodynamic design. To obtain global deviation parameters the mean values of these local variations are calculated in sections using a relation defined exemplary in equation (3) for the bending angle:

where з is the structural span of the wing box. Mean values of twist [AQ] and elastic angle of attack [Aael ] are obtained in the same way. To assess the contri­bution of the variations of torsional and bending angles to the deviation of elastic angle of attack, the local values Aw'(y)/w’max and AQ(y)/Qmax, multiplied with the parameters’ terms w’max/aeJlca sin ф and Qmax/a^cosф from equation (2), are integrated by means of eq.(3). These "transformed" values [Aw’]/r and [AQ}tr are also used within the present work to estimate the effect of variation within struc­tural stiffness properties caused by modeling or stochastic uncertainties on the load distribution. The difference between the term [Aael] = [AQ}tr — [Aw’]tr calculated from the global bending and torsion deformations and the mean value resulting from integration of the local values A ael (y) / a^ directly obtained from the structural response is between 0.01% and 0.03%.

Because nonlinear behavior of aeroelastic problems is highly depending on the local flow conditions as well as on local stiffness characteristics of the wing struc­ture, the change in equilibrium state angle of attack A aEqSt / aEqSt cannot be pre­dicted using a mean value [Aael ] for a complex structure in a direct way. Never­theless, as will be shown within the following sections, the parameter [Aael ] is a suitable indicator to estimate the deviation tendency of wing aerodynamics due to the variation within the stiffness properties of the wing.