A COMPARISON BETWEEN THE AERODYNAMIC PRESSURE FACTORING AND THE AERODYNAMIC DERIVATIVES FACTORING METHODS FOR THE DOUBLET LATTICE PROGRAM

Transcription

1 A COMPARISON BETWEEN THE AERODYNAMIC PRESSURE FACTORING AND THE AERODYNAMIC DERIVATIVES FACTORING METHODS FOR THE DOUBLET LATTICE PROGRAM Paper Reference No. 1-1 by Emil Suciu* ABSTRACT Graphical and analytical means are used to compare the pressure factoring method with the aerodynamic derivatives factoring method as they are applied to a lifting surface with control surface and tab experiencing motion in a general mode of vibration. Errors, advantages and disadvantages of each method are discussed. A variation of the aerodynamic derivatives factoring scheme expected to bring a further refinement to the generalized aerodynamic forces is described here. Proceedings of the MSC.Software rd Worldwide Aerospace Conference and Technology Showcase, April 8-1,, Toulouse, France *Contract Engineer, EMBRAER, Sao Jose dos Campos, Sao Paulo, Brazil Dynamic and aerodynamic models drawings used with permission from EMBRAER 1

2 List of Symbols a i, b i, c i, d i (i=1,) = aerodynamic correction factors c = reference chord DLM = Doublet Lattice Method h = heave displacement k = ω*c/(*v), reduced frequency L h, L α, L β, L δ = lift due to h, α, β and δ motion M = Mach Number M h, M α, M β, M δ = moment about reference axis due to h, α, β and δ motion T h, T α, T β, T δ = control surface hinge moment due to h, α, β and δ motion Q h, Q α, Q β, Q δ = tab hinge moment due to h, α, β and δ motion L, M, T, Q = factored lift, moment and control surface and tab hinge moment using the derivatives factoring method q = (1/)?V = dynamic pressure V = airspeed x = chordwise coordinate α = pitch angle β = control surface rotation angle δ = tab rotation angle?cp h,?cp α,? Cp β,?cp δ = pressure distribution coefficients due to h, α, β and δ motion ε = error? = air density ω = circular frequency of oscillation C.H.L. = control surface hinge line L.E. = leading edge P.M. = pressure method T.E. = trailing edge T.H.L. = tab hinge line Abbreviations

3 1. Introduction The discussion which follows is influenced by the perspective of the practicing flutter analyst with experience in subsonic and transonic civil transport aircraft, using standard analytical methods with their capabilities and limitations as they are generally understood. The presentation is intended to be as self-sufficient as possible. At the present time, the most widely used method for calculating unsteady aerodynamic forces and moments and control surfaces and tabs hinge moments for use in flutter analysis is the Doublet Lattice Method (DLM), Reference 1, in its implementation in MSC.Nastran (Reference ). The flutter analyst is chasing a nonlinear problem with linear methods and trying to account quantitatively for some nonlinear effects missing from the theory (primarily viscosity and compressibility) through empirical factoring of the aerodynamic solution. Qualitative features such as shocks cannot be introduced through factoring if they are not present in the basic theory (Reference ). All factors lump together the effects of both compressibility and viscosity and factoring takes place with factors derived at each Mach Number used in the flutter analysis. From the flutter analyst s standpoint, the necessity of factoring calculated aerodynamic forces and moments is not under debate: Advisory Circular.9-1 unambiguously recommends the practice, though not the method of factoring, which is left for the user to choose. For the purpose of illustrating the calculation of the aerodynamic derivatives and of correction factors, it is found very useful to examine the oscillatory inviscid and incompressible two-dimensional pressure distributions for a flat plate with control surface and tab due to simple motions as seen in Figure 1. Leading edge balance surfaces are neglected for this discussion. Reference contains the analytical expressions of the -D analytical pressure distributions from the potential functions derived by Theodorsen and Garrick in Reference. Theodorsen s and Garrick s pressure distributions are identical with the pressures independently calculated with Kussner s theory (Reference ). See again Figure 1, which shows the real and imaginary parts of the -D pressure distributions for a flat plate with control and tab for four simple motions: unit rigid heave, unit rigid pitch, unit rigid control surface rotation and unit rigid tab rotation at the reduced frequency shown. Once we have calculated a pressure distribution resulting from a unit simple motion at zero frequency, we can integrate it and obtain a lift and a moment for the section, a control surface hinge moment and a tab hinge moment according to the definition of the aerodynamic derivatives, Reference 7. A total of 1 derivatives can be obtained from the

4 four elementary motions shown. Correction factors for each calculated quantity are obtained by dividing the experimentally-obtained values with the calculated ones. At zero frequency the heave-related derivatives are nil. Since the correction factors are good for the steady-state solution, the assumption is made that the same correction factors will apply to the unsteady solution. For k, the heave motion is similar to the pitch motion and the correction factors for heave pressures & derivatives are the same as for the pitch derivatives. The control surface and tab motions and pressure distributions are similar, though the correction factors are different from each other and from the correction factor derived for the pitch motion. It is important to note that once a set of correction factors has been selected, the same set of factors will be applied to all modes of motion. The DLM can duplicate the -D pressure distributions, whether at k= or for any k case. The -D pressure distributions calculated by the DLM on a -D wing-control-tab have the same character as the -D pressures, with spanwise interference accounted for. The discussion which follows applies to any spanwise strip of the DLM aerodynamic surface. One of the problems faced by the flutter analyst is that the average aircraft lifting surface structure refuses to exhibit modal motion consisting only of pure heave or pure pitch or pure control surface rotation or pure tab rotation appearing in a predictable sequence but instead the average modal motion the analyst has to contend with is a combination of all of the above, as seen in Figure, which shows a typical mode of vibration for the horizontal stabilizer of a jet transport aircraft. The symmetric modal motion shown is very general: the lifting surface and elevator have elastic bending, heave (from the fuselage vertical bending), root attachment pitch, control surface rotation and tabs rotation. In order to complete the discourse on general modal motion, we introduce at this point Figure, which shows an MSC.Patran rendition of the same mode as in Figure, but interpolated at the DLM aerodynamic surface. Note that MSC.Patran displays the aerodynamic surface and displacements in the same fashion as it displays the structural part. Figure will help with the understanding of the problem under consideration. The task still remains to apply the aerodynamic correction factors obtained from pure motions to the unsteady aerodynamic solution resulting from general structural motion pictured in Figures and. Reference will be made again and again to Figures and, since the calculated general modal motion shown is ideally suited for the discussion which follows and the popularculture expression paraphrased here as take a mode, any mode is appropriate in this case.

5 . The Pressure Factoring Method A thorough discussion of the pressure (or premultiplying) and the downwash (or postmultiplying) factoring methods is presented in Reference and methods for obtaining correction factors are outlined: a number of calculated aerodynamic quantities are constrained to minimize the error from the experimental values and a set of correction factors for the theoretical pressure distribution are obtained. Reference provides the option to use the pressure factoring method with the MSC.Nastran DLM but the choice and application of correction factors is left to the user. In the industry there are a variety of implementations of the pressure factoring procedure in the context of MSC.Nastran and the list below is limited only by the author s experience: (1) simple multiplications of the pressure distribution at each box with a correction factor on the main surface ahead of the control surface, another factor applied between the control surface leading edge and tab leading edge and another factor on the tab (the most widely used technique); () distributed sets of factors based on pressure ratios; () distributed least-squares-derived factors based on the work of Reference.. The Aerodynamic Derivatives Factoring Method Reference 8 introduced a general aerodynamic derivatives factoring method, which will be outlined here using graphical means; the same graphical means will also serve to facilitate comparison with the pressure-factoring method. Let us take a streamwise cut through the center of any strip of the horizontal stabilizer of Figure, making sure that the section contains a tab. Figure shows this section with camber deformation neglected for the sake of clarity; in Reference 8 it is shown that streamwise camber deformation can be accounted for without difficulty for the purpose of factoring (or at least for inclusion). At the top of Figure, we can see the real part of the pressure distribution calculated by the DLM for the scrambled motion and this is what we have to work with when the pressure method of factoring is used. The imaginary part of the pressures are not shown for lack of space on the page. The structural motion is linear so we can descramble the general elastic mode shape into four simple or elementary mode shapes: heave (or bending), pitch (or torsion), control surface rotation relative to the main surface and tab rotation relative to the control surface. If we perform this motion descrambling at the center of every

6 spanwise station or strip, instead of one wing executing general motion we now have four wings, each executing a simple or descrambled motion: bending (or heave), elastic torsion, control surface rigid rotation and torsion, and tab rigid rotation and torsion. Using the stored aerodynamic influence coefficient matrix, we can calculate four sets of pressures, each corresponding to one of the descrambled motions. In Figure, the descrambled real part of the oscillatory pressures are shown at the representative section. The aerodynamics is also linear, so if we add the descrambled pressures we will reproduce the pressure distribution calculated in a single pass through the scrambled motion as seen at the top of Figure. Since we have the four descrambled sets of pressures, we can calculate the lifts, moments, control surface hinge moments and tab hinge moments for each set. The forces and moments calculated from the descrambled pressures will add up to the lift, moment, control surface hinge moment and tab hinge moment calculated directly from the scrambled pressure distribution as seen at the top of Figure. Generalized response aerodynamic forces are calculated using the unfactored descrambled forces and moments and compared with the generalized forces calculated through integration of the scrambled unfactored pressures and mode shapes (Reference 1). The two sets of generalized aerodynamic forces are identical and this is a very important and nontrivial verification step. The effort of going through descrambling the general modal motion and the calculation of the descrambled forces and moments was not put forth just for the purpose of simply looking at them. Once we have the descrambled aero forces, we can factor them as shown below: L = L h *a 1 + L α *a + L β *a + L δ *a M = M h *b 1 + M α *b + M β *b + M δ *b T = T h *c 1 + T α *c + T β *c + T δ *c Q = Q h *d 1 + Q α *d + Q β *d + Q δ *d (1) where a 1, a, etc. are the correction factors obtained at zero frequency. If experimental derivatives data is available at each spanwise station, the correction can take place with sets of factors tailored for every strip, thus making sure that the spanwise load distribution is matched also. Otherwise, all the strips of each panel of any lifting surface of the aircraft will be factored with a panel set of factors and the DLM-calculated spanwise load distribution is accepted as being correct.

7 The generalized aerodynamic forces are now calculated using the factored aerodynamic forces and moments and the flutter solution will proceed with this modified matrix.. A Comparison Between the Pressure Factoring and Derivatives Factoring Methods; Error Calculation For the purpose of illustration and for simplicity, only the factoring of the lift is discussed. Let us consider the most widely used factoring method in the industry with the pressure method, the application of a factor on the main surface, a different factor on the control surface before the tab and a third factor on the tab. These factors can be distributed or they can be uniform over their range of application. Factor F1 is based on matching the tab hinge moment due to tab rotation. Factor F is based on matching the control surface hinge moment due to control surface rotation taking into account the presence of factor F1. Factor F is based on matching the lift of the entire surface due to pitch, taking into account the presence of factors F1 and F. Figure shows the application of these factors along the chord of any lifting surface section. Let us also remember that the three factors are different from each other. It is important to remember that the same three factors will be applied to all the modes of vibration. The first thing we notice in Figure is that if we apply the three factors F1, F and F to the scrambled pressure distribution resulting from the general mode of vibration, we automatically apply the same three factors to all the component pressure distributions. Let us then examine the effect of applying these three factors to the component pressure distributions as shown in Figure as they relate to the section lift. Factor F1 (fitted to match the tab hinge moment due to tab rotation) is applied incorrectly for all four component pressure distributions. Factor F (fitted to match the control surface hinge moment due to control surface rotation and taking into account the presence of F1) is applied incorrectly for all four component pressure distributions. Factor F (fitted to match the section lift due to pitch and taking into account the presence of F1 and F) is applied incorrectly for the pressure distributions resulting from control surface rotation and from tab rotation; their application to the pressure distributions resulting from the heave and pitch motions is accepted as being correct. Even if the structural mode consists of a simple motion, say tab rotation, the application of the three correction factors for the representative section will result in zero out of three factors being applied correctly. This is a very low success rate. 7

8 Similar errors are introduced due to the pressure factoring just described when the moment about the reference axis and the control surface and tab hinge moments are calculated. We are faced with the situation where the only thing we can state with certainty is that for a wing-control-tab configuration, for a general mode of vibration we have no idea what the value of the lift is, what the value of the moment about the reference axis is, what the value of the control surface hinge moment is and what the value of the tab hinge moment is at k= and also at any k. Flutter analysis will proceed with a generalized aerodynamic forces matrix formed with components which are incorrect. In actuality these errors are not quite arbitrary but they are bound by the magnitude of the correction factors. The errors introduced by the pressure factoring method can be calculated for each mode of vibration for all the forces and moments with the help of the descrambled pressure distributions at any frequency. As discussed, each mode has a different modal motion mix. Only the error for the lift is presented here in analytical form. The derivatives factoring method calculates the exact lift at zero frequency for any modal motion. From Equation 1, for any mode of vibration, for unit strip width, at k= (and within the factoring assumption, at any k ), T. E. L exact = q* ( a1 * Cph + a * Cpα + a * Cpβ + a * Cpδ )* dx () L. E. with a 1 =a () The lift factored with the pressure method (L P.M. ) using the three factors F1, F and F as discussed is: C. H. L. L P.M. = q* F1* ( Cp Cp Cp Cp ) h + + β + δ * dx L. E. T. H. L. α + q* F*( Cp + Cp + Cp + Cp C. H. L. T. E. h )* dx α β δ + q* F*( Cph + Cpα + Cpβ + Cpδ )* dx () T. H. L. It is evident that L exact L P.M. unless by pure chance. The error ε is: %ε= 1.*(L exact L P.M. )/ L exact () 8

9 The errors introduced through the use of the pressure factoring method can be calculated for the moment about the reference axis and for the control surface and tab hinge moments in the same fashion. The same formulas can be used to calculate the errors introduced by distributed factors. In contrast, examine Equations 1 and and Figure and note that the derivatives approach results in exact factoring of all derivatives and that the factoring is exact for all modes regardless of the motion composition since general motion descrambling into elementary motions is exact.. A Modified Aerodynamic Derivatives Factoring Scheme A variation of the aerodynamic derivatives factoring scheme expected to bring a further refinement to the generalized aerodynamic forces is described here. First the pressures are corrected and then the derivatives are corrected if necessary. If at any Mach Number of interest experimental chordwise and spanwise pressure distributions for the four simple unit motions at zero frequency exist, the calculated pressure distributions based on the four simple unit motions can be corrected to match the experimental distributions. The 1 derivatives based on corrected pressure distributions can then be calculated and correction factors for these 1 derivatives can be obtained in the same fashion as previously for the four simple unit motions. It is reasonably expected that the derivatives correction factors calculated based on matched pressures will be much closer to unity than the factors calculated with the theoretical pressure distributions. The correction scheme then proceeds as follows: the modal motion is descrambled and the descrambled pressures are calculated and factored based on the steady-state pressure distributions based on simple motions; if the calculated steady-state derivatives match the experimental ones to an acceptable level, four sets of generalized air force matrices can be calculated directly by integrating the factored descrambled pressure distributions and the final generalized air force matrix is the sum of the four descrambled ones. If the steady-state derivatives calculated with the factored pressures do not match the experimental ones closely enough, all unsteady derivatives are calculated and factored with the factors derived using matched experimental pressures, in other words applying the regular aerodynamic derivatives correction scheme. It is expected that this variation of the aerodynamic derivatives factoring scheme will bring a further improvement to the accuracy of the calculated generalized forces. 9

10 7. Conclusions Regarding pressure factoring, in Reference the first conclusion is: (1) One Set Of Correction Factors Is Not Good For All Modes. Application of correction factors, determined from one mode, to other modes has not met with much success. Specifically, correction factors obtained using a pitch mode cannot be applied to pressures due to control surface deflections. The converse is also true. This conclusion is as valid today as it was in 197 and analytical and graphical proofs of this conclusion have been provided here. In the author s experience, the pressure factoring method is good for a lifting surface without control surfaces. The errors discussed in the previous sections are introduced when control surfaces and tabs are present. The pressure factoring method is easy to implement. Some trends could be predicted, with the understanding of the errors introduced. The aerodynamic derivatives factoring method, due to exact motion descrambling, is capable of factoring up to derivatives per mode independently of each other if camber deformation is included. This makes it exact for all modes, regardless of the motion composition of each mode. Any flutter analyst who has performed a wing-control-tab flutter analysis (and occasionally chase a flutter incident for such a configuration) will have the understanding and the appreciation for the importance of being able to factor the direct derivatives and cross derivatives for control surface and tab independently of each other and to perform sensitivity studies for individual derivatives, which is not possible with the pressure factoring method. If distributed spanwise experimental derivatives data is available, the aerodynamic derivatives factoring scheme can factor every strip with a different set of factors thus insuring correct spanwise load distributions, otherwise all the strips within a panel are factored with the same panel set of factors and accepting the DLM-calculated spanwise load distributions as being correct. The aerodynamic derivatives factoring method requires an off-msc.nastran FORTRAN code to implement the necessary steps but if the geometric data from the CAERO1 cards is used directly, the additional input requirements will be small. A potential difficulty is being able to obtain all the experimental derivatives which the method can handle, but this is not a deficiency of the aerodynamic derivatives factoring method. To summarize, the aerodynamic derivatives factoring method permits pin-point-accurate factoring of up to forces, moments and hinge moments per strip independently of each other for wing-control-tab configurations with arbitrary modal motion. Having precise control over every individual aerodynamic force and moment will increase the confidence in the results of a flutter analysis and decisions for 1

11 hardware modifications based on flutter considerations, if called for, will be made based on realistic aerodynamic data. 8. References 1. Giesing, J.P., Kalman, T.P., and Rodden, W.P., Subsonic Unsteady Aerodynamics for General Configurations, Part I, Vol. I Direct Application of the Nonplanar Doublet Lattice Method. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-71- Part I, Vol. I, Rodden, W.P., and Johnson, E.H., User s Guide V8, MSC.Nastran Aeroelastic Analysis, The MacNeal-Schwendler Corporation, Los Angeles, CA, Giesing, J.P., Kalman, T.P., and Rodden, W.P., Correction Factor Techniques for Improving Aerodynamic Prediction Methods, NASA CR-197, NASA Langley Research Center, May Postel, E.E., and Leppert, E.L. Jr., Theoretical Pressure Distributions for a Thin Airfoil Oscillating in Incompressible Flow, Journal of Aeronautical Sciences, August 198, p. 8.. Theodorsen, T., and Garrick, I.E., Nonstationary Flow about a Wing-Aileron-Tab Combination Including Aerodynamic Balance, NACA Report 7, 19.. Kussner, H.G., and Schwarz, L., The Oscillating Wing with Aerodynamically Balanced Elevator, NACA TM 991, Smilg, B., and Wasserman, L.S., Application of Three-Dimensional Flutter Theory to Aircraft Structures, AAF Technical Report 798, July Suciu, E., Glaser, J., and Coll, R., Aerodynamic Derivatives Factoring Scheme for the MSC.Nastran Doublet Lattice Program Including Elastic Streamwise Camber Deformation, Presented at the MSC.Nastran World Users Conference, Los Angeles, California, March

12 h Real -D Theodorsen D Cp Due to Main Surface Heave; Imaginary -D Theodorsen D Cp Due to Main Surface Heave; k=.7; Calculate L h, M h, T h, Q h 1 k=.7; Calculate L h, M h, T h, Q h 1 1 Imag Dcp 8-1 α Real -D Theodorsen D Cp Due to Main Surface Pitch; Imaginary -D Theodorsen D Cp Due to Main Surface Pitch; k=.7; Calculate La, Ma, Ta, Qa k=.7; Calculate La, Ma, Ta, Qa 1 Imag Dcp β 1 9 Real -D Theodorsen D Cp Due to Control Surface Rotation; Hinge Line at 7% Chord; No Aerodynamic Balance Surface; k=.7; Calculate Lb, Mb, Tb, Qb Imag Dcp - - Imaginary -D Theodorsen D Cp Due to Control Surface Rotation; Hinge Line at 7% Chord; No Aerodynamic Balance Surface; k=.7; Calculate Lb, M b, Tb, Q b - δ 1 9 Real -D Theodorsen D Cp Due to Tab Rotation; Hinge Line at 8% Chord; No Aerodynamic Balance Surface; k=.7; Calculate Ld, M d, Td, Qd Imag Dcp Imaginary -D Theodorsen D Cp Due to Tab Rotation; Hinge Line at 8% Chord; No Aerodynamic Balance Surface; k=.7; Calculate Ld, M d, Td, Qd Figure 1. Two-Dimensional Oscillatory Pressure Distributions on Flat Plate Due to Four Elementary Unit Motions. 1

13 MSC/PATRAN Version 9. -May-1 :: Deform: _BOW_11_KG.SC1, Mode :Freq.=.977, Eigenvectors, Translational, (NON-LAYERED) Z Y X default_deformation : Max 199 Figure. General Structural Mode Shape MSC/PATRAN Version 9. 1-Jun-1 8:1: Deform: SC AEROSGD HYBRID DYNAMIC MOD, A1:Mode : Freq. =.977, Eigenvectors, Translational, (NON-LAYERED) Z Y X default_deformation : 98 Figure. General Structural Mode Shape at Aerodynamic Surface 1

14 h α β δ 1 Real -D Theodorsen D Cp Due to General Motion; Hinge Lines at 7% and 8% Chord; No Aerodynamic Balance Surfaces; k=.7; Calculate L, M, T, Q L = Lh + La + L β + L δ M = M h + M a + Mβ + Mδ T = T h + Ta + T β + T δ Q = Q h + Qa + Q β + Q δ = = Real -D Theodorsen D Cp Due to Main Surface Heave; k=.7; Calculate L h, M h, T h, Q h h Reference Axis Real -D Theodorsen D Cp Due to Main Surface Pitch; α 1 k=.7; Calculate La, Ma, Ta, Qa + + Real -D Theodorsen D Cp Due to Control Surface Rotation; Hinge Line at 7% Chord; No Aerodynamic β Balance Surface; k=.7; Calculate Lb, Mb, T b, Qb + + Real -D Theodorsen D Cp Due to Tab Rotation; Hinge Line at 8% Chord; No Aerodynamic Balance Surface; k=.7; Calculate L d, Md, Td, Q d 1 δ 9 Figure. Descrambling of General Elastic Motion into Pure Bending, Torsion, Control Rotation and Tab Rotation. Streamwise Camber deformation Is Neglected. Real Pressure Distribution Only Shown. 1

15 Real -D Theodorsen DCp Due to General Motion; Hinge Lines at 7% and 8% Chord; No Aerodynamic Balance Surfaces; k=.7; Calculate L', M', T', Q' L' = a1*lh + a *L a + a*l b + a*ld h α β δ 1 F F F1 = = Real -D Theodorsen D Cp Due to Main Surface Heave; k=.7; Calculate L h, M h, T h, Q h h Reference Axis 1-1 F a 1 *L h F F1 + + α 1 Real -D Theodorsen D Cp Due to Main Surface Pitch; k=.7; Calculate La, Ma, Ta, Qa F a *La F F1 + + Real -D Theodorsen DCp Due to Control Surface Rotation; Hinge Line at 7% Chord; No Aerodynamic 1 Balance Surface; k=.7; Calculate L b, Mb, T b, Qb β F a *Lb F F1 + + Real -D Theodorsen D Cp Due to Tab Rotation; Hinge Line at 8% Chord; No Aerodynamic Balance Surface; k=.7; Calculate Ld, M d, T d, Qd δ 1 9 F a *Ld F F1 Figure. Application of Correction Factors for Pressure Distribution for General Mode of Vibration Using the Pressure Method and the Aerodynamic Derivatives Factoring Method. Only Factored Lift is Shown.