An Investigation of the Sensitivity of F-16 Fighter Flutter Onset and Limit Cycle Oscillations to Uncertainties Jeffrey P. Thomas, Earl H. Dowell, and Kenneth C. Hall Duke University, Durham, NC 27708 0300 and Charles M. Denegri Jr. U.S. Air Force SEEK EAGLE Office, Eglin Air Force Base, Florida 32542-6865 A computational investigation of flutter onset and limit cycle oscillations of the F-16 fighter using a nonlinear frequency-domain harmonic-balance approach is presented. In this latest study, we examine the sensitivity of computed aeroelastic behavior to characteristics and parameters of the structural and fluid dynamic model. Three different F-16 weapons and stores configurations are considered. Results indicate that the flutter onset Mach number is very sensitive to the structural natural frequencies, and, more specifically, the difference between the natural frequencies, of the first two antisymmetric structural mode shapes. Wing mean angle-of-attack is also shown to be very important. The results presented herein may prove useful and provide insight to other researchers modeling F- 16 fighter aeroelastic behavior, and who also may be observing large sensitivities in their computational solutions. I. Introduction This paper is a follow-on to the research efforts reported in Thomas, Dowell and Hall 1,2 where the harmonic balance (HB) technique for modeling nonlinear unsteady aerodynamics (see Hall et al. 3 and Thomas, Dowell, and Hall 4 6 ) is used to determine the limit cycle oscillation (LCO) behavior of the F-16 fighter aircraft. The goal of the present investigation is to study the sensitivity of F-16 LCO behavior to characteristics and parameters of the structural and fluid dynamic models. In recent years, a number of researchers have begun modeling the flutter onset and limit cycle oscillation behavior of the F-16 fighter. e.g. Denegri and Dubben 7 9 have been active in doublet-lattice methods for flutter onset and transonic small-disturbance methods for flutter onset and limit cycle oscillation prediction. Parker and Maple 10 have recently studied the effects viscosity and modeling external stores for computed F-16 limit cycle oscillations. Lieu et al. 11 have recently been active in developing reduced order models for F-16 flutter analysis. Pranata et al. 12 have recently used a time-domain coupled computational fluid dynamic (CFD) and modal based structural method for modeling limit cycle oscillation response of the F-16. II. Configurations Table 1 shows the weapons and stores arrangements for the three different F-16 fighter configurations examined in this paper. Note that configuration number one has no AIM-9P missiles. We studied configuration number one in our two recent papers that detail the use of the HB method for modeling flutter onset Research Assistant Professor, Department of Mechanical Engineering and Materials Science, Senior Member AIAA. William Holland Hall Professor, Department of Mechanical Engineering and Materials Science, and Dean Emeritus, School of Engineering, Honorary Fellow AIAA. Julian Francis Abele Professor and Department Chairperson, Department of Mechanical Engineering and Materials Science, Associate Fellow AIAA. Lead Flutter Engineer, Engineering Division, Certification Division, 205 West D Avenue, Suite 348. Senior Member AIAA. 1 of 8
Stn. Configuration 1 Configuration 2 Configuration 3 1 LAU-129 launcher AIM-9L missile/lau-129 launcher LAU-129 launcher 2 AIM-9P missile/lau-129 launcher AIM-9L missile/lau-129 launcher AIM-120 missile/lau-129 launcher 3 Air-to-ground missile Air-to-ground missile General-purpose bomb 4 Empty 370-gal fuel tank Half-full 370-gal fuel tank Quarter-full 370-gal fuel tank 5 Empty station Empty station Empty station 6 Empty 370-gal fuel tank Half-full 370-gal fuel tank Quarter-full 370-gal fuel tank 7 Air-to-ground missile Air-to-ground missile General-purpose bomb 8 AIM-9P missile/lau-129 launcher AIM-9L missile/lau-129 launcher AIM-120 missile/lau-129 launcher 9 LAU-129 launcher AIM-9L missile/lau-129 launcher LAU-129 launcher Table 1. Configurations and LCO response of the F-16 (Thomas, Dowell and Hall 1,2 ). Some of the main conclusions reached in those two papers include: As consistent with flight test, the F-16 flutters antisymmetrically, and as such, only the antisymmetric structural modes need to be included in the aeroelastic computational model. The flutter onset and finite amplitude LCO response aeroelastic mode shapes are dominated by the wing first bending and first twisting mode shapes. As such, one can obtain well converged computation aeroelastic solutions using just these two structural mode shapes in the structural model portion of the aeroelastic solver. The computed flutter onset conditions are very sensitive to the wingtip geometry. In our present analysis, we are only modeling the F-16 wing. In an attempt to roughly model the presence of the wingtip launchers, we extended the wing tips by six inches. We found this leads to a significant reduction in the computed flutter onset Mach number at a constant altitude. Namely, from M 0.9 to M 0.7 at 2000 feet altitude. Thus, accounting for wingtip external stores in the computational fluid dynamic model appears to be very important. Including viscous effects in the computational fluid model is crucial for computing finite amplitude limit cycle oscillations. Mode Configuration 1 Configuration 2 Configuration 3 First Antisymmetric Bending (f 1ab ) 8.167878 Hz 5.470121 Hz 6.500493 Hz First Antisymmetric Twisting (f 1at ) 8.671919 Hz 5.735396 Hz 7.320433 Hz Second Antisymmetric Bending (f 2ab ) 10.89219 Hz 7.868404 Hz 8.369000 Hz Second Antisymmetric Twisting (f 2at ) 12.31600 Hz 8.007840 Hz 8.969988 Hz f 1at f 1ab 0.504041 Hz 0.265275 Hz 0.819940 Hz Table 2. F-16 Fighter Configuration Natural Frequencies (via NASTRAN) Table 2 shows the computed (via NASTRAN) structural natural frequencies for the first and second, antisymmetric, bending and twisting structural mode shapes for the three F-16 fighter configurations studied in this paper. Note the small difference in natural frequencies (< 1 Hz) between the first antisymmetric bending and first antisymmetric twisting mode shapes. III. Flutter Onset Table 3 shows the trend in computed flutter onset Mach number and frequency with respect to changes in the natural frequencies of the first two antisymmetric mode shapes (modes two and four of the finite element structural model) for F-16 configuration number one. Shown in the first row of Table 3 are the computed 2 of 8
Mode 2 Mode 4 Flutter Onset f 2 (Hz) f 4 (Hz) Mach f (Hz) Baseline 8.168 8.672 0.9012 8.421 Modified 1 8.168 8.622 0.8613 8.398 Modified 2 8.168 8.572 0.8122 8.377 Modified 3 8.168 8.472 0.7032 8.331 Modified 4 7.868 8.372 0.8854 8.123 Modified 5 7.868 8.322 0.8427 8.102 Table 3. Sensitivity of Computed F-16 Flutter Onset Mach Number and Frequency to Structural Natural Frequencies at 2000 feet Altitude and a Mean Angle-of-Attack of ᾱ 0 = 1.5 degrees. flutter onset results for the baseline (nominal) computational structural and fluid dynamic models for the F-16 configuration. The second and third columns list the natural frequencies of the first two antisymmetric mode shapes (f 2 and f 4 ). Columns four and five show the computed flutter onset Mach number and frequency, respectively, in this case for an altitude of 2000 feet and a wing mean angle-of-attack of ᾱ 0 = 1.5 degrees. For the nominal configuration computational model, this results in a flutter onset Mach number of M = 0.9012 and a flutter onset frequency of f = 8.421 Hz. Flight test experiments indicate that flutter onset occurs at a Mach number of approximately M = 0.85 with a frequency of approximately f = 8.1 Hz. Here we investigate the sensitivity of the flutter onset solution to apparently small changes in the structural model, i.e. sensitivities to the structural natural frequencies. Rows two through six of Table 3 show the computed flutter onset conditions for five different modifications of the structural frequencies. The first three modifications are reductions in the natural frequency of the second antisymmetric mode shape (mode four). i.e. 0.05, 0.10, and 0.20 Hz, respectively. As can be seen, the flutter onset Mach number changes significantly as a result of these modifications to the mode four natural frequency. Each 0.05 Hz reduction in the mode four natural frequency causes approximately a 0.05 reduction in the flutter onset Mach number. We believe this is an important finding and an indication that an extremely accurate structural model is necessary in order to obtain computed flutter onset and LCO results which correlate well with flight test. This may also explain, at least in part, the reported sensitivity from one aircraft to another in regards to flutter onset and LCO response. It can also be seen that the flutter onset frequency is roughly the arithmetic mean of the natural frequencies of modes two and four. For the first three modifications to the structural natural frequency of mode four, it can be seen that the flutter onset frequency is still somewhat high compared to the flight test value of approximately f = 8.1 Hz. Thus as a fourth modification, the natural frequencies of both modes two and four are reduced by 0.3 Hz. This results in a flutter onset natural frequency of f = 8.123 Hz, which is much closer to flight test, however the flutter onset Mach number of M = 0.8854 is still somewhat high compared to flight test. So as a fifth modification, we reduce the natural frequency of mode two by 0.3 Hz and the natural frequency of mode four by 0.35 Hz. This results in a flutter onset frequency of f = 8.102 Hz and a flutter onset Mach number of M = 0.8427, both quantities are now very close to the flight test values. A. Flutter Onset Altitude Figure 1 shows the computed flutter onset altitudes for the three different F-16 fighter configurations. There is much scatter in the flight test data in regards to flutter onset altitude versus Mach number, and thus clear flutter onset conditions are difficult to discern. However, it can be noted from available flight test data that configuration number one has the tendency experience flutter onset in the Mach number range of 0.75 M 0.85 for altitudes of 2000, 5000, and 10000 feet, with there being a general trend for flutter onset to occur at a slightly higher Mach number with increasing altitude. Also for altitudes of 2000, 5000, and 10000 feet, configuration number two has the tendency to experience flutter onset in the Mach number range of 0.70 M 0.80, and configuration number three has the tendency to experience flutter onset in the Mach number range of 0.50 M 0.70. As can be seen from Fig. 1, the computed flutter onset altitudes for configurations two and three are much lower than observed in flight test. We believe one 3 of 8
Flutter Onset Altitude, h x 10 3 (feet) 20 10 0 10 20 30 40 50 60 70 80 90 Configuration #1 Configuration #3 Configuration #2 100 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Figure 1. Computed F-16 Fighter Flutter Onset Altitude. of the main reasons for the discrepancies is due to the fact that geometric details of the wingtip launcher and missile are absent in the CFD portion of our aeroelastic model. In fact, Fig. 2 shows the computed flutter onset altitudes for F-16 fighter configuration number two with a six inch extension of the wing tip to approximately account for the presence of the wingtip launcher and missile. Notice the dramatic change in flutter onset altitude. Thus the details of the wingtip geometry appear to be very important for predicting flutter onset. Figure 3 shows the computed flutter onset altitudes for configurations one and three for three different angles-of-attack. As can be seen, computed flutter onset Mach number at fixed altitude can change by nearly 0.1 Mach for only a single degree of mean angle-of-attack change. Thus accurately accounting for mean angle-of-attack, say by solving the six degree-of-freedom (DOF) aircraft trimmed flight dynamic equations, may be important for precise flutter onset results. Finally, Fig. 4 shows the computed flutter onset altitudes for configurations one and three for two different alterations in the first antisymmetric twisting structural mode shape frequency. As can be seen, computed flutter onset Mach number at fixed altitude can also change by nearly 0.1 Mach for only a 0.10 Hz change in structural natural frequencies. Thus an accurate structural model may important for precise flutter onset computations. Furthermore, this may explain the wide variation in flutter onset behavior among various F-16s with identical weapons and stores configurations. B. Limit Cycle Oscillations Figure 5 shows the computed and flight test LCO response of the F-16 forward wingtip launcher accelerometer versus Mach number for an altitude of 2000 feet and a mean angle-of-attack of ᾱ 0 = 1.5 degrees. In this case, we consider the baseline configuration, and the modification number five configuration, as per Table 3, where we have reduced the natural frequency of mode two by 0.3 Hz and mode four by 0.35 Hz. As can be seen, with these very small changes in the natural frequencies of the structural modes, the computational results for the limit cycle oscillation response are in much better agreement with the flight test data. Both the flutter/lco onset Mach number and subsequent LCO response amplitude for increasing Mach numbers 4 of 8
Flutter Onset Altitude, h x 10 3 (feet) 20 10 0 10 20 30 40 50 60 70 80 90 Configuration #2 with Tip Extension Configuration #2 100 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Figure 2. Computed F-16 Fighter Flutter Onset Altitude. 20 Flutter Onset Altitude, h x 10 3 (feet) 10 0 10 20 Configuration #1 α 0 =2.0 o, 1.5 o, 1.0 o Configuration #3 30 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Figure 3. Computed F-16 Fighter Flutter Onset Altitude Angle of Attack Effect. 5 of 8
20 Flutter Onset Altitude, h x 10 3 (feet) 10 0 10 20 f 1at = 0.10 Hz f 1at = 0.05 Hz Configuration #1 f 1at = 0.10 Hz f 1at = 0.05 Hz Configuration #3 30 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Figure 4. Computed F-16 Fighter Flutter Onset Altitude Structural Frequency Effect. 3.5 3.0 Wing Tip Launcher LCO Response, g 2.5 2.0 1.5 1.0 Flight Test Modified Structrual Frequencies Nominal Structrual Frequencies 0.5 0.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 Figure 5. Computed and Flight Test F-16 Forward Wingtip Launcher Accelerometer LCO Response Level Versus Mach Number for an Altitude of 2000 feet and a Mean Angle-of-Attack of ᾱ 0 = 1.5 degrees. 6 of 8
correlate better with the flight test data than for the baseline computational configuration. Amplitude Phase (deg) Mach g Response ξ 4 /ξ 2 ξ 4 /ξ 2 ξ 4 /ξ 2 0.901 0.0274 (0.400,-4.30) 4.31-84.7 0.903 0.534 (0.418,-4.18) 4.20-84.3 0.915 1.15 (0.702,-3.57) 3.64-78.9 0.946 1.41 (0.585,-2.95) 3.01-78.8 0.960 1.54 (0.470,-2.41) 2.46-79.0 0.994 1.58 (0.375,-1.98) 2.01-79.3 1.052 1.73 (0.336,-1.82) 1.85-79.5 Table 4. LCO Response, 2000 Feet Case, Nominal Structural Frequencies. Amplitude Phase (deg) Mach g Response ξ 4 /ξ 2 ξ 4 /ξ 2 ξ 4 /ξ 2 0.843 0.0347 (1.536,-5.72) 5.93-75.0 0.845 0.690 (1.483,-5.70) 5.89-75.4 0.863 1.58 (0.978,-5.31) 5.40-79.6 0.886 2.00 (0.315,-4.54) 4.55-86.0 0.904 2.07 (0.099,-3.53) 3.53-88.4 0.917 2.14 (0.363,-2.92) 2.94-82.9 0.943 2.20 (0.260,-2.51) 2.52-84.1 0.965 2.24 (0.078,-1.91) 1.91-87.7 1.010 2.44 (0.050,-1.66) 1.66-88.3 Table 5. LCO Response, 2000 Feet Case, Alternate Structural Frequencies. Tables 4 and 5 show the computed g response levels for the results shown in Fig. 5 along with the amplitude and phase relation between the first antisymmetric twisting mode and first antisymmetric bending mode. Note how the phase angle is nearly -90 degrees. This means the node line of the wing will be moving substantially during a cycle of wing motion at flutter onset and for the finite amplitude LCO conditions. IV. Conclusions F-16 flutter onset and LCO response is very sensitive to model input parameters. Wing tip geometry, as well as small changes in angle-of-attack and structural mode natural frequencies can greatly influence the aeroelastic solutions. Computational models of ever increasing fidelity appear to be required for precise flutter onset and LCO response computations. V. Acknowledgments This work is being supported by AFOSR grant, Theoretical Predictions of Limit Cycle Oscillations for Flight Flutter Testing. The program manager is Dr. Neal Glassman and the key technical point of contact in the SEEK EAGLE office at Eglin AFB is Dr. Charles Denegri. References 1 Thomas, J. P., Dowell, E. H., Hall, K. C., and Charles M. Denegri, J., Further Invesitgation of Modeling Limit Cycle Oscillation Behavior of the F-16 Fighter Using a Harmonic Balance Approach, AIAA Paper 2005-1917, Presented at the 46th 7 of 8
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials (SDM) Conference, Austin, TX. 2 Thomas, J. P., Dowell, E. H., Hall, K. C., and Charles M. Denegri, J., Modeling Limit Cycle Oscillation Behavior of the F-16 Fighter Using a Harmonic Balance Approach, AIAA Paper 2004-1696, Presented at the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials (SDM) Conference, Palm Springs, CA. 3 Hall, K. C., Thomas, J. P., and Clark, W. S., Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique, AIAA Journal, Vol. 40, No. 5, May 2002, pp. 879 886. 4 Thomas, J. P., Dowell, E. H., and Hall, K. C., Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit Cycle Oscillations, AIAA Journal, Vol. 40, No. 4, April 2002, pp. 638 646. 5 Thomas, J. P., Dowell, E. H., and Hall, K. C., Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach, AIAA Paper 2002-1414. 6 Thomas, J. P., Dowell, E. H., and Hall, K. C., A Harmonic Balance Approach for Modeling Three-Dimensional Nonlinear Unsteady Aerodynamics and Aeroelasticity, ASME Paper IMECE-2002-32532. 7 Charles M. Denegri, J. and Dubben, J. A., F-16 Limit Cycle Oscillation Analysis Using Transonic Small-disturbance Theory, AIAA Paper 2005-2296, Presented at the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials (SDM) Conference, Austin, TX. 8 Charles M. Denegri, J. and Dubben, J. A., In-Flight Wing Deformation Characteristics During Limit Cycle Oscillations, AIAA Paper 2003-1426. 9 Charles M. Denegri, J., Limit Cycle Oscillation Flight Test Results of a Fighter with External Stores, Journal of Aircraft, Vol. 37, No. 5, 2000, pp. 761 769. 10 Parker, G. H. and Maple, R. C., The Role of Viscosity in Store-Induced Limit-Cycle Oscillation, AIAA Paper 2005-1916, Presented at the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials (SDM) Conference, Austin, TX. 11 Lieu, T. and Farhat, C., Adaptation of Pod-Based Aeroelastic ROMs for Varying Mach Number and Angle of Attack: Application to a Complete F-16 Configuration, AIAA Paper 2005-7666, U.S. 2005 Air Force T&E Days, Nashville, TN. 12 Prananta, B. B., Kok, J. C., Spekreijse, S. P., Hounjet, M. H. L., and Meijer, J. J., Simulation of Limit Cycle Oscillation of Fighter Aircraft at Moderate Angle of Attack, Tech. Rep. NRL-TP-2003-526, National Aerospace Laboratory NRL, 2003. 8 of 8