Three-Dimensional Aeroelastic Solutions via the Nonlinear Frequency Domain Method

Introduction It has previously been shown that aeroelastic solutions may be obtained efficiently using a frequency-domain flow solver; the approach was demonstrated in two and three dimensions using the Harmonic Balance... [ view full abstract ]

Introduction

It has previously been shown that aeroelastic solutions may be obtained efficiently using a frequency-domain flow solver; the approach was demonstrated in two and three dimensions using the Harmonic Balance technique [1-3], and in two dimensions using the Nonlinear Frequency Domain (NLFD) [4] and Time-Spectral [5] methods. The goal of the present work was to extend the NLFD approach for computing aeroelastic solutions to three dimensional inviscid flows. Specifically, the intent of this work was to provide a framework for the computation of the flutter boundary of aircraft components.

Governing Equations: Flow and Structural Solvers

For the sake of brevity, the governing equations of the flow and structural solvers are intentionally omitted in this abstract. For more information, the reader is referred to the work of Kachra and Nadarajah [4], since the equations and procedures utilized here are very similar, except that they are formulated in three dimensions. In the present work, however, the structural solver is based on a one-dimensional finite element model, comprising six degrees of freedom per node.

Flutter Speed Search Procedure

In order to determine the flutter boundary of a structure, the flutter speed has to be computed at various Mach numbers. Unlike typical time-accurate solvers, frequency-domain solvers require an oscillation frequency a priori. Therefore, an iterative approach for computing the reduced frequency (ωr) and the flutter speed index (Vf) has been devised. To obtain limit-cycle oscillations for a specific Mach number (M), the reduced frequency and speed index are converged using Newton's method, driven by the satisfaction of two criteria: between two successive periods, the motion amplitude and phase must not change. To approximate the flutter speed, which is usually provided by a linear analysis, the amplitude of the limit-cycle oscillations are chosen to be small in order to limit nonlinearities in the flow.

Preliminary Results

Aeroelastic results were obtained for the AGARD I.-wing 445.6 weakened model 3 [6]. The 32 beam elements of the structural model were placed along the elastic axis of the wing, and their mechanical properties were determined based on the data provided by Yates [6]. Figure 1 shows the employed computational mesh and structural model. Table 1 reports the first four natural frequencies calculated using the structural solver, and comparison with previous work is provided [6,7]. Figure 2 shows the computed mode shapes of the first four modes compared to those of Yates [6]; a good shape comparison is obtained, considering the simplicity of the structural model. For the aeroelastic computations, the initial reduced frequency and speed index were chosen as those observed in the experimental results, and were subsequently converged to their final values using the aforementioned technique. Lift coefficient time history and pressure contours at Mach number 0.901 are shown in figure 3 and 4. Preliminary results for the flutter boundary are shown in figure 5, and comparison with previous work is provided. Updated and more complete results are to be included in the final paper, including the consideration of other three-dimensional aeroelastic cases. A two-dimensional plate bending structural solver is also under development and results will be included in the final paper.

References

[1] J. P. Thomas, E. H. Dowell, and K. C. Hall. Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations. AIAA Journal, 40(4):638-646, April 2002.

[2] J. P. Thomas, E. H. Dowell, and K. C. Hall. A Harmonic Balance Approach for Modeling Three-Dimensional Nonlinear Unsteady Aerodynamics and Aeroelasticity. In ASME 2002 International Mechanical Engineering Conference and Exposition, number IMECE-2002-32532, New Orleans, LA, November 2002.

[3] J. P. Thomas, E. H. Dowell, and K. C. Hall. Three-Dimensional Transonic Aeroelasticity Using Proper Orthogonal Decomposition-Based Reduced-Order Models. Journal of Aircraft, 40(3):544-551, May-June 2003.

[4] F. Kachra and S. K. Nadarajah. Aeroelastic Solutions using the Nonlinear Frequency-Domain Method. AIAA Journal, 46(9):2202-2210, September 2008.

[5] N. L. Mundis and D. J. Mavriplis. An Efficient Flexible GMRES Solver for the Fully-coupled Time-spectral Aeroelastic System. In 52nd AIAA Aerospace Sciences Meeting, number 2014-1427, National Harbor, MD, January 2014.

[6] J. E. Carson Yates. AGARD Standard Aeroelastic Configurations for Dynamic Response. Candidate Configuration I.-Wing 445.6. Technical Memorandum 100492, NASA, 1987.

[7] R. M. Kolonay. Unsteady Aeroelastic Optimization in the Transonic Regime. PhD thesis, Purdue University, 1996.