CORRELATION OF ANALYTICAL AND EXPERIMENTAL ELASTIC MODE SHAPES FOR A SMALL ASPECT RATIO WING USING MODAL DESCRAMBLING

Transcription

1 CORRELATION OF ANALYTICAL AND EXPERIMENTAL ELASTIC MODE SHAPES FOR A SMALL ASPECT RATIO WING USING MODAL DESCRAMBLING James Senter 1, Evan Martin 1, Michael Rakieski 1 and Emil Suciu 1 1 Loads & Dynamics Analyst, L-3 Communications Platform Integration Division, 7500 Maehr Drive, Waco, Texas, USA SUMMARY The Modal Assurance Criterion is used to compare scrambled and descrambled experimental and calculated mode shapes for a rectangular flat steel plate of AR2. KEYWORDS GVT, DLM, Modal Descrambling, Elastic Streamwise Camber Deformation, MAC

2 1: INTRODUCTION The subject matter discussed in this paper is related to aircraft flutter analyses, in particular the process of improving the analytical structural dynamic model by comparing its results to the GVT results. The GVT results are available as frequencies and mode shapes defined at specified points. The user attempts to name or identify modes of vibration in order to understand the physical problem. Some flutter analysts refer to the mode naming process as an art. It can be argued that stick model modes can be consistently identified by making use of the beam directional stiffness properties and the strain energy summation method (Reference [1]). Full finite element structural dynamic models do not easily lend themselves to such easily quantifiable processes as shown in Reference [1] for stick models, therefore naming modes still remains an art for such models. This is especially true for complicated modes such as the mode shown in Figure 1 for a transport aircraft at the Doublet Lattice Method (DLM, Reference [2]) aerodynamic surfaces. Figure 1: Scrambled Mode Shape at the Aerodynamic Surfaces of Transport Aircraft. The mode shape shown in Figure 1, whether GVT-measured or calculated is called scrambled, since it is a linear combination of 5 simpler and always the same elementary modes: pure bending, pure torsion, pure control surface rotation, pure tab rotation and pure elastic streamwise camber deformation (Reference [3]). Any measured or calculated scrambled elastic mode of vibration can be descrambled into the five elementary modes listed above. A convenient way to do this is to transfer the structural motion to the DLM aerodynamic surface and to perform the modal descrambling at this surface, see Figure 2. So far, this method has only been used to correct unsteady aerodynamic forces and moments for flutter analyses (Reference [4]).

3 Figure 2: Schematic of Descrambling Process of General Wing-Control Surface- Tab Mode Shape at Any Strip of the DLM Aerodynamic Surface. The modal descrambling process will be illustrated for a rectangular steel flat plate wing of AR2, shown in Figure 3. No control surface and tab are present. Figure 3: Flat Plate Steel Wing of AR2 to Be Analyzed and Tested. For any scrambled mode of vibration, experimental or calculated, the Modal Assurance Criterion (MAC, Reference [5]) comparison between the scrambled mode and its descrambled modal components will provide a numerical indicator which will help identify or name the mode: mostly bending or mostly torsion or mostly camber. The existence of elastic streamwise camber deformation especially as it occurs in real aircraft lifting surfaces is still debated by a dwindling number of flutter analysts. Pairs of scrambled or descrambled experimental vs. calculated mode shapes can now be compared one-to-one, visually and numerically. The Modal Assurance Criterion will be calculated for each corresponding pair of compared modes, scrambled and descrambled.

4 This paper therefore has a triple purpose: 1. To improve the understanding of any lifting surface mode of vibration by descrambling it into its simpler components and to obtain numerical values of the degree of correlation between the scrambled mode and its descrambled components; numerical guidance to aid in naming modes will result. 2. To improve the process of correlating the analytical and experimental modes of vibration of a lifting surface structure by obtaining numerical values of the correlation between the corresponding pairs of scrambled and descrambled experimental and calculated modes. 3. To somewhat curtail the poetic license we can take in identifying and naming modes of vibration. A better understanding of the mode shape (and implicitly of the structure of the test article) will be achieved. 2: DETAILS OF THE GROUND VIBRATION TEST (GVT) AND ANALYTICAL MODEL a. Details of the GVT A modal survey of a 12 inch by 24 inch steel plate was performed using impact excitation. The plate was 3/16 inch thick and weighed 14.6 lbs. To properly separate the elastic modes from the rigid body modes (to obtain nearly free-free boundary conditions), the steel plate was placed on top of several layers of bubble wrap and foam padding. For testing, the plate was divided into 7 chordwise and 11 spanwise test nodes for a total of 77 test nodes. Using the roving accelerometer method of modal testing, 4 separate tests were performed with the accelerometers moved to a different set of nodes for each test as shown in Figure 4. Considerable care was exercised to avoid accepting FRFs when double impact hammer strikes occurred. The results were then combined into a single data set and modes were extracted from the resulting Frequency Response Functions (FRF). To perform the test, Smart Office version V4.1 B3096 CD8.01 by MP International was used to record measurement data and extract the modes of the plate. Six locations were impacted for each test: at 3 corners, 1 location along both the short and long edges, and in the center of the plate. These locations were chosen to ensure that all flexible modes of the plate were adequately excited. The 1st through 3rd bending, torsion and camber modes of the plate were successfully measured. The highest rigid body mode was measured to be 14 Hz and the first bending mode frequency was 65 Hz. With this significant separation, the

5 rigid and elastic modes do not interfere and provides better results for the purposes of this effort. Figure 5 shows the FRFs for two different pairs of excitation and response locations. In Figure 6, a sample of experimental mode shapes as displayed with Smart Office software is shown. Note the presence of elastic streamwise camber deformation visible in the scrambled bending and torsion modes. This is due to the high degree of mass and stiffness coupling. The verification of the modes identified in the modal survey test is discussed in Section 3. Test #1 Test #2 Test #3 Test #4 Figure 4: Modal Survey of Steel Plate. FRF - Reference 6Z, Response 6Z FRF - Reference 39Z, Response 39Z Figure 5: Frequency Response Functions at Two Different Pairs of Excitation/Response Points.

6 Experimental 1 st Bending, 2-View Experimental 1 st Torsion, 2-View Experimental 1 st Camber, 2-View Figure 6: A Sample of Experimental Mode Shapes at the GVT Accelerometer Locations. b. Details of the Analytical Model Three different plate FEMs of varying plate finite element (CQUAD) densities were created in MSC.PATRAN and analyzed with MSC Nastran SOL103 to study the convergence of the normal modes solution. A test mesh consisting of a 6 x10 element array was created with node locations coinciding with the accelerometer placements on the test article. A coarser mesh and a finer mesh were also created which consisted of a 4 x 6 and 12 x 20 element array, respectively. The dimensions of each FEM were identical with the tested plate, and the material properties of A36 steel were used to define the elements. The free-free boundary condition was used for the normal modes solution. Table 1 shows the modes of each FEM compared to the modes identified in the modal survey test. As the element density increased, the analytical frequencies began to converge on the test frequencies. The fine mesh FEM showed the best comparison to the test data with the percent differences of 3% or less and the analytical results obtained with the fine mesh are considered to be converged. The most dramatic convergence occurred for the camber mode. Figure 7 shows the 1 st camber mode displacement of each mesh at the structural grid points. Since the converged analytical modes compared so well to the test modes, the free-free boundary condition was considered sufficient, and no attempt was made to model the bubble wrap used to support the test article. This confirms that the rigid and elastic modes are well separated and the rigid body modes do not interfere with the elastic ones. Table 1 shows that the converged MSC Nastran structural dynamic solution with high-density mesh simulates the test results extremely closely. In Section 3 it will be shown that the visual and MAC comparison of the experimental and

7 analytical, scrambled and descrambled mode shapes is also very good. This is due to the simple geometry and homogeneous properties of the plate. As complexities are added eventually culminating in an aircraft structure, the discrepancies between measured and analytical mode shapes will be larger. Table 1: Analytical Modes Comparison to Test Data; Convergence of Analytical Solution with Structural Mesh Density. Figure 7: Analytical 1 st Camber Mode of Plate with Varying Structural Mesh Densities. c. Structural Motion Transfer to the Aerodynamic Surfaces and Modal Descrambling The experimental mode shapes were obtained at 77 grid points, as previously shown in Figure 6. An aerodynamic surface box arrangement of NSPAN10

8 and NCHORD6 for the steel flat plate wing was chosen so that the corners of the boxes coincide with the accelerometer locations. The measured mode shape displacements are then inputted into MSC Nastran SOL145 at the specified corner grid points locations using the DMIG input format for all the modes of interest. This structural motion is then transferred to the DLM aerodynamic surface through the surface splining feature of SOL145. The mode shapes at the aerodynamic set are then available for descrambling and SOL145 is then stopped, since it is not expected that the flutter results for this steel plate will bring any further enlightenment. After descrambling the modes using a FORTRAN program, the descrambled displacements at the centers of the aerodynamic boxes are plotted using MATLAB. A procedure to display and animate the descrambled mode shapes using MSC PATRAN is in work. The analytical mode shapes obtained with the fine structural mesh are also transferred to the same aerodynamic surface with NSPAN10 and NCHORD6 using NASTRAN SOL145. The calculated mode shapes at the centers of the aerodynamic boxes are now available for descrambling and for graphical display at exactly the same locations as the measured modes. 3: COMPARISON OF THE GVT AND ANALYTICAL MODE SHAPES Figures 8 through 12 show the experimental and analytical scrambled and descrambled plots of the first 5 mode shapes of the steel plate of AR2 sideby-side for easy visual comparison. The viewing angle and magnification factors are the same for each corresponding pair of scrambled or descrambled modes. These comparisons are self-explanatory. The observation must be made that the analytical results show perfect symmetry about 2 axes, while the descrambled experimental results show the likely unavoidable extremely small non-symmetries. These non-symmetries appear in the plots, but are very, very small, as shown by the maximum normalized displacement accompanying each plot.

9 Experimental 1 st Bending 65.9 Hz Analytical 1 st Bending 66.2 Hz Scrambled Mode Bending Torsion Camber Figure 8: Modes. Scrambled and Experimental and Analytical 1 st Bending

10 Experimental 1 st Torsion 81.1 Hz Analytical 1 st Torsion 82.9 Hz Scrambled Mode Bending Torsion Camber Figure 9: Mode. Scrambled and Experimental and Analytical 1 st Torsion

11 Experimental 2 nd Torsion Hz Analytical 2 nd Torsion Hz Scrambled Mode Bending Torsion Camber Figure 10: Scrambled and Experimental and Analytical 2 nd Torsion Mode.

12 Experimental 2 nd Bending Hz Analytical 2 nd Bending Hz Scrambled Mode Bending Torsion Camber Figure 11: Scrambled and Experimental and Analytical 2 nd Bending Mode.

13 Experimental 1 st Camber Hz Analytical 1 st Camber Hz Scrambled Mode Bending Torsion Camber Figure 12: Scrambled and Experimental and Analytical 1st Camber Mode.

14 Mode Number Mode Number Scrambled Mode Number Scrambled Mode Number CORRELATION OF ANALYTICAL AND EXPERIMENTAL ELASTIC The MAC was also calculated to compare the scrambled and descrambled modes for both the analytical and experimental mode shapes. Those are shown in Table 2 for the experimental modes and Table 3 for the analytical modes. In each case, the numbering convention is X-1 for descrambled bending, X-2 for descrambled torsion and X-3 for descrambled camber, X being the scrambled mode number. Table 2: MAC of Experimental Scrambled and Modes Table 3: MAC of Analytical Scrambled and Modes Scrambled Mode Number Scrambled Mode Number Note that the MAC comparison between the scrambled mode and its descrambled components is a good indicator of the contribution of each descrambled component to the scrambled mode. This numerical indicator can be used to complement and augment the mode-naming process. Table 4 shows the MAC comparison between corresponding pairs of scrambled and descrambled experimental and analytical mode shapes. The comparison is very good, confirming the visual agreement between these pairs of modes. All values above 0.25 were plotted in Table 4 to provide some information on commonality among experimental and analytical scrambled and descrambled modes. It should be noted that at certain points along the diagonal, points that would be expected to be 1 are not shown. This is due to the pure motions of the analytical model which may have no bending or torsion component for a given mode.

15 Analytical Modes CORRELATION OF ANALYTICAL AND EXPERIMENTAL ELASTIC Table 4: MAC of pairs of Experimental and Analytical Scrambled and Modes Experimental Modes : COMPARISON OF MODE SHAPES FOR A PLATE OF AR2 AND AR0.5 Figure 13 shows the rectangular flat plate of AR2 next to the plate of AR0.5 with coordinate systems and arrow showing wind direction and reference axes. AR 2 AR 0.5 Figure 13: Flat Plate Steel Wings of AR2 and AR0.5.

16 The plate of AR0.5 is identical in every respect with the plate of AR2 except its orientation and the aerodynamic coordinate system. Figure 14 shows the scrambled mode shapes of the two steel plates with the names assigned for each. AR 2 AR st Bending 1 st Camber 1 st Torsion 1 st Torsion 2 nd Torsion 2 nd Torsion 2 nd Bending 2 nd Camber 1 st Camber 1 st Bending Figure 14: Side-by-Side Comparison of the Scrambled Mode Shapes of Wing of AR2 and Wing of AR0.5.

17 With respect to a coordinate system fixed at a corner of the plate of AR2, the modes of vibration of the plate are invariant with the location and angular orientation of the plate in the global coordinate system. However, the steady and unsteady aerodynamic forces on the wing of AR0.5 will be very different than the aerodynamic forces for the wing of AR2. Due to the flutter analyst s need to understand the physical phenomenon, the modes are named quite differently for the plate of AR0.5 than for the plate of AR2. 5: CONCLUSIONS The structural dynamic characteristics of a flat steel plate of AR2 of uniform properties were tested using impact hammer excitation and Smart Office software and also analyzed with MSC Nastran SOL103 and SOL145. The experimental and the converged analytical frequencies and mode shapes were compared. The comparison is very good due to the simple geometry and homogeneous properties of the steel plate. The experimental and analytical structural mode shapes were interpolated to their respective aerodynamic surfaces having the same box arrangement. The scrambled modal motions at the centers of the aerodynamic boxes were then descrambled and the scrambled and descrambled mode shapes were then compared. If the MAC is calculated between a scrambled mode and its descrambled components, a numerical guide arises which helps with naming the mode, thus offering the possibility to augment the mode naming art with something more precise. The visual and MAC comparisons between corresponding pairs of scrambled and descrambled measured and analytical modes offers the possibility to gage the analytical model and calculate which descrambled component contributes the most to disagreement, if any, between experiment and analysis. The drawback of this correlation method using descrambled modes is the fact that for a real airplane, a minimum of 2 to 3 times the number of accelerometers would be required for the average lifting surface as compared to a conventional GVT in order to define the mode shape with the purpose of properly descrambling it. Until it will be economical to routinely use such large numbers of accelerometers, the practice of naming modes as an art form will continue as such (see comparison between wing of AR2 and wing of AR0.5).

18 6: ACKNOWLEDGEMENTS The authors are indebted to Mr. John Corbett, who wrote the MATLAB mode shape plotting procedure. 7: REFERENCES 1. Suciu, E., and Buck, J., Postprocessor for Automatic Mode Identification for MSC/NASTRAN Structural Dynamic Solutions with Emphasis on Aircraft Flutter Applications, presented at the 1998 MSC/NASTRAN America s Users Conference, Los Angeles, CA, October 5-8, Rodden, W.P., and Johnson, E.H., User s Guide V68, MSC.Nastran Aeroelastic Analysis, The MacNeal-Schwendler Corporation, Los Angeles, CA, Panza, J.L., and Suciu, E., A Closer Look at the Elastic Streamwise Camber Deformation of Swept and Unswept Wings, Presented at the International Forum for Aeroelasticity and Structural Dynamics 2005 as Paper No. IF-090, Munich, Germany, June 28 July 1, Suciu, E., Stathopoulos, N., Dickinson, M. and Glaser, J., The T-Tail Flutter Mechanism Revisited, Paper No. IFASD , Presented at the International Forum for Aeroelasticity and Structural Dynamics, Paris, France, June 26-30, Allemang, R.J., "The Modal Assurance Criterion (MAC): Twenty Years of Use and Abuse", Presented at IMAC-XX: Conference & Exposition on Structural Dynamics, 2002.