26 A. Singh et al. 4.2 Quasi-Static Modal Analysis Following the preload step, the next step in the QSMA process is to compute the linearized modes of vibration about the preloaded state. The modal analysis step was accomplished using the Sierra Structural Dynamic (Sierra/SD) [29] finite element code, which is able to import the deformation and stress state directly from the Sierra Solid Mechanics preload simulation. Within Sierra/SD, tied multipoint constraints can be defined based on the normal contact traction magnitudes calculated from the preload step. It should be noted that only fully tied interfaces were used within the modal analysis, i.e., the preloaded joints were constrained in both normal and tangential directions. For the linearized modal analysis step, the chosen normal contact pressure cutoff value was 0 psi. Figure 9 depicts the mode of interest for the morphing wing structure, referred to as the first stiff direction bending mode, with a natural frequency of 166 Hz. The final step of the QSMA process is to apply a body force to the structure proportional to the shape of the bending mode in Fig. 9. The modal force was appliedto the structure such that the tip had a positive displacement, opposite of the tip preload step. The solver was allowed to settle for 50 additional iterations prior to applying the modal force to ensure that the model was in the equilibrium state. After the modal filter was applied to the displacement fields, the nonlinear frequency and damping curves were computed and shown in Fig. 10. The modal force amplitude ranges were chosen such that the linearized model had a tip displacement in a prescribed range. In the results shown here, two separate simulations are shown for a tip displacement range of 0.005– 0.05 inch, and 0.01–0.1 inch. The frequency and damping curves are plotted against the tip displacement (peak) to quantify Fig. 9 Stiff bending mode of the morphing wing structure Fig. 10 Frequency (left) and damping (right) as a function of displacement amplitude estimated using QSMA
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