264 I. Stanciulescu et al. 1 2 3 4 5 6 7 8 9 0 10 20 30 40 50 60 70 Rise [mm] Critical load [N] panel (limit point) panel (bifurcation point) orthogrid (limit point) orthogrid (bifurcation point) Fig. 24.8 Load at which loss of stability occurs at various rises for the orthogrid and constant thickness panels 24.3 Dynamic Analysis To understand the stability behavior under transient loading, it is of interest to identify the dynamic snap-through boundary. This boundary separates the small amplitude non-snap from large amplitude post-snap vibrations. Obtaining it requires a high computational cost because of the extensive parametric studies that are involved. For this section we consider again a simple curved cylindrical panel but shallower than the benchmark example (rise 5.08 mm). All other material and geometrical properties are the same as for the benchmark panel (Fig. 24.2) discussed in Sect. 24.2.2. Distributed load is applied statically and dynamically. Figure 24.9a shows the primary equilibrium path obtained, on which, except for the limit points, there are no other critical points. This indicates that no branching takes place for this system and that snap-through will always be controlled by the limit point. For the transient analysis we investigate the loading space by performing simulations with different forcing frequency and amplitude. The snap-through boundary obtained from sweeping the parameter space is shown in Fig. 24.9b. Pairs of forcing amplitude and frequencies above the snap-boundary (gray) correspond to loading cases that result in responses exhibiting snap-through, and the ones below the snap-boundary (white) to responses that do not experience snapthrough. Figure 24.10a shows the small amplitude (no snap) response for a forcing amplitude of PD213.348 Pa and a frequency of ¨D116.3018 rad/s (blue square in Fig. 24.9b) with a displacement range in the interval [ 1.5, 1] mm. Similarly, Fig. 24.10b shows the large amplitude response for a forcing amplitude of PD240.0165 Pa and the same frequency of ¨D116.3018 rad/s (red circle in Fig. 24.9b). The range of the displacement in this case is [ 12, 5] mm, or approximately seven times larger, indicating that the system is visiting the region of remote equilibria (snap-through behavior). Also note that the snap through events are persistent in this case and the system does not settle into small amplitude oscillations. Similar parametric investigations of shallow panels with varied geometrical properties and identical material properties and boundary conditions indicate that the V shape of the boundary in Fig. 24.9b is typical and that the boundary scales with the change in geometry (and the induced changes in the critical load value and natural frequencies). Work is currently in progress to identify the regimes for which such scaling is possible, which may offer opportunities for a more efficient computation of the dynamic stability boundaries. 24.4 Concluding Remarks Critical points and all postbuckling responses are identified for curved panels using a numerical procedure that requires no prior knowledge of the bifurcation modes and uses the same mesh to compute all secondary paths. The performance of orthogrid panels is then compared to that of constant thickness panels, with similar length and width, and the same volume
RkJQdWJsaXNoZXIy MTMzNzEzMQ==