98 J. Korbar et al. Round Robin Structure Case Study A case study of the proposed framework was performed on the Round Robin structure, addressing the Thin & Thick Wing challenge, where the objective is to estimate the dynamics of the Frame & Thick Wing assembly by decoupling the Thin Wing from the Frame & Thin Wing assembly and coupling the Thick Wing. For brevity, the Frame, Thin Wing and Thick Wing substructures are denoted as A, TS, and B, respectively. Frame & Thin Wing and Frame & Thick Wing assemblies are respectively denoted as ATS and AB. The substructuring procedure can be conceptually outlined in the equation form: ATS−TS+B=AB. The substructures involved in this case study are shown in Fig. 1, along with the sensor placement. Both Thin and Thick Wing are mounted to the Frame at four connection points. Three triaxial accelerometers are placed in the proximity of each connection point to facilitate the application of M-VPT. A VP is located at each connection point for a total of four VPs, depicted as green spheres in Fig. 1, with six rigid IDMs per VP. The four triaxial accelerometers in the corners of TS and B were also included in the substructuring procedure, therefore, the extended interface formulation was considered. (a) − (b) + (c) = (d) Fig. 1 Substructures used in the Thin & Thick Wing challenge: (a) ATS, (b) TS, (c) B, (d) AB. To clarify the substructuring procedure, the matrices W, Bm, and generalized coordinates ηare written in a block form as: W="Φ (TS) b + 0 0 Φ(TS) b +#, Bm="Φ(ATS) b −Φ(TS) b 0 0 Φ (TS) b −Φ(B) b # , η = η(ATS) η(TS) η(B) . (16) The proposed mode selection framework is performed by removing a single row in Φ (TS) b and the corresponding rows in Φ(ATS) b and Φ(B) b . The number of modes taken into consideration for modal substructuring for substructures ATS, TS, and B, are denoted as M(ATS), M(TS), and M(B), respectively . The results of the mode selection framework are shown in Fig. 2, where two distinct subsets of modes S1 and S2 are tested, which are defined as follows: S1 = M(ATS) =25 M(TS) =22 M(B) =22 and S2 = M(ATS) =30 M(TS) =18 M(B) =18 . (17) Modal substructuring is performed on each subset of modes with two distinct interface representations, namely the equivalent multi-point connection (EMPC), where the constraints are defined directly on the physical set of coordinates, as well as the M-VPT representation, where the constraints refer to the VP DoFs. It can be seen that the more stable eigenfrequency results are generally in line with the reference eigenfrequencies, which are not known in practice. The stability of the eigenfrequencies can therefore be used as a guide whether the subset of modes used within modal substructuring is appropriate to estimate the assembly dynamics in the frequency range of interest. The results also allow to compare the EMPC and the M-VPT approaches for describing the interface DoFs. One significant advantage of M-VPT over the EMPC is the improved accuracy of rigid-body dynamics. All substructures in this study have free boundary conditions, therefore, each substructure has six rigid-body vibration modes. The M-VPT approach accurately predicts six rigid-body modes for the coupled structure, while the EMPC approach typically predicts only four
RkJQdWJsaXNoZXIy MTMzNzEzMQ==