130 F. M. Gruber et al. Table 11.4 Modes used for reduction and resulting size of the reduced assembled system Craig/Ni’s method Liu/Zheng’s method Substructure α β γ δ α β γ δ Kept eigenmodes 15 (17) 10 (15) 13 (15) 12 (15) 15 (17) 10 (15) 13 (15) 12 (15) Attachment modes 2 4 4 2 4 8 8 4 Rigid body modes 0 2 4 3 0 2 4 3 DOFs reduced system 59 (71) 59 (71) Liu/Zheng’s reduction basis with de Kraker/van Campen’s method primal assembly Third order reduction Substructure α β γ δ α β γ δ α β γ δ Kept eigenmodes 15 10 13 12 15 10 13 12 15 10 13 12 Attachment modes 4 8 8 4 4 8 8 4 6 12 12 6 Rigid body modes 0 2 4 3 0 2 4 3 0 2 4 3 DOFs reduced system 71 71 71 The numbers in brackets are used for a comparison based on the identical size of the reduced system as shown in Fig. 11.15 eigenvalues. This also holds in Fig. 11.15, where the number of kept eigenmodes for Craig and Ni’s method and Liu and Zheng’s method is increased. Liu and Zheng’s reduction basis with primal assembly as well as the third order reduction lead in both Figs. 11.14 and 11.15 to lower relative errors compared to Craig and Ni’s method, de Kraker and van Campen’s method and Liu and Zheng’s method. Overall, the third order reduction achieves the most accurate approximation. 11.6 Conclusions In this paper, a derivation and comparison of three existing free interface substructuring methods for viscously damped systems were provided and two new approximation approaches were proposed. The three existing investigated methods are Craig and Ni’s method, Liu and Zheng’s method, and de Kraker and van Campen’s method. Liu and Zheng’s method can be considered as a second-order extension of Craig and Ni’s method. We extended Liu and Zheng’s method to a third-order approximation and generalized it further to arbitrarily higher orders. Moreover, a new method combining the reduction basis of Liu and Zheng’s approach with the primal assembly procedure applied by de Kraker and van Campen was derived. The new method combining Liu and Zheng’s reduction basis with primal assembly and the third-order method give the best results and are recommended for the approximation of arbitrarily viscous damped substructured systems. The third-order method in particular shows the very best results, but increases the size of the reduced system compared to Craig and Ni’s and Liu and Zheng’s method. Liu and Zheng’s reduction basis with primal assembly outperforms de Kraker and van Campen’s method, while both methods generate the same size of the reduced system. The five methods were applied to three different beam structures. In the future, we want to apply the method to bigger problems with a larger number of DOFs. The examples used in this paper are very illustrative, allow for comparison to results in the literature and demonstrate all critical points for the application of the suggested methodology. However, they are too small for a meaningful comparison in terms of computational time. For this purpose, it is necessary to consider additional numerical examples to further examine the performance of the proposed methods. References 1. Craig, R.R., Kurdila, A.J.: Fundamentals of Structural Dynamics. Wiley, New York (2006) 2. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968). https://doi.org/10.2514/3. 4741 3. MacNeal, R.H.: A hybrid method of component mode synthesis. Comput. Struct. 1(4), 581–601 (1971). https://doi.org/10.1016/00457949(71)90031-9 4. Rubin, S.: Improved component-mode representation for structural dynamic analysis. AIAA J. 13(8), 995–1006 (1975). https://doi.org/10. 2514/3.60497 5. Rixen, D.J.: A dual Craig-Bampton method for dynamic substructuring. J. Comput. Appl. Math. 168(1–2), 383–391 (2004). https://doi.org/10. 1016/j.cam.2003.12.014
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