Walter D’Ambrogio and Annalisa Fregolent It can be observed that ˆZB and ˆZAB differ in only the upper left cc block, i.e. that relative to the coupling DoFs, and they are conveniently written as: ˆZAB =⎡ ⎣ ˆZAB cc ˆZB ci ˆZB ic ˆZB ii ⎤ ⎦ ˆZB =⎡ ⎣ ˆZB cc ˆZB ci ˆZB ic ˆZB ii ⎤ ⎦ (28) Note that, when standard interface is considered, only the upper left cc block exists because i is an empty set. By looking at Eq. (44), it can be noticed that [ZB] uu must be inverted. [Z B] uu is the dynamic stiffness matrix of the residual subsystemBwith master (interface) DoFs grounded, and it is singular at its own resonant frequencies. Therefore, det([ ˆZB]) tends to infinity at the resonant frequencies of [ZB] uu: at those frequencies, det([ ˆHB]) tends to zero and [ ˆHB] is singular. Similarly, det([ ˆZAB]) tends to infinity at the resonant frequencies of the coupled structureABwith master (interface) DoFs grounded: at those frequencies, [ ˆHAB] is singular. By looking at Eq. (45), where the matrix to be inverted is a block diagonal matrix including [ZB] uu and [Z A] aa, it can be noticed that the resonant frequencies of the residual substructure B, with interface DoFs grounded [ZB] uu, are a subset of the resonant frequencies of the coupled structure ABwith interface DoFs grounded. References 1. D’Ambrogio, W., Fregolent, A.: Promises and pitfalls of decoupling procedures. In: Proceeding of 26th IMAC. Orlando (U.S.A.) (2008) 2. D’Ambrogio, W., Fregolent, A.: Decoupling procedures in the general framework of frequency based substructuring. In: Proceedings of 27th IMAC. Orlando (U.S.A.) (2009) 3. D’Ambrogio, W., Fregolent, A.: The role of interface dofs in decoupling of substructures based on the dual domain decomposition. Mechanical Systems and Signal Processing 24(7), 2035– 2048 (2010). Doi:10.1016/j.ymssp.2010.05.007, also in Proceedings of ISMA 2010, pp. 18631880, Leuven (Belgium) 4. Jetmundsen, B., Bielawa, R., Flannelly, W.: Generalised frequency domain substructure synthesis. Journal of the American Helicopter Society 33(1), 55–64 (1988) 5. de Klerk, D.: Dynamic response characterization of complex systems through operational identification and dynamic substructuring. Ph.D. thesis, TU Delft (2009) 6. de Klerk, D., Rixen, D.J., Voormeeren, S.: General framework for dynamic substructuring: History, review, and classification of techniques. AIAA Journal 46(5), 1169–1181 (2008) 7. Sjo˜vall, P., Abrahamsson, T.: Substructure system identification from coupled system test data. Mechanical Systems and Signal Processing 22(1), 15–33 (2008) 8. Vormeeren, S.N., Rixen, D.J.: A dual approach to substructure decoupling techniques. In: Proceeding of 28th IMAC. Jacksonville (U.S.A.) (2010) 9. Vormeeren, S.N., Rixen, D.J.: A family of substructure decoupling techniques based on a dual assembly approach. In: P. Sas, B. Bergen (eds.) Proceedings of ISMA 2010 - International Conference on Noise and Vibration Engineering, pp. 1955–1968. Leuven (Belgium) (2010) 76
RkJQdWJsaXNoZXIy MTMzNzEzMQ==