A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies ⎡ ⎢⎢⎢ ⎢⎣ Z (1) ii 0 0 Z (1) ib L(1) b Z (1) bi 0 −B (1) T b Z (1) bb L(1) b 0 Z (2) ii 0 Z (2) ib L(2) b 0 Z (2) bi −B (2) T b Z (2) bb L(2) b ⎤ ⎥⎥⎥ ⎥⎦ ⎡ ⎢⎢ ⎣ u (1) i u (2) i λ uγ ⎤ ⎥⎥ ⎦ =⎡ ⎢⎢⎢ ⎣ f (1) i f (1) b f (2) i f (2) b ⎤ ⎥⎥⎥ ⎦ , (9) where three lines of zeros have dropped out of the equation, resulting in a square systemof equations. As can be seen this expression for the assembled systemis non symmetric, therefore in practice this formof the assembled equations is often not very useful. Pre-multiplication with this transformation matrix does solve this, as it eliminates the Lagrange multipliers and results in the primal assembled systemobtained in (8). The reason we still show this type of assembly is that a similar transformation is used in the case of mixed assembly. Note that the name of this type of assembly refers to the way the assembled systemcan be solved, namely using the Gauss-Seidel method. This leads to so-called Dirichlet-Neumann iterations known fromdomaindecomposition theory [27]. 2 1 ` wdD K ` w1D K ` 2 1 ` 7ZAR $ Z $Bt .wdD .w1D .wdD .w1D Fig. 4: Simplification of the three-field formulation for stiffness assembly using to Dirichlet-Neumann assembly. 2.2 Receptance Assembly In this section we will address the assembly of two subsystems expressed in terms of flexibility. These substructure models can for example result fromdirect measurements or a modal synthesis froman FEmodel or measurement. For such systems, the equations of motion are as follows: u(s)(ω)=Y(s)(ω) f(s)(ω)+g(s)(ω) (10) The true degrees of freedomfor this problemare not the displacements u, but the unknown coupling forces g. Hence we can write the equations of motion as: Y(s)g(s) =u(s) −Y(s)f(s) =u(s) −p(s) Here the vector pcan be interpreted as a displacement excitation vector resulting fromthe externally applied forces. However, since by definitiong is only non-zero at the boundary DoF, the problemreduces to the interface DoF: Y (s) bb g (s) b =u(s) b −p (s) b This indicates that in order to assemble receptance FRF models, only information at the interface DoF is needed. Mathematically, this is due to fact that the receptance FRF matrix is the inverse of the dynamic stiffness matrix and thereby implicitly a “condensed” matrix. In contrast to the dynamic stiffness matrix, the receptance matrix is therefore not a sparse matrix but is in general non-zero at all of its entries. Physically, this means that every single receptance FRF contains all of the structure’s dynamics and as a result, the interface receptance FRFs are sufficient for an FBS analysis. In case one is also interested in the responses at internal points in the structure after assembly, these responses can be reconstructed once the interface behavior is computed. Based on the above equation we can now set up the Lagrangian of the receptance assembly problemas: L g (s) b ,u(s) b ,uγ =∑ s 1 2 g (s) T b Y (s) bb g (s) b +p (s) T b g (s) b +g (s) T b L (s) b uγ−u (s) b (11) 335
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