As before we now consider the assembly of two subsystems for the sake of illustration, as shown schematically in figure 1 (b). Taking again the variation of this expression to the free variables, one obtains the assembled equations of motion as: ⎡ ⎢⎣ Y (1) bb 0 L (1) b 0 Y (2) bb L (2) b L (1) T b L (2) T b 0 ⎤ ⎥⎦ ⎡ ⎢⎣ g (1) b g (2) b uγ ⎤ ⎥⎦= ⎡ ⎢⎣ −p (1) b −p (2) b 0 ⎤ ⎥⎦ (12) Note that although we started from a three field formulation, the above equation is actually the dual assembled form for flexibility type of substructures, as can be seen by comparing eqs. (12) and (2.2). In contrast to the dynamic stiffness assembly of the previous section, here the dual DoF are the interface connection forces, while the intermediate interface displacements constitute the unique field. The compatibility conditions for the components are stated implicitly in the first and second rows; in case some formof filtering is applied to the FRFs (e.g. modal synthesis and truncation), compatibility is required in a weakened formonly. The third row can be recognized as the equilibriumcondition. Fromthe above set of assembled equations, the only simplification to be made is to go to a true primal systemas in the previous section. As in the dual assembly of the previous section, we again choose the interface forces in the form: g (s) b =−B (s) T b λ (13) As before, this gives the following transformation: ⎡ ⎢⎣ g (1) b g (2) b uγ ⎤ ⎥⎦= ⎡ ⎢⎣ −B (1) T b 0 −B (2) T b 0 0 I ⎤ ⎥⎦ λ uγ Again, we first substitute the above transformation in eq. (12). This introduces the unique interface force field λ and eliminates the g (s) b , hence the equilibriumcondition is satisfied. Pre-multiplication then eliminates the interface displacement field (illustrated in figure 5) and gives the primal assembled systemin terms of the unknown interface force intensities λonly: B (1) b Y (1) bb B (1) T b +B (2) b Y (2) bb B (2) T b λ=− B (1) b p (1) b +B (2) b p (2) b It should be noted that this exactly the same expression as would be obtained fromthe elimination of the Lagrange multipliers in . When this expression for λ, combined with eq. (13), is then used in the subsystemmodels (10), one essentially ends up with the assembled equation as found fromtheLMFBSmethod. 7ZAR $ Z $Bt _T8Zj $}j${s $Bt 2 1 /w1D K / wdD K ` 2 1 ` 2 1 Vw1D VwdD Vw1D VwdD Vw1D VwdD Fig. 5: Simplification of the three-field formulation of the flexibility assembly case. 2.3 Mixed Assembly In this section we consider the case of mixed assembly, where one subsystemis expressed in dynamic stiffness formwhile the other is represented using receptance FRFs. This is for instance the case when directly assembling anFEmodel (possibly with frequency dependent behavior) and a measured component model. To derive the assembled equations of motion in this case, S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 336
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