11 Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring 113 From Eq. (11.70), the transformation matrix (s) 2 can be derived, which expresses the generalized DOFs p (s) a depending on the physical DOFs z (s) b [32]: p (s) kept p (s) a = I (s) kept 0 (s) kept − (s) a,b −1 (s) kept,b (s) a,b −1 - .+ , (s) 2 p (s) kept z (s) b . (11.71) The final reduction follows from multiplying reduction matrix (s) KC and transformation matrix (s) 2 : z(s) = z (s) i z (s) b ≈ (s) KC (s) 2 p (s) kept z (s) b = (s) kept,i − (s) a,i (s) a,b −1 (s) kept,b (s) a,i (s) a,b −1 0(s) b I (s) b p (s) kept z (s) b . (11.72) It can be seen that only the internal DOFs are transformed into generalized coordinates and that the interface displacement and velocity DOFs are present in physical coordinates. Thus, it is possible to assemble the substructures in a primal way after the transformation and superelements are created. For the assembled reduced matrices, the following dimension properties apply: Ared ∈C(n (α) kept+n (α) r +n (β) kept+n (β) r +n (α) a )×(n (α) kept+n (α) r +n (β) kept+n (β) r +n (α) a ) and Bred ∈C(n (α) kept+n (α) r +n (β) kept+n (β) r +n (α) a )×(n (α) kept+n (α) r +n (β) kept+n (β) r +n (α) a ) . By keeping the boundary displacement and velocity DOFs, the size of the assembled system increases with the number of attachment modes n (α) a and n (β) a compared to Craig and Ni’s method and Liu and Zheng’s method. 11.4 Improvements of Free Interface Substructuring Approaches 11.4.1 Third and Higher-Order Reduction Interface Flexibility Representation The determination of higher-order attachment modes as in the reduction according to Liu and Zheng can be extended arbitrarily. Thus, the influence of the truncated modes is represented more accurately. However, the size of the reduced system also increases. Substructures Without Rigid Body Modes In the case of substructures without rigid body modes, the attachment modes of higher order are generated by a,k =QkFa (11.73) with Qk =(−1) k−1 Ge(AGe) k−1 + kept −k kept T kept for k =1, . . . , h. Here, hcorresponds to the highest order of the reduction. Substructures with Rigid Body Modes For substructures with rigid body modes, the attachment modes of higher order are generated by a,k =QkFa (11.74)
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