11 Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring 109 This finally results in the reduced overall system Ared ˙pkept +Bredpkept =0 (11.46) with Ared = T 2,u (α) CN T A(α) (α) CN 0 0 (β) CN T A(β) (β) CN 2,u and Bred = T 2,u (α) CN T B(α) (α) CN 0 0 (β) CN T B(β) (β) CN 2,u. 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) kept+n (α) r +n (β) kept+n (β) r ) and Bred ∈C(n (α) kept+n (α) r +n (β) kept+n (β) r )×(n (α) kept+n (α) r +n (β) kept+n (β) r ) This shows that the interface displacement DOFs and the interface forces are completely eliminated during assembly according to Craig and Ni. This is similar to the assembly procedure of MacNeal’s method for the undamped case. 11.3.2 Liu and Zheng’s Method [17] 11.3.2.1 Reduction Basis The second-order reduction according to Liu and Zheng (LZ) [17] represents an extension of the first-order reduction according to Craig and Ni described in the previous section. For this reason, only the differences to the procedure in Sect. 11.3.1.1 are explained below. The exact representation (no reduction) of the state-space vector z considering all eigenmodes φk is z = n ! k=1 φkpk = nkept ! k=1 φkept,kpkept,k + n ! k=nkept+1 φtrunc,kptrunc,k. (11.47) The basic idea of model order reduction is the approximation by keeping only a reduced number nkept of eigenmodes φk. Vector φkept,k in Eq. (11.47) denotes one kept eigenmodeφk andvector φtrunc,k denotes one truncated eigenmode. According to Craig and Ni’s reduction basis, the influence of the truncated modes on the dynamic behavior is statically approximated by means of the attachment modes. The extension in the second-order reduction according to Liu and Zheng refers to the additional determination of second-order attachment modes. In comparison to Craig and Ni’s first-order reduction, a dynamic part is retained with the second-order attachment modes. Thus, the inertia and damping effects of the truncated modes are considered in the following [17]. The kept eigenmodes kept used for the reduction are identical to the kept modes according to Craig and Ni’s reduction basis. Substructures Without Rigid Body Modes In the case of substructures without rigid body modes, the attachment modes according to Liu and Zheng are defined as [17] a1,LZ =Q1Fa and a2,LZ =Q2Fa (11.48) with Q1 =B−1 + kept −1 kept T kept and Q2 =−B−1 AB−1 + kept −2 kept T kept. (11.49) Utilizing the relation Ge =B−1, Eq. (11.49) can be rewritten as Q1 =Ge + kept −1 kept T kept =Gres and Q2 =−GeAGe + kept −2 kept T kept. (11.50) From Eq. (11.50), it can be seen that the first-order attachment modes a1,LZ according to Liu and Zheng are identical to the attachment modes a,CN as defined by Craig and Ni in Eq. (11.24):
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