110 F. M. Gruber et al. a1,LZ = a,CN (11.51) The difference between this and the reduction according to Craig and Ni is the extension of the first-order attachment modes by the second-order attachment modes a2,LZ of Eq. (11.48). For the dimension of the attachment modes a,LZ = a1,LZ a2,LZ according to Liu and Zheng, a1,LZ ∈ Cn×nb,u, a2,LZ ∈ Cn×nb,u and thus a,LZ ∈ Cn×2nb,u apply. The reduction of substructures without rigid body modes results in zLZ ≈ kept a1,LZ a2,LZ - .+ , LZ ⎡ ⎣ pkept pa ˙pa ⎤ ⎦ . (11.52) Substructures with Rigid Body Modes In the case of substructures with rigid body modes, the attachment modes are defined as [17] a1,LZ =GresFa = a,CN a2,LZ =−GeAGeFa. (11.53) In Eq. (11.53), the matrices Gres andGe are identical to the definition according to Craig and Ni in Eq. (11.33). In comparison to the attachment modes of substructures without rigid body modes, it can be seen that for substructures with rigid body modes, the dynamic term kept −2 kept T kept is not considered for the computation of second-order attachment modes a2,LZ by Liu and Zheng [17]. This results in the reduction of substructures with rigid body modes according to Liu and Zheng as zLZ ≈ r kept a1,LZ a2,LZ - .+ , LZ ⎡ ⎢⎢ ⎣ pr pkept pa ˙pa ⎤ ⎥⎥ ⎦ . (11.54) 11.3.2.2 Assembly Procedure (Elimination of Interface DOFs) For assembly of the reduced substructures according to Liu and Zheng, the interface displacement compatibility (11.36) and the interface force equilibrium (11.37) are extended to the state-space form [17]: u (α) b ˙u (α) b = u(β) b ˙u(β) b , (11.55) g (α) b ˙g (α) b + g (β) b ˙g (β) b =0 (11.56) Equations (11.55) and (11.56) show that for the assembly according to Liu and Zheng an additional velocity compatibility and an impulse equilibrium at the interface are used. Thus, it is possible to eliminate all interface DOFs. From the extended equilibrium of forces in Eq. (11.56), the state-space equilibrium condition p (α) a ˙p (α) a + p (β) a ˙p (β) a = 0 (11.57) follows. By inserting the approximation of Eq. (11.52) into the extended compatibility condition of Eq. (11.55), the following is obtained: (α) kept,b p (α) kept + (α) a1,b p(α) a + (α) a2,b ˙ p(α) a = (β) kept,b p (β) kept + (β) a1,b p(β) a + (β) a2,b ˙ p(β) a (11.58) From Eq. (11.58) follows under utilization of Eq. (11.57) and rearrangement (α) a1,b + (β) a1,b (α) a2,b + (β) a2,b p (α) a ˙p(α) a = − (α) kept,b (β) kept,b p (α) kept p (β) kept . (11.59)
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