11 Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring 121 Ce =0.0002Ke Ce =0.0004Ke Ce =0.0002Ke Ce =0.0001Ke 1.08m 0.72m Substructure β 8 beam elements n(β) =36 Substructure α 12 beam elements n(α) =52 20 beam elements, n=84 Fig. 11.6 Free-free beam structure divided into two substructures [17]. The beam consists of 20 Euler-Bernoulli beam elements (Young’s modulus 7.0· 10 10 Nm−2, density 2.7· 10 3 kgm−3, cross-section 5.0· 10−5 m2, moment of inertia 1.04· 10−10 m4) and has n = 84 DOFs in the statespace. The total length of the beam is 1.8 m. The length of substructure α is 1.08 m and the length of substructure β is 0.72 m. The damping for elements 1–5 and 13–16 is Ce =0.0002Ke, for elements 6–12 Ce =0.0004Ke and for elements 17–20 Ce =0.0001Ke −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 ·105 −8000 −6000 −4000 −2000 0 2000 4000 6000 8000 Real part [rads−1] Imaginary part [rads−1] Fig. 11.7 Exact eigenvalues of the coupled unreduced system of Fig. 11.6 Figure 11.7 shows all 84 eigenvalues of the unreduced system in the complex plane. It can be seen that compared to the previous example in Fig. 11.2 significantly more real eigenvalues occur without imaginary parts. Altogether, there are 20 complex conjugate eigenvalue pairs. Thus, there are 40 complex eigenvalues with imaginary parts. In contrast, there are 40 real eigenvalues without imaginary parts. Additionally, four rigid body modes with zero eigenvalue occur. Furthermore, it can be seen that the maximum amount of the real parts is larger by two orders of magnitude compared to the previous example in Fig. 11.2, whereas the imaginary part is smaller by approximately two orders of magnitude. Figure 11.8 shows the exact eigenvalues of the substructures. For substructure α there are 10 eigenvalue pairs and 28 real eigenvalues as well as 4 rigid body modes with zero eigenvalue. For substructure β there are 9 eigenvalue pairs and 14 real eigenvalues as well as 4 rigid body modes with zero eigenvalue. The damped free-free beam is reduced and assembled using the substructuring methods described in Sects. 11.3 and 11.4. For the reduction of substructure α, n (α) kept =11 eigenmodes belonging to the 11 eigenvalues with the lowest absolute value
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