11 Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring 117 −6000 −5000 −4000 −3000 −2000 −1000 0 −2 0 2 ·10 5 Real part [rads−1] Imaginary part [rads−1] (a) −6000 −5000 −4000 −3000 −2000 −1000 0 −2 0 2 ·10 5 Real part [rads−1] Imaginary part [rads−1] (b) Fig. 11.3 Exact eigenvalues of the unreduced substructures of Fig. 11.1. (a) Substructure α. (b) Substructure β Table 11.1 Modes used for reduction and resulting size of the reduced assembled system Craig/Ni’s method Liu/Zheng’s method de Kraker/van Campen’s method Liu/Zheng’s basis with primal assembly Third order reduction Substructure α β α β α β α β α β Kept eigenmodes 20 15 20 15 20 15 20 15 20 15 Attachment modes 2 2 4 4 4 4 4 4 6 6 Rigid body modes 0 3 0 3 0 3 0 3 0 3 DOFs reduced system 38 38 42 42 42 eigenmodes belonging to the 15 eigenvalues with the lowest absolute value are kept. These are seven complex conjugate pairs and one real eigenvalue without imaginary parts. Additionally, for the reduction of substructure β, the n (β) r =3 rigid body modes are used. For the reduction according to Craig and Ni, both attachment modes are determined for substructure α according to Eq. (11.24) and for substructure β according to Eq. (11.34). For the reduction according to Liu and Zheng and for the reduction according to de Kraker and van Campen, the four attachment modes are determined for substructure α according to Eqs. (11.48) and (11.63), respectively, and for substructure β according to Eqs. (11.53) and (11.67), respectively. For the third order approach, six attachment modes are determined according to Eq. (11.73) for substructure α and according to Eq. (11.74) for substructure β. After assembly, the reduced system according to Craig and Ni’s method (CN) and Liu and Zheng’s method (LZ) has nred,CN =nred,LZ =38 DOFs. According to de Kraker and van Campen’s method (KC), Liu and Zheng’s reduction basis with primal assembly (LZ,KC), and the third order approach (TO), the reduced system has nred,KC =nred,LZ,KC =nred,TO =42 DOFs. Table 11.1 summarizes the number of used modes and the size of the reduced assembled systems. In order to quantify the differences of the individual methods, the relative error of the real parts and imaginary parts of the eigenvalue λk is used in the following. The eigenvalues of the full unreduced systemλfull,k are set in relation to the eigenvalues of the assembled reduced systemλred,k, which leads to the relative errors εrel, ,k of the real part and the relative errors εrel, ,k of the complex parts: εrel, ,k = λred,k − λfull,k λfull,k and εrel, ,k = λred,k − λfull,k λfull,k (11.82) Figure 11.4 shows the relative errors of the real and imaginary parts corresponding to the 34 eigenvalues with the lowest absolute value for the various methods.For better distinguishability, only the relative errors εrel >3· 10−9 are depicted in the following figures. The relative errors of both eigenvalues belonging to a complex-conjugate pair are represented. The absence of a relative error of the imaginary part εrel, implies that the associated eigenvalue is purely real and has no imaginary part. Relative errors increase with increasing eigenvalue. In addition, the imaginary parts show a slightly better agreement than the
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