38 R.N. Coppolino “instrumented” DOFs (see Eq. 5.12). For “modified” Guyan reduction and “general” reduction, an indirect, least-squares approximation is utilized (see Eqs. 5.17 and 5.22), and an inherent singularity causes residual kinetic energy to have an upper bound that may exceed 1.0. Nevertheless, when “instrumented” DOFs are adequate to map modes of interest, the [RKE] column sums will be substantially below 1.0 (or 100 %). When the “instrumented” DOFs do not adequately map a mode of interest, the non-negligible terms of the particular [RKE] column indicate DOF locations required to complete the mapping. 5.6 Branched Shell Illustrative Example 5.6.1 Model Description and Dynamic Characteristics The shell structure, shown in Fig. 5.1, serves as the illustrative example in this paper. It consists of five substructures, namely (1) a lower cylindrical skirt (fully fixed at its base), (2) a lower hemispherical bulkhead, (3) lower cylindrical section, (4) upper cylindrical section, and (5) upper hemispherical bulkhead. The overall dimensions of the aluminum structure are length, LD100 inches, radius, RD20 inches, and wall thickness, hD0.4 inches. Fig. 5.1 Multi-segmented shell structure The shell’s finite element model, fixed at the lowest skirt station, is described in terms of 5616 “free” degrees of freedom (936 grid points). The lowest 146 normal modes of the model includes 22 modes characterized by “body” and “nD0, 1 breathing” modes, whose characteristics are summarized below in Tables 5.1 and 5.2. Table 5.1 Shell “body” and “nD0, 1 breathing” mode substructure energy characteristics SKIRT DOME1 SHELL1 SHELL2 DOME2 SKIRT DOME1 SHELL1 SHELL2 DOME2 1 1 Y-Bending 122.2 1 1 11 36 51 65 0 27 8 1 2 2 X-Bending 122.2 1 1 11 36 51 65 0 27 8 1 3 11 RZ-Torsion 316.9 4 6 22 43 26 59 0 31 9 0 4 14 Y-Bending 377.6 12 36 28 9 16 67 5 15 11 2 5 15 X-Bending 377.6 12 36 28 9 16 67 5 15 11 2 6 24 Z-Axial 469.4 3 7 18 35 37 56 1 29 11 3 7 49 Y-Bending 708.9 3 37 14 27 19 10 21 37 23 9 8 50 X-Bending 709.3 3 37 15 27 19 10 21 37 23 9 9 63 RZ-Torsion 851.9 19 5 35 38 3 11 12 37 34 6 10 76 Y-Bending 1005.7 17 19 27 25 13 20 14 29 35 1 11 77 X-Bending 1011.5 18 18 27 25 11 20 14 30 35 1 12 86 N=0 Domes 1036.2 8 52 10 4 27 24 28 11 20 17 13 103 N=1 1228.2 19 4 36 36 6 20 4 35 35 7 14 104 N=1 1228.3 19 4 36 36 6 20 4 35 34 7 15 111 N=1 1244.3 0 0 46 53 0 0 0 46 53 0 16 112 N=1 1244.3 0 0 46 53 0 0 0 46 53 0 17 121 N=0 Skirt 1302.3 96 0 2 1 0 95 1 2 1 0 18 128 N=1 1378.2 2 1 46 49 1 3 2 45 48 2 19 129 N=1 1378.4 2 1 46 49 1 3 2 45 48 2 20 140 N=1 1475.4 14 23 19 21 23 15 22 19 22 23 21 141 N=1 1476.0 13 20 22 22 23 14 19 23 23 22 22 146 N=0 Shells 1507.2 8 2 44 43 2 8 4 43 43 3 Kinetic Energy (%) Strain Energy (%) Body Mode System Mode Description FREQ (Hz)
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