12 Expansion of Coupled Structural-Acoustic Systems 133 Fig. 12.1 Coupled model with structural and acoustic domains (left) and chosen subsets of the structural and acoustic domains for analysis (right) a total of 120 modes. The results of this solver are given as complex-valued modes that contain both a real and an imaginary portion. After the solution was found, subsets of the structural and acoustic domains were chosen to perform the expansion analysis. The internal surface of the aluminum cylinder and the external surface of the acoustic domain were the chosen subsets. The chosen subsets of the acoustic and structural domains will be referred to as the n-set of the model. A sample of 50 points on the solid cylinder and 160 points on the acoustic domain within the n-set points were then chosen to create the a-set points. The n-set points are shown in Fig. 12.1 as three-dimensional scatter plots of each domain, and the a-set points are marked as black dots. The a-set points will be considered as the “measurement” points while the n-set points will be considered as the “full-field” response. To make the a-set “measurements” more representative of actual measurements, low-level random noise at 1% of the response mean was added to the a-set responses to simulate imperfect data. Three modes will be used to study expansion in this paper: a mode dominated by structural response, a mode dominated by acoustic response, and a mode with somewhat equal contributions of structural and acoustic responses. The modes were chosen by plotting the contributions of structural and acoustic responses calculated in the FE result file, as shown in Fig. 12.2. The contribution of each type of response to each mode was given as a value from zero to one, with higher values indicating a higher contribution to the mode. From the contribution factors shown in Fig. 12.2, mode 12 was chosen to be the representative structurally-dominated mode, mode 38 was chosen to be the representative acoustically-dominated mode, and mode 66 was chosen to be the representative coupled mode. The real and imaginary portions of each mode are shown, respectively, in Figs. 12.3, 12.4, and 12.5. Take note of the color scale in each plot, as some responses look significant but are actually very small with respect to the dominant motion. In the structural mode in Fig. 12.3, both the real and imaginary structural modes have a strong response while the acoustic response is small and has a similar shape to the structural mode. In the acoustic mode in Fig. 12.4, imaginary part of the acoustic mode dominates the response. In the coupled mode in Fig. 12.5, the real part of the structural mode and the imaginary part of the acoustic mode both contribute strongly to the response. To perform the expansion, a transformation matrix was required that could map between both structural and acoustic modes. All solution modes, including rigid body modes, were included in the expansion. The transformation matrix was computed by assembling each mode shape variable (x, y, and z for the structural response and p for the acoustic pressure response) into a universal mode shape matrix, and then calculating the transformation matrix using Eq. 12.4. The results of expansion are shown in the next three figures. The structure-dominated mode 12 expansion is shown in Fig. 12.6, with the major contributors to the mode highlighted. The structural expansion matches closely with the actual structural mode in shape and magnitude. The expanded acoustic mode, however, does not resemble the actual acoustic mode and the expanded acoustic response has a higher amplitude than the actual acoustic amplitude. This is likely due to the noise that was added to make the response more realistic. The acoustic
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