# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 4 Validation of Reduced-Order Models
\begin{table}
\begin{tabular}{c c c c} \hline
**Blade** & \(\delta_{n}^{b}\)**(\%)** & **Disc Sector** & \(\delta_{n}^{d}\)**(\%)** \\ \hline
1 & 1.351 & 1 & 0.675 \\
2 & \(-\)2.342 & 2 & \(-\)0.632 \\
3 & 1.070 & 3 & 1.561 \\
4 & 1.246 & 4 & \(-\)1.614 \\
5 & \(-\)1.553 & 5 & \(-\)0.211 \\
6 & 1.763 & 6 & \(-\)0.412 \\
7 & 1.447 & 7 & 0.921 \\
8 & 1.009 & 8 & 1.026 \\
9 & \(-\)0.377 & 9 & \(-\)1.088 \\
10 & \(-\)0.070 & 10 & \(-\)0.035 \\
11 & \(-\)2.1667 & 11 & \(-\)0.184 \\
12 & 2.053 & 12 & 0.509 \\
13 & 1.842 & 13 & 0.728 \\
14 & \(-\)0.825 & 14 & 0.886 \\
15 & \(-\)0.421 & 15 & \(-\)0.772 \\
16 & \(-\)3.228 & 16 & 0.623 \\
17 & 3.123 & 17 & 0.535 \\
18 & \(-\)1.272 & 18 & \(-\)1.167 \\ \hlineFigure 4 shows the comparisons of the first 100 natural frequencies of the tuned assembly using the full-order model and cyclic symmetric sector model. It can be seen in the figure that the two cases agree well, with the maximum relative error in natural frequency being less than 0.01%. Therefore, cyclic symmetry analysis is also feasible for bladed discs mounted on flexible shafts. Figure 5 shows the relative errors in eigenvalues of the reduced-order model at different rotating speeds, where the first 20 are included and the mistuning in the blade and disc is considered. It can be seen in the figure that the relative error grows with rotating speed but within a relatively small range. Although the relative errors increase for higher-order eigenvalues, the overall maximum relative error is still less than 0.1% when the rotating speed is less than 2000 rad/s. Thus, the presented model order  
Figure 4: Comparisons on the first 100 natural frequencies of the tuned blade–disc–shaft assembly using the full-order model and cyclic symmetry sector model. Figure 3: Comparisons of the forced responses of the excitation nodes of the assembly using the normal modes of the mistuned case and those of its tuned counterpart with excitations of (**a**) one engine order (1 EO), (**b**) 2 EO, and (**c**) 3 EO. reduction technique can give accurate results in eigenvalues for a mistuned blade-disc-shaft assembly in spite of the significant reduction in model order. Figures 6 and 7 illustrate the comparisons of forced responses of the assembly between the full-order model (FOM) and reduced-order model (ROM) with mistuning in the blade and disc, respectively, where the 1 EO, 2 EO, and 3 EO excitations are presented. As can be seen in the figures, for both cases of blade and disc mistuning, the two models give quite similar results in the entire speed range with different EOs of excitations. Thus, it can be concluded that the reduced-order model can give quite accurate results in forced responses compared with those of the full-order model. Figure 5: The relative errors in eigenvalues of the reduced-order model of a mistuned blade-disc-shaft assembly at different rotating speeds. Figure 6: Comparisons of the forced responses of the excitation nodes of the assembly with blade mistuning between the full-order model and reduced-order model with excitations of (**a**) 1 EO, (**b**) 2 EO, and (**c**) 3 EO. The full- and reduced-order models contain 103,563 and 7179 degrees of freedom (DOFs), respectively. In order to illustrate the efficiency of the presented method, the computational time of the reduced-order model in computing natural frequencies and forced responses is compared with that of the full-order model, which is listed in Table 3. The specifications of the employed computer during the analysis were as follows: Intel Core i7-8700 central processing unit (CPU) with 3.20 GHz; size and speed of the cache memory = 32 GB and 2666 MHz, respectively; size of the solid-state drive (SSD) hard disk = 512 GB. The presented values are all the averages of 100 analysis cases. It can be seen that the presented model order reduction technique could significantly increase the computational efficiency. Based on the results presented in this section, the effectiveness and efficiency of the presented method can be validated.