# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 5 Conclusions  
In this paper, an effective reduced-order model was presented for the vibration analysis of a mistuned blade-disc-shaft assembly. Three-dimensional finite element models, which can provide accurate modeling and quantitative analysis for complex structures, were introduced to obtain the governing equations of motion, where the Coriolis force, centrifugal stiffening, and spin softening effects due to rotation were all taken into account. An efficient model order reduction technique, based on the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Analysis Type** & **FOM (s)** & **ROM (s)** & **Ratio** \\ \hline First 100 natural frequencies & 31.79 & 0.135 & 235 \\ Forced response at a rotating speed & 8.94 & 0.00895 & 999 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison on the computational efficiency between full-order model (FOM) and reduced-order model (ROM).  
Figure 7: Comparisons of the forced responses of the excitation nodes of the assembly with disc 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 DOFs, respectively.  
coordinate projection by normal modes of tuned assembly and cyclic symmetry analysis, was developed for a mistuned blade-disc-shaft assembly. As the basis for employing cyclic symmetry analysis, the criterion of whether one matrix could be incorporated in the analysis, as well as rigorous proof, was given. It was shown that the Coriolis, centrifugal stiffening, and spin softening matrices satisfy this criterion and can be converted into block-circulated forms in cyclically symmetric coordinates. In the presented model order reduction method, the mistuning in the blade and disc was taken into account and dealt with independently to reflect the realistic conditions of blade-disc-shaft assemblies. Regarding the coupling issue of the disc mistuning matrix, an effective mistuning projection approach was developed by using coordinate expansion and projecting the mistuning matrix onto a free tuned sector model of the disc. Detailed deductions on the theoretical basis of coordinate expansion were presented. The employed example of a blade-disc-shaft assembly illustrates the effectiveness of the presented method in free and forced vibration analysis in both static and rotating conditions.  
As cyclic symmetry analysis was employed in this paper, the presented method required the rotating systems to be cyclically symmetric. In some circumstances, such requirements may not be satisfied due to the anisotropy in supports, such as the anisotropy in bearings and foundations, which may lead to periodically time-variant terms and more complicated vibration characteristics. Then, further development of this method should be done, which was not included in this paper and remains to be conducted in the future.  
[PERSON] and [PERSON] conceptualized the presented idea; [PERSON] developed the mathematical formulas, performed the simulation, and wrote the manuscript; [PERSON] contributed to the finite element models; [PERSON] and [PERSON] critically revised the manuscript.  
This work was supported by the Project of National Natural Science Foundation of China (No. 51805130), the Natural Science Foundation of Anhui Province (No. 1808085 QE138), and the China Postdoctoral Science Foundation funded project (No. 2018M632521).  
The authors declare no conflicts of interest.