# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 4 Validation of Reduced-Order Models  
Validation of the developed order reduction method is presented in this section. As the accuracy of the reduced-order model in vibration analysis for practical structures relies on the accuracy of its counterpart full-order model, the effectiveness of the developed method can be verified by comparing the results of the reduced-order model with those of the full order model. Figure 1 illustrates the finite element model of a blade-disc-shaft assembly, which is used as an example in this section, where the DOFs of the whole model and the sector model are equal to 103,563 and 7179, respectively. The whole model contains 18 sectors. The reduced-order model contains 216 DOFs with 12 modes per ND. During the projection of blade and disc mistuning, the retained modes of the cantilevered blade and free disc are 20 and 50, respectively. The excitation is applied at the tip of the blades with an amplitude of 2 N. The structural damping is assumed to be Rayleigh damping and \(\mathbf{C}_{d}=\beta\mathbf{K}_{e}\), where \(\beta\) is set to \(3\times 10^{-6}\) in the subsequent analysis. The material of the blade and disc is titanium alloy, and that of the shaft is steel, where the properties of these materials are listed in Table 1.  
Generally, the mistuning in the blade and disc is random and hard to measure. Herein, a simpler mistuning model, the commonly used proportional mistuning model, is used to validate the developed method. Small random variations in [PERSON]'s modulus are introduced as  
\[\left\{ \begin{array}{l}E_{n}^{b}=\left(1+\delta_{n}^{b}\right)E_{0}\\ E_{n}^{d}=\left(1+\delta_{n}^{d}\right)E_{0}\end{array} \right.,\quad n=1,2, \cdots,N, \tag{25}\]  
where \(E_{n}^{b}\) and \(E_{n}^{d}\) denote the [PERSON]'s modulus of the \(n\)th sector models of the blade and disc, \(E_{0}\) represents the nominal Young's modulus, and \(\delta_{n}^{b}\) and \(\delta_{n}^{d}\) are random variables representing the small variations. Specifically, the employed proportional mistuning patterns of the blade and disc in the subsequent analysis are listed in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Blade and Disc** & **Shaft** \\ \hline Density & 7850 kg/m\({}^{3}\) & 4420 kg/m\({}^{3}\) \\ Poisson’s ratio & 0.3 & 0.31 \\ Young’s modulus & 210 GPa & 210 GPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the blade, disc, and shaft. Figure 1: The finite element model of a blade–disc–shaft assembly and its sector model. Figure 2 illustrates the natural frequencies versus nodal diameters of the tuned blade-disc-shaft assembly in a static condition. Some of the 0 ND and 1 ND modes are shaft-dominated and not involved in the lines. With this diagram, the excited modes can be easily obtained by comparing the exciting frequency and the natural frequencies. Figure 3 shows the comparisons of the forced responses of the mistuned blade-disc-shaft assembly using the normal modes of the mistuned case and those of its tuned counterpart. The presented results are the responses of the excitation nodes, which are located at the tip of each blade. In the subsequent analysis, the results of these nodes are also employed. It can be seen in the figures that the results agree very well. The maximum relative error in amplitude is less than 0.1%. Therefore, the normal modes of the tuned blade-disc-shaft assembly can accurately represent the modes of mistuned assembly.