# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## 5 Conclusions  
In this paper, the method of Chebyshev orthogonal polynomials is utilized to compute the steady-state response of a rotor with uncertainties. To enhance computational efficiency in high-dimensional scenarios involving multiple uncertain parameters, the sparse-grid numerical integration method is employed. A dynamically similar rotor of a turboshaft engine is selected as the study's subject, and the influences of multiple uncertain parameters on the rotor's dynamics are analyzed. The main conclusions are as follows:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **FTG Method** & **MCS Method** & **Proposed Method** \\ \hline Computational cost & 16,384 & 1000 & 680 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of computational efficiency.  
Figure 13: Comparison of mean and standard deviation of steady-state response.  
1. The uncertainty in support stiffness significantly affects the steady-state response and critical speed, causing the resonance frequency, originally characterized by a single-peak pattern, to transform into a broad resonance band; this is accompanied by shifts in the resonance peak.
2. The upper and lower bounds of the steady-state response are symmetric around the deterministic curve under the influence of mass-unbalance uncertainty, with these bounds widening at high rotational speeds, although the critical speed remains unchanged. Additionally, the phase of the uncertainty unbalance significantly influences the dynamic response of the rotor, resulting in substantial increases in vibration amplitude at critical speeds.
3. The effects of multiple uncertainty parameters together on the rotor's steady-state response are multifaceted, characterized by broad amplitude bands, resonance peaks, frequency shifts, and fluctuations in amplitude magnitude.  
In summary, the method proposed in this paper integrates Chebyshev orthogonal polynomial approximation with sparse-grid numerical integration techniques, targeting the analysis of high-dimensional random variables. By utilizing sparse-grid integration, the approach addresses the computational challenges associated with the \"curse of dimensionality\", allowing for efficient processing without compromising accuracy. The combination of these techniques not only streamlines the computation but also enhances the robustness of the results, making it a valuable method for engineers and researchers dealing with high-dimensional problems in rotor dynamics and related fields.  
Future research will concentrate on the experimental validation of uncertainties in rotor systems to enhance the connections between theoretical models and practical applications. Systematic experimental protocols will be developed to elucidate how various uncertainty factors influence rotor dynamic performance, thereby providing empirical evidence to confirm the accuracy of these models. Furthermore, a strong emphasis on robust optimization methods is crucial, as it integrates results from uncertainty analysis to create designs that are both efficient and resilient to variations in operational conditions.  
Writing-review and editing, software, B.X.; writing--original draft preparation, P.N.; visualization, G.W.; supervision, C.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [National Natural Science Foundation of China] grant number [No. 52366002], [Project on Enhancement of Basic Research Ability of Young and Middle-Aged Teachers in Guangxi's Universities] grant number [No. 2023 KY0827], and [GUAT Special Research Project on the Strategic Development of Distinctive Interdisciplinary Fields] grant number [No.TS2024411].  
Not applicable.  
The data presented in this study are available in the article.  
The authors declare no conflict of interest.