# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## 2 The Proposed Method
### Uncertainty Analysis Flow Based on Chebyshev Orthogonal Polynomials  
The process of interval uncertainty analysis using Chebyshev orthogonal polynomials consists of four primary steps. First, the random variables are specified, and one-dimensional interpolation points are generated using Gauss integration nodes. These nodes serve as the foundation for the subsequent steps in the analysis. Second, \(n\)-dimensional interpolation points are created from each one-dimensional point using either the tensor product or sparse-grid integration method. The choice between these two methods de  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(n\) & \(\mathbf{L}\) & **SG** & **FTG** & \(\mathbf{L}\) & **SG** & **FTG** \\ \hline
5 & 1 & 11 & 32 & 2 & 66 & 243 \\
7 & 1 & 15 & 128 & 2 & 113 & 2187 \\
10 & 1 & 21 & 1024 & 2 & 231 & 59,049 \\
13 & 1 & 27 & 8192 & 2 & 243 & 1,594,323 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the number of integration nodes for SG and FTG.  
pends on the complexity of the problem and the desired balance between accuracy and computational efficiency. Once the \(n\)-dimensional interpolation points are established, the system response is calculated at each point, providing a comprehensive set of data for the uncertainty analysis.  
In the third step, the coefficients of the Chebyshev orthogonal polynomials are computed to construct an orthogonal polynomial approximation model. This model serves as a surrogate for the original system, allowing for a more efficient uncertainty analysis. Finally, the steady-state response bounds of the rotor systems are calculated using the constructed approximation model. Once the Chebyshev orthogonal polynomial approximation is established, it can be used as a proxy model for the system, enabling efficient and accurate estimation of the steady-state response bounds through random sampling of the standard random variables using the MCS approach. Compared to the original finite element model, the use of Chebyshev orthogonal polynomials significantly enhances the efficiency of uncertainty analysis by reducing the computational burden while maintaining a high level of accuracy. Figure 2 provides a visual representation of this procedure, illustrating the flow of data and the key steps involved in the interval uncertainty analysis using Chebyshev orthogonal polynomials, thereby facilitating a clearer understanding of the methodology employed.  
The accuracy and predictive quality of the Chebyshev orthogonal polynomial surrogate model can be assessed using the relative generalization error, denoted as \(\varepsilon_{r}\). The formula for calculating \(\varepsilon_{r}\) is as follows:  
\[\varepsilon_{r}=\frac{\mathbb{E}\Big{[}\Big{(}\mathcal{M}(\mathbf{\mathcal{X}}_{ \text{val}})-\mathcal{M}_{\text{Chebyshev}}(\mathbf{\mathcal{X}}_{\text{val}}) \Big{)}^{2}\Big{]}}{\text{Var}[\mathcal{M}(\mathbf{\mathcal{X}}_{\text{val}})]} \tag{14}\]  
where \(\mathbb{E}[\cdot]\) denotes the expectation operator, while \(\text{Var}[\cdot]\) represents the variance operator. \(\mathcal{M}\) represents the finite element model of the rotor system (also referred to as the original model or source model), \(\mathcal{M}_{\text{Chebyshev}}\) denotes the Chebyshev orthogonal polynomial, and \(\mathbf{\mathcal{X}}_{\text{val}}\) represents the random validation set.  
Figure 2: Uncertainty analysis procedure for the steady-state response of rotor systems by Chebyshev orthogonal polynomials.