# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## 1 Introduction  
The rotor system is a vital component of rotating machinery such as aero engines and gas turbines. In the field of rotor dynamics, numerical simulators are commonly used to predict the dynamic behavior of the rotor system. However, uncertainties inherent in actual rotor systems stem from operational conditions and geometric parameters that may vary, including loose connections [1] or cracks [2]. These variations can lead to deviations from the expected steady-state response. Consequently, a steady-state response of the rotor system which initially meets design specifications may exceed acceptable limits in practice. A major challenge for engineers and researchers is to analyze and quantify the impacts of these uncertainties on the steady-state responses of complex rotor systems [3, 4, 5]. There are four commonly used methods for uncertainty analysis: the stochastic finite element method, probabilistic models, fuzzy methods, and interval methods. The stochastic finite element method assesses uncertainty by establishing stochastic differential equations for the system's response based on stochastic matrix theory or [PERSON]'s formula [6]. [PERSON] [7] applied this method to analyze a multi-degree-of-freedom rotor system in both time and frequency domains, investigating dynamics such as axial trajectory and random amplitude and frequency responses. [PERSON] [8] developed a stochastic finite element model of a rotor-bearing system considering radial clearance randomness and employed the orthogonal polynomial method to examine the effects of factors like mean and coefficient of variation of radial clearances, eccentricity, and rotational speeds on the system's dynamicperformance. However, the stochastic finite element method's disadvantage is found in the difficulty and time consumed in deriving embedded equations when analyzing complex rotor systems. Probabilistic methods necessitate complete probability distributions of uncertain parameters, which are often challenging to obtain [9]. Fuzzy methods employ fuzzy variables to represent uncertain parameters, offering a robust framework for capturing the inherent ambiguity in a system's characteristics. The Fuzzy Stochastic Finite Element Method (FSFEM) is particularly advantageous, as it integrates fuzzy logic with stochastic modeling, allowing for a more comprehensive analysis of uncertainties in dynamic systems. [PERSON] [10; 11] utilized the FSFEM to investigate flexible rotor dynamics under uncertain parameters, examining the impacts of variables such as pad radius, oil viscosity, and radial clearance on tilting-pad journal bearings. However, the derivation of membership functions in fuzzy methods can be complex, presenting challenges when applied to the intricate task of rotor uncertainty analysis. Additionally, the effectiveness of the FSFEM may be limited by the presence of multi-source uncertainties, which complicates the modeling process and can hinder accurate assessments of system behavior. Interval analysis is widely applied in uncertainty analysis. Although variations in system parameters are typically random, their range is usually finite, described as \"unknown but bounded\", indicating that the parameters fluctuate within definable intervals [12]. In the interval analysis of structural dynamics, uncertain parameters are represented as interval vectors, and differential equations governing rotor motion are formulated by incorporating these interval vectors. The solutions to these equations are derived using interval mathematics and regression theory, which facilitates the determination of the range of the system's steady-state response [13]. [PERSON] [14; 15] proposed a Bound Correction Interval Analysis Method (BCIM) based on Chebyshev expansion and modal superposition for the analysis of uncertain rotors. [PERSON] [16] combined interval regression analysis with regularization to introduce a computational inverse method for identifying bearing loads.