## 2 Research Methodology
### Modeling Methods Improvements
To compare solution performance with that of the classical methods, the solution results were also compared with classical ones, i.e., the linearized equation method (LEM), nonlinear equation reduction method (NREM), and empirical selection method (ESM). For verification, classical nonlinear dynamic equations, such as [PERSON]'s equation  
\[\ddot{x}+2\dot{\zeta}x+\alpha x+\varepsilon x^{3}-Fcos\omega_{F}t=0, \tag{46}\]  
[PERSON]'s equations  
\[\left\{\begin{aligned} \ddot{x}_{1}+x_{1}-\lambda\left(-2\mu_{1} \dot{x}_{1}+\gamma_{1}x_{1}^{2}x_{2}\right)&=0;\\ \ddot{x}_{  
In [91] the authors present a detailed comparison of two non-intrusive uncertainty-quantification (UQ) frameworks: stochastic finite element (FE) analysis with Karhunen-Loeve discretization (KL), and fuzzy finite elements, solved by \(\alpha\)-level optimization for a flexible rotor mounted on hydrodynamic bearings. They built a complete FEM model of the shaft-disk assembly, injected variability, either as Gaussian random fields or as triangular fuzzy numbers and propagated the scatter to frequency-response functions and steady-orbit shapes. The convergence tests (Figure 32) show that the stochastic branch needs roughly forty KL terms and a few hundred Monte Carlo samples, whereas the fuzzy branch requires dozens of differential evolution searches per \(\alpha\)-cut but dispenses with any assumed probability distribution. Both approaches predict almost identical upper- and lower-bound envelopes, and all laboratory measurements fall inside those envelopes, demonstrating that each scheme can deliver reliable, experimentally validated safety margins. The authors highlighted that the stochastic route is computationally lighter, while the fuzzy route is preferable when statistical data are scarce or epistemic. At the same time, they acknowledge several limitations: the parameters are treated as independent and mildly dispersed; large epistemic uncertainties or strong correlations remain unexplored; fuzzy analysis can be computationally intensive; and the Gaussian assumption in the stochastic case may misrepresent non-Gaussian reality. The study, therefore, recommends adaptive or hybrid UQ strategies for future robust rotor design and monitoring, particularly when scalability to high-dimensional uncertainty spaces or fault-induced nonlinearities becomes critical. Overall, the chapter offers a rare, head-to-head validation of stochastic versus fuzzy UQ on real hardware while clearly outlining the trade-offs that still need to be addressed. In the study by [92], a hybrid framework that conducts non-intrusive PCE with harmonic balance (HB) for linear cases and the asymptotic numerical method (ANM) for weakly nonlinear ones targets fast uncertainty analysis of rotor systems. A ten-element Timoshenko shaft with two rigid disks and linear bearings was perturbed by randomness in Young's modulus, bearing stiffness, and shaft density, while a cubic bearing stiffness term emulated nonlinearity. The PCE coefficients were fitted on a [PERSON] sparse grid requiring no more than 25 model evaluations, after which the surrogate yielded full frequency-response curves and a \"stochastic Campbell diagram\". It reproduced 95% of envelopes of \(10^{5}\)-sample Monte Carlo runs yet cut the runtime by up to two orders of magnitude; the nonlinear test still matches the brute-force results while capturing fold points and  
Figure 31: Comparison of numbers of convergences in two experiments with initial values obtained from four methods for the Duffing equation [90]: (**a**) artificially induced crack under moderate excitation conditions; (**b**) naturally developed damage under severe operational conditions. backward whirls. This speed enables rapid what-if studies and provides a differentiable surrogate for robust optimization.