## 2 Research Methodology
### Modeling Methods Improvements
Limitations remain, however: The accuracy degrades beyond four or five random variables, grid and polynomial order need expert tuning, no experimental validation or real-fault scenarios are included, and the nonlinearity is confined to a single cubic term (Figure 33). Even so, the work convincingly shows that PCE-accelerated HBM/ANM can make stochastic rotor dynamics tractable and sets a clear agenda for scaling to higher-dimensional uncertainty and richer defect models. The authors proposed an interval uncertainty framework that approximates a multi-variable rotor model with third-order Chebyshev orthogonal polynomials and evaluated  
Figure 33: Deterministic linear and nonlinear FRFs. \[\varphi=h_{s}\Delta\omega_{K_{2}}\left[n_{3}+\frac{K_{2}(M_{c}-1)}{2}\right] -arg\left[y_{K_{2}}(n_{3})\right]+\frac{\pi}{2} \tag{43}\]0 Convergence analysis for Karhunen–Loeve (KL) expansion (**a**,**c**) and Monte Carlo simulation (MCS) (**b**,**d**) for the FRF (**a**,**b**) and the orbit (**c**,**d**). Adopted from [91]. the coefficients on a Smolyak sparse grid in the study of [93]. A three-disk test stand was analyzed with seven uncertain inputs--two bearing stiffnesses, three unbalance magnitudes, and two phases--and the surrogate requires only 680 finite element runs versus \(1.6\cdot 10^{4}\) for a full tensor grid or \(1\cdot 10^{3}\) for Monte Carlo (Figure 34). The method reproduces Monte Carlo envelopes for frequency-response curves and time-domain orbits within 1% except for a single anti-resonance while lowering the CPU costs by roughly 96%. It also revealed how bearing stiffness broadens resonant \"bands\" and how an unbalanced phase can triple the peak amplitudes, providing helpful information for robust threshold settings. Key advantages are the absence of probability-density assumptions and rapid scalability to a moderate number of interval variables. The drawbacks include a conservative widening of the bounds, neglect of the parameter correlations, and sensitivity to the chosen polynomial order and grid level; moreover, no physical defects such as cracks or rub- impact are modeled. The study demonstrates that Chebyshev-sparse-grid surrogates can make interval analysis practical for steady-state rotor responses, yet further work is needed to handle higher-dimensional uncertainties and defect-induced nonlinearities. Deterministic linear and nonlinear frequency-response functions (FRFs). In the work by [94], the authors introduce the sub-interval similarity (SIS) metric, designed to quantify how well simulated and experimental data agree when those data are expressed as either probability distributions or simple value intervals. The idea is to split each interval into a data-driven number of sub-intervals, compute a bounded similarity score for every slice, and average the result, thereby retaining information that would be lost if only the outer limits were compared. SIS is embedded, on the one hand, in a Bayesian Markov-chain Monte Carlo (MCMC) scheme for stochastic model updating and, on the other, in a particle swarm optimizer for interval updating, and is validated on a 3-DOF mass-spring model as well as on 55 steel plates with uncertain elastic moduli. In both cases, the new metric cuts the parameter error levels below 1% while using far fewer samples than the Bhattacharyya distance or KL divergence and runs roughly 500\(\times\) faster than the high-order probabilistic distances in nine-dimensional space (Figure 35). Its main strengths are universality, equally compatible with probabilistic and non-probabilistic data, and efficiency when experimental samples are scarce.