## 1 Introduction  
Rotary machines (pumps, compressors, turbines, turbopump units, etc.) ensure power equipment's performance, reliability, and energy efficiency, especially in power generation, the oil and gas industry, and renewable energy. This article uses the terms \"classical methods\" and \"modern methods\" to distinguish between fundamentally different approaches to fault diagnosis. The distinction between classical and modern methods in fault diagnosis essentially relies on the computational approach taken and the level of automation involved. Classical methods primarily employ analytical modeling and numerical calculations, often requiring complex physical models and substantial computational power to solve complex mathematical equations. Modern approaches either extend classical methods with artificial intelligence (AI) and machine learning (ML) techniques or are mainly based on them. Thus, contemporary methods are used not just for fault prediction and classification but also to cut down on the computational complexity of analytical techniques. Certainly, classical methods are still an essential and powerful tool for solving diagnostics problems, but they have a number of limitations, such as high sensitivity to input parameters, scatter or noise, modeling errors, and initial settings. Other limitations are rapid computational cost growth, vulnerability to variations in initial conditions, and difficulty scaling. Moreover, because of these limitations, the mathematical models that describe the system are often simplified, which inevitably diminishes the accuracy of the predictions and the range of conditions under which the approach can be reliably applied. The study by [1] proposes a new index for detecting blade-to-case contact (rub-impact), demonstrating effectiveness in numerical experiments. However, the model on which it is tested is oversimplified: the rotor stiffness is constant, the bearings are perfectly symmetrical, and temperature- or friction-induced nonlinearities are not considered. As a result, the method is limited to a narrow set of cases and is unlikely to be scalable to complex multi-defect scenarios. The research work by [2] is also a prime example of the successful use of finite element modeling but with similar limitations. The algorithms process \"pure\" signals without considering noise, temperature effects, or the gradual deterioration of structural elements. The work by [3] presents techniques for damage detection in rotary systems. The methods rely on a simple beam model that, while useful for the initial detection of defects, does not account for nonlinearities, asymmetric mass distributions, or variations of elastic moduli characteristic of wear that exist in real machines. The research by [4] presents a detailed model of a local defect in the inter-rotor bearing. The model considers nonlinear dynamic effects, but symmetrical rotor structures with constant rotation speed are used. All boundary conditions are fixed and do not consider the actual mechanical wear or thermal expansion of the shaft--that is, the model is isolated from variables typical of industrial systems. The work by [5] assesses the capability of the fast Fourier transform (FFT) for simple fault detection scenarios, demonstrating fast spectral evaluation and easy implementation. However, FFT exhibits low sensitivity to complex, nonstationary signals. The authors do not address multi-fault discrimination. The paper by [6] deals with linear and nonlinear performance analyses of hydrodynamic bearings. The authors use models based on the Reynolds equation but do not consider the effects of variable lubricant viscosity, thermal effects, or load asymmetry. Moreover, all material properties (stiffness, damping) are assumed to be constant. Thus, although the modeling is detailed from a geometrical point of view, it is limited in physical accuracy. The work by [7] presents a complex nonlinear model of a rotor-disk system that considers large deflections and even chaotic regimes.