## 2 Research Methodology
### Modeling Methods Improvements
It expands the classical system equivalent reduction expansion process (SEREP) with an iterative procedure for determining the optimum number of eigenmodes, where calculations on the reduction model will be stabilized from excessive reduction, and finally, cost-cutting by using only the eigenmodes used within a particular frequency range that is required for accuracy in calculations. Conceptually, this method can be further classified into a few processes. The Sturm sequence check is the first performed to understand how many eigenvalues are in the specified frequency range. This is followed by computing eigenvalues and eigenvectors within the identified range through the bisection method. Finally, the system's dynamic behavior is calculated based on the limited dataset consisting of eigenmodes, allowing for an accurate dynamic response. However, the accuracy and computation efficiency of the results have much to do with the proper selection of the frequency range. Differential equations are common in rotor dynamics. Many methods exist to solve such equations analytically and numerically, but some are hard or even impossible when using classical approaches. The study by [89] explores a technique for solving differential equations using a genetic algorithm (GA). As a specific case, the authors selected the equation describing an electron in the Coulomb field of two protons. After imposing initial conditions and choosing an approximate solution form, the author obtained a vector of unknown parameters, including the distance between protons, the system's energy, and differentiation constants. This vector was treated as a chromosome in the genetic algorithm. The objective function was formulated by rearranging the differential equation terms to  
Figure 30: Standard velocity deviation for the complete and reduced systems’ responses. Adopted from [88]. one side, and the genetic algorithm was applied to minimize the triple integral of the squared objective function. This formulation naturally follows the physical nature of the process described by the equation. The GA-based solution was compared with published experimental results addressing the same problem. The relative error in determining the proton-proton distance was approximately 2%, whereas the relative error in the system's energy estimation was around 30%. Many numerical methods come down to iterations. One of the drawbacks of this method is the possibility of iteration divergence. Article [90] explores the use of neural networks (NN) for finding optimal initial conditions in the incremental harmonic balance method (IHBM). The harmonic balance method (HBM) is based on the harmonic linearization of nonlinear equations, where periodic functions are represented as a Fourier series. The expected solution is approximated by the fundamental harmonic and specific higher harmonics with unknown amplitudes and initial phases. If the sum is then replaced in the nonlinear differential equation (NDE) and equated with the coefficient of corresponding harmonics, a system of algebraic equations emerges. The accuracy of the solution is characterized by the number of selected harmonics (it is exact for infinite harmonics), but the larger the number of utilized harmonics, the greater the degree of computational complexity of the analytical solution. The harmonic balance method incrementally solves the ensuing algebraic system with [PERSON]'s numerical iterative method. The technique, however, suffers from the possibility of iteration divergence if the starting parameters are not suitably chosen. The authors have tackled this problem using a neural network (NN) to identify the initial parameters. The authors enhanced the Newton-Raphson algorithm by incorporating [PERSON]'s gradient iteration to improve convergence stability. A neural network was constructed and trained in different convergence difficulty equation systems. Various network structures were attempted, and two optimal structures were identified.