## 3 Modeling, Implementation, and Numerical Simulation  
A test rig dynamic analysis was performed using Transmission3D (T3D) software [49], the aim of which was the extraction of displacements to further analyze vibrations and their spectra to predict damage. The test rig can be configured in different regimes that can be applied to the system. The traditional FEM describes the macroscopic deflection of components, while T3D solves analytically the contact between bodies. Generally, a mesh refinement, to provide convergence and properly simulate a contact, is required in the traditional approach. Moreover, in order to generate good numerical results and obtain converged simulations, the flanks in the region of the gear teeth, where the main analysis take place, must be discretized with a very fine mesh. This is usually performed due to the contact position of the gears and the variation of their engaged surfaces during operation, which increases the number of equations to be solved [50]. Thus, this problem can be solved in the T3D environment. As an example, in the approach applied by [51], the FEM was used as the second stage of product development. It was performed due to the fact that lumped parameters (e.g., stiffness of gearbox components) were determined to support the analysis of the analytical model. This type of numerical simulation, which is a highly effective technique for this type of problem, provides an accurate inspection of the mesh stiffness model. In this way, several engagement positions need to be simulated to perform a consistent analysis related to the contact of the gears and acquire a mean value related to them. This is due to the variation in the stiffness during the engagement of the gears. For accurate results, a mesh sensitivity analysis should also be performed for the contact regions so that the elements and nodes in these areas can activate a simulation that converges properly. Subsequently, the complexity of creating feasible finite element (FE) models of geared systems, including the test rig presented in this research, and the expansion time of their simulations make the traditional FEM approach unfeasible in some cases. Moreover, even being completely useful for some systems, it can be replaced by a more efficient one. Thus, each contact during the engagement was analytically solved in this method through a few equations, without using a robust mesh refinement. This benefits the entire process objective through a hybrid FEM-analytical approach and effectively reduces the time of each simulation. The simulation of transmission systems, in which characteristics can be complex and widely extensive in geometrical matters, is performed through an approach specifically developed for this, a hybrid FEM-analytical approach. All the internal components of the system (e.g., gears, shafts, and bearings) are modeled entire; thus, accuracy with respect to system response can be achieved. Since the deformations originated by contacts between rotating machine components were approached based on the Hertzian theory, the method used in this research paper adopts a solution different from traditional FE, which refers to the macroscopic deflection of the components under analysis. Given this, the constant contacts of the rotating components under analysis, such as gear teeth and internal bearings, need mesh refinement. This is due to the fact that a FE simulation of this magnitude needs to guarantee convergence so that the software can correctly simulate the desired system. Furthermore, mesh refinement is also required when the contacts and couplings of the variable rotating components are analyzed in order to obtain convergence with the simulated system. On the other hand, the model presented in this research paper solves, for each contact of the acting components, an additional equation through