# Design of a Spin Test System for Burst Phenomen Considering Nonlinear Rotordynamic Effects
## 2 Materials and Methods
The test rig and the main rotor shaft will be connected through bolt fastening, allowing for the testing of discs with different geometries. The test specimens and the drive shaft to which they are connected in the spin test rig are presented in detail in Figure 3. The first specimen weighs 10 kg and will be tested at 30,000 RPM, while the other specimen weighs 4 kg and will be tested at 60,000 RPM. Figure 1: Algorithm of this study. Figure 2: Definition of the system (**a**) spin test rig (**b**) assembly of shaft. Rotodynamic calculations have been performed separately for each of these specimens. Boundary condition definitions for the drive shaft at the inner race of the bearing have also been provided in detail. The DM of the shaft represents a flexible body as a combination of a rigid-body model and a deformation model. The rigid-body model accounts for the body's rotation and translation movements without any deformation. This model calculates shape changes at specific points on the body, which are derived using the rigid-body model. Achieving accurate results with this method relies on the spatial distribution of the geometry's mass, inertia, stiffness, and damping properties. The DM divides the body into a mesh consisting of numerous nodes, each with six degrees of freedom. The details of the DM are depicted in Figure 4.  
Figure 3: Diameters of Test Specim (**a**) 4 kg specimen drawings (**b**) 10 kg specimen drawings. In the mathematical model of the spin shaft, the main rotor shaft is modeled as flexible. Using the developed model, it is also possible to size the rotor shaft under relevant constraints with optimal values. Figure 4 illustrates the main rotor shaft and its bearings. For a shaft supported by bearings at two points and free at one end, the translational and rotational movements of the center of mass along two different axes are defined as follows:  
The mathematical model of the spin shaft supported by bearings in two regions is presented in Figure 5. Based on this mathematical approach, the 3D rotordynamics model developed in the MATLAB Simulink is detailed in Figure 4.  
Figure 3.0  
Figure 3.1  
Figure 3.2  
Figure 3.3  
Figure 3.4 Discrete model of the shaft. Figure 3.5 Mathematical model of spin shaft. Based on this, the equations of motion are derived using the Euler-Newton method as follows:  
Figure 3.6  
Figure 3.7  
Figure 3.8  
Figure 3.9  
Here, \(J_{Tr,rs}\) is the transverse mass moment of inertia and \(J_{Prs}\) is the polar mass moment of inertia, and they are defined by the following expressions. Figure 2.0  
The system matrices in the equation of motion consist of the global mass matrix \(\underbrace{M}_{\text{ }}\) the global gyroscopic matrix \(\underbrace{G}_{\text{ }}\) and the global stiffness matrix \(\underbrace{K}_{\text{ }}\) as shown in Equation (1). The generalized external force vector for the global system, denoted by \(\underbrace{F}_{\text{ }}\) encompasses all external disturbance forces and torques that affect the dynamics of the global system. All system matrices and vectors are organized in the same sequence as the nodal displacements in vector \(\underbrace{q}_{\text{ }}\) as seen in Equation (2). Figure 2.1  
Figure 2.2  
In the rotor dynamics studies, bearing stiffness values were not treated as constants in either the radial or axial directions; instead, the stiffness characteristics obtained from tests were utilized as input data. The nonlinear stiffness behavior curves for both radial and axial directions are presented in Figure 2.3 When applying bearing stiffness values in rotor dynamics calculations, the selection was based on the allowable unbalance load for radial stiffness, while mounting loads were considered for axial stiffness. This approach incorporates the nonlinear stiffness effects of the bearings into the calculations. As a result, the rotor dynamics calculations yield results that are more consistent with the experimental data.