## 4 Results
### Unbalanced Disc in the Rotor Central Position (Rotor05 un)  
This configuration is commonly used in rotor dynamics studies to analyze the effects of unbalance and system oscillations [17; 21]. The unbalanced mass causes rotor oscillation during rotation. The magnitude and direction of this oscillation depends on various factors, such as the mass and location of the unbalanced mass, rotational speed, stiffness, and damping of the bearings [21]. Figure 5 shows the configuration of a numerical model of a Laval rotor with a centrally mounted disc and the additional unbalanced mass (m). The rotor is mounted in two bearings (depicted by squares) on both sides. The scatter plot Figure 6 shows the value of the critical speed with respect to individual runs of the analysis. The left vertical axis of the scatter plot represents the value of the critical angular velocity in units of [rad/s], while the right vertical axis indicates the corresponding value of this speed in units of rotational frequency [Hz]. The histogram describes the probability distribution of the rotor's critical speed values, which range from 780 to 830 [rad/s], representing a spread of approximately 6%. The probability distribution corresponds \"approximately\" to a unimodal distribution. Figure 7a represents a 3D projection of the probability of the shaft's orbital position when passing through the critical speed. The base axes represent the horizontal and vertical positions of the shaft in a plane perpendicular to the axis of rotation. The vertical axis in this case represents the probability value of the shaft's position at individual points. For better visualization, the probability range is defined by a color scale. In the 2D projection (Figure 7b), or top view, the probability distribution appears circular. The  
Figure 5: Schematic display of Rotor05 un configuration. Figure 6: Critical speed analysis for Rotor05 un configuration. (**Left**): Scatterplot of critical speeds across multiple simulation trials. (**Right**): Histogram showing the probability distribution of critical speeds. highest probabilities are concentrated in a central ring pattern clearly depicted in yellow to white colors. The contour plot (Figure 8) illustrates the probability distribution of the shaft's orbital position at critical speed. The circular pattern indicates a predominantly circular orbit, with probability density represented by color gradients. Concentric rings in the distribution suggest non-uniform positioning across the orbit. Contours effectively delineate borders between probability intervals, allowing for a detailed analysis of probability gradients across the orbital space. This visualization depicts the complex spatial distribution of the shaft's orbital behavior, showing subtle features such as slight asymmetry in the highest probability region, which points to a potential bias in shaft motion. Figure 8: Contour plot of shaft orbital position probability for the Rotor05 un configuration at critical speed. Figure 7: Probability of shaft orbital position for Rotor05 un configuration: (**a**) 3D view of the shaft orbital probability distribution; (**b**) top view of the probability distribution. To evaluate a histogram of the shaft orbital, the cross-section is added in the horizontal and vertical positions (Figure 9). Cross-sectional analysis locations are marked by lines A-A (horizontal) and B-B (vertical), which will be used to generate histograms of the orbital distribution. Four quadrants (I, II, III, IV) are identified, with circular highlights at key positions, enabling a detailed examination of the shaft's behavior at different orbital locations. Figure 10 represents the distribution of shaft positions along the horizontal cross-section A-A, for the left and right sides of the orbital, respectively. The central scatterplot in each figure shows individual points from multiple shaft orbitals. The horizontal axis represents the position along A-A, while the vertical axis shows the deviation from A-A.