Probabilistic Analysis of Orbital Characteristics of Rotary Systems with Centrally and Off-Center Mounted Unbalanced Disks

Abstract

Rotor dynamics plays a crucial role in the performance and safety of rotating machinery, with disk position and unbalance significantly impacting system behavior. This study investigates the dynamic characteristics of two rotor configurations: a centrally mounted unbalanced disk (Rotor05un) and an off-center unbalanced disk (Rotor025un). Using numerical simulations and Monte Carlo analysis, we examined critical speeds and orbital patterns for both configurations. Probability distributions of shaft orbital positions revealed distinct patterns for each configuration. Quantile analysis revealed approximate linear trends for Rotor025un, suggesting higher system stiffness and more predictable behavior near critical speeds. Cross-sectional analyses of the orbits provided insights into the complex interactions between disk position, gyroscopic effects, and system natural frequencies. These findings provide valuable insights for rotor system design, particularly for applications with non-ideal mass distributions. The study goes beyond traditional critical speed analysis to examine orbital patterns and point on orbit occurrence from a probabilistic perspective. Based on the simulation of the orbits, an orbital is determined that allows the probability of the shaft occurring at the analyzed distance from the origin to be determined. The paper also offers insights into the complex interaction behavior of chosen rotor configurations and highlights the importance of considering disk position in predicting and optimizing rotor dynamic behavior, contributing to the development of more robust and efficient rotating machinery.

1. Introduction

Rotordynamics, a specialized branch of applied mechanics, plays a crucial role in the design and operation of rotating machinery across various industries, including energy, aerospace, and manufacturing. The study of rotor behavior, particularly at critical speeds, is fundamental to ensuring the safety, efficiency, and longevity of these systems. Recent advancements in computational methods and experimental techniques have enabled more sophisticated analyses of rotor dynamics, allowing engineers to better predict and mitigate potential issues [1].

One of the most significant challenges in rotordynamics is the accurate prediction and control of rotor behavior in the presence of unbalance and asymmetry [2]. While traditional models often assume ideal conditions, real-world rotors frequently exhibit imperfections that can significantly affect their dynamic response [3]. The impact of these imperfections becomes particularly critical as rotors operate at or near their critical speeds, where even small imbalances can lead to large amplitude vibrations [4].

This study focuses on two specific rotor configurations: one with a centrally mounted unbalanced disk (Rotor05un) and another with an off-center unbalanced disk (Rotor025un). By comparing these configurations, we aim to investigate how the position of an unbalanced mass affects the rotor’s critical speed and orbital behavior. This comparison is important for understanding the complex interplay between rotor geometry, mass distribution, and dynamic response [5].

The research employs a combination of numerical simulations and statistical analysis to provide a comprehensive view of rotor behavior under different conditions. By utilizing Monte Carlo simulations, we account for the inherent variabilities in rotor systems, offering a more realistic prediction of their performance [6]. This probabilistic approach represents a significant advancement over deterministic methods, allowing for a more nuanced understanding of rotor dynamics.

Recent studies have also shown the potential of methods based on machine learning in rotordynamics analysis. Saucedo-Dorantes et al. [7] presented a novel condition-monitoring methodology for induction motors, leveraging a high-dimensional set of hybrid features extracted from vibration and stator current analyses. The proposed approach utilizes artificial intelligence and machine learning techniques to optimize and reduce the feature space, enhancing the identification of multiple and combined faults occurring simultaneously. The effectiveness of the methodology is validated through a comprehensive experimental dataset encompassing healthy, single-fault, and combined-fault conditions.

Zhao et al. [8] developed a deep learning approach for rotor orbit reconstruction and fault diagnosis. Their method uses convolutional neural networks to analyze rotor orbit images, achieving high accuracy in fault classification.

Other authors, like Cavalini Jr et al. [9], used operational modal analysis for the online identification of critical speeds in rotating machinery. This method does not require system shutdown and can provide real-time data on rotor behavior.

The scientific article presented by Hongjun W. and colleagues [10] introduces a method for diagnosing rotor system faults using orbit data. This method converts vibration data into a high-dimensional space and then reduces it using the ISOMAP algorithm. The resulting data are used to identify different rotor operating conditions. The method effectively differentiates between normal, misaligned, and rub-impact states without requiring prior knowledge of specific fault characteristics.

Yao et al. [11] examined the vibration characteristics of pre-twisted rotating Ti-SiC composite airfoil blades. They employed a combination of the Mori-Tanaka method and Finite Element analysis to calculate material properties, ensuring reliability of their calculations. Their probabilistic approach to analyzing various parameters (such as rotating speed and aspect ratio), and their well-established model, accurately predict the dynamic behaviors of the rotating system. This has significant guiding value for the optimization of materials and geometric sizes in rotor design.

Yang et al. [12] studied vibration in a four-disk flexible hollow shaft rotor system under disk unbalance and bearing pulse excitation. They developed a simple finite element model using Lagrangian methods and Timoshenko beam theory, verified against 3D FEM. Their analysis of disk position effects on vibration sensitivity and the impact of in-shaft damping provides valuable insights for aero-engine rotor design and vibration control.

The Monte Carlo method used in the article offers a robust way to account for uncertainties in rotor parameters, providing a probabilistic view of rotor behavior. This is particularly useful for understanding the range of possible outcomes in complex rotor systems. The machine learning technique offers a more general approach for non-linear systems, potentially applicable to a wider range of machinery but requiring a substantial amount of training data, prolonging the analysis process [13]. Modal analysis and wavelet transform methods provide specific insights into rotor behavior, with modal analysis offering real-time capabilities and wavelet transforms, in conjunction with FFT, excelling at detecting transient phenomena.

The main aim of this work is to quantify and compare the differences in critical speeds and orbital patterns between the two rotor configurations. By doing so, we provide insights that can inform more robust design practices for rotating machinery, particularly in applications where off-center or variable mass distributions are unavoidable.

Our findings highlight the significant impact of disk position on rotor behavior, demonstrating how off-center placement can lead to more complex orbital patterns and shifted critical speeds. These results have important implications for the design and operation of rotating machinery, particularly in applications where the precise control of rotor dynamics is crucial for performance and safety [14]. The choice between the methods mentioned above would depend on the specific application and the available resources. If a comprehensive analysis of multiple and combined faults is required, the approach by Saucedo-Dorantes et al. [7] might be more suitable. However, if computational resources are limited, or the focus is on vibration-based fault detection, the method presented in this article could be a viable alternative.

2. System Equations of Motion

The Laval/Jeffcott rotor is a fundamental model in rotordynamics, comprising a flexible shaft with a centrally located rigid disk (Figure 1). Despite its simplicity, this model effectively captures critical dynamic behaviors such as critical speeds, unbalance response, and whirling [15]. As a foundational element in the field, it serves as a basis for understanding more complex rotor systems and is extensively employed in both theoretical and practical engineering applications.

Assuming the rotor disk has no impact on the stiffness of the weightless shaft, the lateral bending stiffness at the midpoint of a uniformly supported beam with consistent properties can be expressed as follows:

$$k={\displaystyle \frac{48EI}{L^3}}$$

(1)

where: E is the Young’s modulus of the shaft,

L is the shaft length between the bearings,

I is the shaft area moment of inertia.

For the off-centered disk, or disk in the arbitrary position on shaft:

$$k={\displaystyle \frac{3EIL}{c^2{d}^2}}$$

(2)

where: c and d are the distances from the left and right bearing, respectively.

A small amount of viscous damping, represented by the constant cs, affects the disk’s lateral movement. This damping results from a combination of factors including shaft flexibility, fluid resistance within the machine, and bearing friction. By applying Newton’s laws to the rotor disk, assuming the shaft has negligible mass, we can derive equations describing the disk’s lateral motion in both the horizontal (z) and vertical (y) directions. These equations, and their inputs visualized in Figure 1 and Figure 2, account for the disk’s inertia and the forces produced by the shaft’s deformation. They can be written as:

$$m{\ddot{z}}_T=-b{\dot{z}}_H-k{z}_H$$

(3a)

$$m{\ddot{y}}_T=-b{\dot{y}}_H-k{y}_H$$

(3b)

where: m is the disk mass,

b is the damping coefficient,

k is the stiffness coefficient,

z_H, y_H, z_T, y_T are the co-ordinates of the geometric center and mass center.

Figure 2. Disk of a single mass Jeffcott/Laval rotor.

Figure 2. Disk of a single mass Jeffcott/Laval rotor.

The positional co-ordinates of the disk’s mass center can be mathematically represented in relation to its geometric center [16]. These co-ordinates are dynamically influenced by the rotor’s rotational angle, which varies over time.

$$z_T=z_H+\epsilon \cos \left(\omega t\right)$$

(4a)

$$y_T=y_H+\epsilon \sin \left(\omega t\right)$$

(4b)

where: ε is the eccentricity,

ω is the angular velocity,

t is time.

By inserting the second-order derivatives derived from Equation (4a,b) into Equation (3a,b), the equations of motion for the Föppl/Jeffcott rotor can be reformulated in terms of the disk’s geometric center [17,18].

$$m{\ddot{z}}_H+b{\dot{z}}_H+k{z}_H=m\epsilon {\omega }^2\cos \left(\omega t\right)$$

(5a)

$$m{\ddot{y}}_H+b{\dot{y}}_H+k{y}_H=m\epsilon {\omega }^2\sin \left(\omega t\right)$$

(5b)

It is important to note that this model simplifies the rotor system by excluding gyroscopic effects. This simplification is due to assumptions of infinitely stiff bearings and a rigid disk without tilt. Consequently, the shaft is fixed and always aligned with the bearing centerline.

3. Methodology

In our previous study [19], we investigated the dynamic behavior of two rotor configurations: a centrally mounted unbalanced disk (Rotor05un) and an off-center unbalanced disk (Rotor025un). We employed a Jeffcott rotor system model, with each rotor simplified as an identical shaft-disc with an eccentric mass. This model comprised a flexible steel shaft with uniform circular cross-section, a rigid disk with a mass imbalance, SKF 61805 deep groove ball bearings, and aluminum shaft shackles. The bearings incorporated stiffness, non-linearity, and realistic contact behavior. The aluminum shaft shackles acted as a shaft adapter between the bearings and the shaft. The study develops a method to estimate critical velocity for flexible rotors with uncertain parameters. It highlights the impact of mass distribution on rotor behavior and demonstrates the value of probabilistic analysis in understanding rotor dynamics. While the method is effective, it is computationally intensive and relies on accurate model assumptions.

Building upon the previously validated numerical model, this study focuses on shaft orbital analysis incorporating uncertainty. Experimental validation was conducted for a rotational speed of 377 [rad/s] (60 [Hz]) with a 6.1 [g] unbalance mass. Comparison of experimental and simulated orbital trajectories (Figure 3) revealed good agreement, indicating that the numerical model captures the essential characteristics of the rotor’s motion reasonably well.

The Monte Carlo simulation process used in this study builds upon the method described in our previous work [19]. This analysis was conducted using Adams Insight, a component of MSC Adams/View software (version 2023.4), which implements the Monte Carlo method for design optimization and uncertainty analysis. This method randomly sets values of the specified design factors for each run of the simulation. The investigation aims to evaluate the impact of real-world variations on the design’s performance.

By conducting over 400 trials, statistical predictions regarding the design’s response can be established. The method’s foundation lies in representing parameters using a Probability Density Function (PDF). The investigation strategy chosen was the Monte Carlo method, and the design type was Full Factorial, which is a comprehensive method. It incorporates every potential combination of levels (number of possible values that can be taken by a design factor) for each design factor. The number of runs required follows a mathematical formula: m^n, where “m” is the number of levels and “n” represents the number of factors. However, it is good to note that, as the values of “m” and “n” escalate, Full Factorial becomes more appropriate for experiments involving a limited number of factors as the number of runs increases exponentially. Based on the specified design type, Adams Insight produces a design matrix. In this study, we identified key design factors affecting dynamics of the analyzed rotor configurations, such as bearing damping, bearing clearance, and mass imbalance. For each parameter, we defined a suitable PDF based on expected real-world variations and manufacturing tolerances. For each run, we recorded key outputs such as critical speeds and orbital characteristics.

Overall methodology can be summarized in these five points:

• Input variables (key design factors) were specified;
• For each input variable, an interval of permissible values was defined;
• Assuming a normal distribution on this interval, a set of values was randomly generated in Adams Insight, which were then put into the computational model;
• A calculation was performed for this set of input variables. The Newmark solver (Adams) was used here because stability of the solution and low computation time were also required with random selection of input variables and repeated computations;
• The resulting set of values was statistically processed in Matlab 2023a in the form of histograms and quantiles.

The probabilistic nature of results allows manufacturers or designers to set more precise tolerances for rotor components. By knowing the likelihood and extent of orbital deviations, they can specify tighter or looser tolerances as needed, potentially reducing costs without compromising performance. This study as a continuation of a previous analysis aims to provide a more comprehensive understanding of rotor dynamics under various conditions and configurations.

4. Results

Two main rotor designs were considered in the analyses, characterized by the position of the disc on the shaft (Figure 4). Specifically, the unbalanced disc located in the central position (Rotor05un) and with the unbalanced disc located off-center position (Rotor025un). These two basic designs are the object of analysis in this paper.

Understanding the correlation between disk placement and system behavior enables informed decisions regarding mass distribution to achieve desired performance objectives.

The rotor unbalance can be considered as a shift of the rotor effective mass by a small distance or as eccentricity from the axis of rotation [18]. This effect could also be considered as the effect of a small weight placed at a radial distance R from the geometric center of the disk. The theory of rotor dynamics and balancing is based on the assumption that the masses causing the unbalance (unbalances) are due to the total mass of the rotor or disk and are quite small [18,20]. For particular rotor configurations where unbalanced mass is present, there will be a matching applied 3 [g] unbalanced mass on a disk radius of 40 [mm] (“radius of unbalance”) in the form of an added M5x20 locking bolt. This weight represents approximately 0.2% of the disc weight.

4.1. Unbalanced Disc in the Rotor Central Position (Rotor05un)

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 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.

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. The top histogram displays the probability distribution of positions along A–A, providing insight into the likelihood of the shaft being at specific horizontal locations. The right-side histogram shows the distribution of vertical deviations from A–A, indicating the spread of points around the cross-section line. To better see the difference between the positions I. and II., Figure 11 illustrates the probability distribution of the shaft’s horizontal positions along the A–A cross-section plane, distinguishing between the left (position I., blue) and right (position II., orange) sides of the orbital. Each side of the distribution roughly follows a normal (Gaussian) pattern, though not perfectly symmetrically. Distributions are asymmetrical. The right side (orange) shows a slightly higher peak probability and a narrower spread compared to the left side (blue).

Figure 12 illustrates the distribution of shaft positions along the vertical cross-section B–B, for the lower and upper semicircles of the orbital, respectively. The central scatterplot in each figure represents individual points from multiple shaft orbitals. Here, the horizontal axis shows deviation from B–B, while the vertical axis represents the position along B–B. The top histogram displays the probability distribution of horizontal deviations from B–B, showing the lateral spread of the shaft’s position. The right-side histogram shows the distribution of positions along B–B, indicating the likelihood of the shaft being at specific vertical locations.

The histogram in Figure 13 shows the probability distribution of the shaft’s vertical positions along the B–B cross-section plane. The upper and lower distributions show notable asymmetry. The lower distribution (yellow) exhibits a broader, more uniform spread, while the upper distribution (purple) is more concentrated with a higher peak probability. The asymmetry and non-uniform spread suggest the presence of anisotropic forces or structural asymmetries in the rotor system. The more uniform spread in the lower semicircle might indicate greater variability in shaft position, possibly due to gravitational effects or system-specific factors.

The probability distributions (Figure 11 and Figure 13) are not perfectly symmetric or identical, which could indicate slight imbalances or asymmetries in the rotor system The similarity between horizontal and vertical distributions suggests a nearly circular orbit. The clear separation between peaks in both directions indicates a well-defined orbital path with minimal deviation.

The plots in Figure 14 assess the normality of the shaft’s position distribution in both horizontal (A–A) and vertical (B–B) cross-sections. This figure presents four normal probability plots, each corresponding to a different quadrant of the shaft’s orbital motion at critical speed.

All four plots in Figure 14 show a generally linear trend, indicating that the shaft’s horizontal and vertical positions approximately follow a normal distribution. Plots also display some deviation from perfect linearity, particularly at the extremes, which is common in real-world data. These deviations might indicate the presence of outliers or suggest that the underlying distributions have slightly heavier tails than a perfect normal distribution.

4.2. Unbalanced Disc in Off-Center Position (Rotor025un)

Figure 15 shows the configuration of a rotor with an unbalanced disk positioned off-center, at a distance of ¼ of the bearing span. As the shaft deflects, the disk tilts relative to the axis of rotation, which may exhibit a stronger influence of the gyroscopic effect. This can potentially change the system’s stiffness and “shift” the critical speed to higher values [17,20,22]. The gyroscopic effect arises due to the misalignment between the disk’s axis of rotation and the shaft’s axis of rotation.

The scatter plot in Figure 16 shows the critical speed calculated independently for numerous individual simulations (“trials”). The horizontal axis of the scatter plot represents the number of individual simulation runs. The critical speed ranges from 1100 to 1200 [rad/s], with an average value of 1146.1 [rad/s], which can be seen in the attached histogram (Figure 16). The dispersion of critical speed is approximately 8.3%, or ±4.15% relative to the average value of 1146.1 [rad/s]. The corresponding rotational frequency, which is approximately 182 [Hz], is shown on the secondary y–axis. Compared to the previous configuration with a centrally located unbalanced disk (Figure 5), this configuration shows greater sensitivity to the unbalanced additional mass by approximately 2% within the range of critical speed values. The histogram exhibits (for the evaluated number of simulation runs) two peaks of critical speed, one around 1140 [rad/s] and the other in the region of 1160 [rad/s], suggesting a bimodal probability distribution (Figure 16).

Figure 17 illustrates the probability distribution of the shaft’s orbital positions for the Rotor025un configuration. The 3D projection (Figure 17a) provides a volumetric visualization of the probability density, with height and color representing probability. The top view (Figure 17b) reveals a ring-like pattern of probabilities, where the axes represent horizontal and vertical displacements in millimeters. The color scale in both figures indicates probability density.

This visualization confirms the circular orbit observed in a previous analysis of Rotor05un (Figure 7) and suggests a stable oscillation amplitude throughout the rotation. The highest probability concentrates in the central circle of the ring, indicating a well-defined orbital path with limited dispersion. In comparison with the Rotor05un configuration, this figure shows a smaller orbital diameter. The probability is more evenly distributed around the ring, likely due to the wider range of calculated critical speeds. This more uniform distribution suggests a more consistent orbital behavior despite the off-center disk placement, highlighting the complex dynamics introduced by the disk’s position.

A contour plot (Figure 18) illustrates the probability distribution of the shaft’s orbital positions for the Rotor025un configuration, where the unbalanced disk is mounted off-center at a distance of ¼ of the bearing span. The overall circular shape indicates that the shaft’s orbit remains predominantly circular, despite the asymmetric disk placement. Interestingly, the distribution appears largely symmetrical, suggesting a balanced behavior that counters intuitive expectations for an off-center configuration. The plot reveals multiple concentric rings of varying probabilities, representing a complex, layered structure in the shaft’s orbital behavior.

For the cross-sectional analysis of the Rotor025un configuration, we applied the same logic as in Figure 9. Four areas were identified (I, II, III, IV), corresponding to the left and right sides of the horizontal cross-section (A–A), and the lower and upper parts of the vertical cross-section (B–B). These areas are evaluated separately to provide a detailed understanding of the shaft’s orbital behavior in different quadrants.

Figure 19 represents the horizontal (A–A) cross-section analysis of the Rotor025un shaft orbital, where the disk is mounted off-center at 1/4 of the shaft length. It shows Position I (a) and Position II (b) of the orbit. Both positions display roughly normal distributions of horizontal positions. Position I spans from about −2.6 [mm] to −2.1 [mm], and Position II ranges from 0.65 [mm] to 1.15 [mm]. The vertical spread is similar for both positions.

Figure 20 presents a side-by-side comparison of the probability distributions for the left (blue) and right (orange) sides of the horizontal cross-section in the Rotor025un configuration. Both distributions show a roughly normal shape, but with a higher spike around 0.85 [mm] and the sign of bimodal distribution of the second (right) position. The right-side distribution appears slightly more skewed compared to the left (to the axis of rotation).

Figure 21 shows the vertical (B–B) cross-section analysis for the lower (Position III) and upper (Position IV) parts of the Rotor025un shaft orbital. Position III ranges from about −2.65 [mm] to −2.15 [mm], while Position IV spans from 0.6 [mm] to 1.1 [mm]. Both exhibit roughly normal distributions with asymmetry. The horizontal distribution remains similar.

Figure 22 represents a side-by-side comparison of the vertical position distributions for the lower (yellow) and upper (purple) parts of the orbit. Both distributions have an approximate shape of normal distribution and are noticeably skewed towards the central axis of rotation.

The characteristic orbital patterns identified for different configurations can aid in fault diagnosis. Unusual orbital behaviors can be more accurately linked to specific issues based on our probabilistic findings.

The boxplots (Figure 23 and Figure 24) reveal a wider orbital pattern for Rotor05un across both horizontal and vertical planes. The median positions are farther from the rotational axis, indicating a larger overall orbital diameter. The data also show a higher degree of symmetry in orbital behavior. Additionally, there is a greater presence of outliers compared to Rotor025un. These characteristics suggest a more dynamic orbital motion for Rotor05un.

In contrast, Rotor025un demonstrates a narrower orbital pattern, particularly in positions I and III. The median positions are closer to the rotational axis, implying a smaller orbital diameter. The data suggest a slight asymmetry in orbital behavior, likely influenced by the off-center disk placement. Outliers are less prominent compared to Rotor05un. Overall, Rotor025un exhibits a more constrained and less variable orbital motion.

Table 1. Quantiles of shaft orbital positions [mm] for Rotor05un and Rotor025un configurations across horizontal (A–A) and vertical (B–B) cross-sections.

Rotor Configuration	Cross-Section and Position	Quantile
		2.5 [%]	25 [%]	50 [%]	75 [%]	97.5 [%]
Rotor05un	A–A (horizontal) Left	−2.5781	−2.4677	−2.4152	−2.3597	−2.2495
	A–A (horizontal) Right	0.7926	0.9022	0.9581	1.01222	1.1244
	B–B (vertical) Lower	−2.6144	−2.5105	−2.4553	−2.4007	−2.2904
	B–B (vertical) Upper	0.7576	0.8663	0.9219	0.9766	1.0806
Rotor025un	A–A (horizontal) Left	−2.5200	−2.4010	−2.3285	−2.2818	−2.1843
	A–A (horizontal) Right	0.7321	0.8291	0.8770	0.9504	1.0725
	B–B (vertical) Lower	−2.5637	−2.4415	−2.3699	−2.3212	−2.2217
	B–B (vertical) Upper	0.6837	0.7849	0.8330	0.9065	1.0273

presents a detailed quantitative comparison of shaft orbital positions for the Rotor05un and Rotor025un configurations across horizontal (A–A) and vertical (B–B) cross-sections.

Each subplot on Figure 25 shows the relationship between quantiles (2.5%, 25%, 50%, 75%, and 97.5%) on the x-axis and the corresponding displacement in millimeters on the y-axis. Both configurations show similar trends across all positions, indicating consistent behavior. Rotor05un generally exhibits larger displacements compared to Rotor025un, particularly at higher quantiles.

Quantile analysis of orbital behavior helps establish more accurate operational limits. For example, the 97.5% quantile values could be used to set vibration alarm thresholds, ensuring safe operation while minimizing false alarms.

5. Conclusions

The Rotor05un configuration, with a centrally mounted disk, exhibits larger orbital diameters across all positions. Its boxplots and quantile analysis show wider ranges and more symmetrical distributions between left/right and upper/lower positions. The interquartile ranges are larger, indicating more variability in shaft positions. Quantile plots reveal curved trends, suggesting non-linear behavior in orbital positions across different percentiles. This configuration demonstrates consistent patterns in both horizontal and vertical planes.

The Rotor025un configuration, featuring an off-center disk, displays smaller orbital diameters in all positions. Boxplots and quantile analysis reveal narrower ranges and slight asymmetries between positions. Interquartile ranges are smaller, indicating more concentrated orbital paths. Quantile plots show more linear trends, particularly in horizontal and lower vertical positions. The upper vertical position exhibits a unique curved trend, suggesting complex dynamics due to the off-center disk placement and gravitational effects.

The comparison between Rotor05un and Rotor025un reveals significant differences in orbital behavior, potentially linked to their critical speeds. Rotor025un consistently shows smaller orbital radii and more concentrated distributions, suggesting a higher critical speed due to increased system stiffness from the off-center disk placement. This configuration’s more linear quantile trends indicate more predictable behavior near the critical speed. Conversely, Rotor05un’s larger orbits and curved quantile trends suggest lower stiffness and potentially lower critical speed. The asymmetries observed in Rotor025un, particularly in the vertical plane, may result from complex interactions between the off-center mass, gyroscopic effects, and the system’s natural frequencies, influencing its response as it approaches and passes through the critical speed.

The novelty of our orbital pattern analysis lies in its comprehensive approach to understanding rotor behavior. We introduce a probabilistic method to visualize and quantify shaft orbital positions, revealing subtle variations often missed by traditional approaches. Our cross-sectional analysis of horizontal and vertical planes uncovers asymmetries and directional dependencies in rotor behavior. We proposed a quantile-based comparison method, providing insights into how disk positioning affects rotor dynamics across different probability levels. We also introduce innovative visualization techniques, including 3D probability distributions and contour plots, for the intuitive interpretation of complex data. This detailed orbital analysis offers a more accurate representation of real-world rotor behavior, enhances our understanding of rotor dynamics, and provides practical tools for optimizing rotor design and predicting performance under various conditions.

The random input variables, i.e., randomly selected for the simulation, were the damping and clearance in the bearings, and the added unbalanced mass to the disc. Other input parameters, such as geometry and material data for the shaft, were deterministic in the presented work. Admittedly, the simulation can be performed with a larger number of random input variables.

Our findings have direct practical applications to rotor system monitoring and design. We determined the orbital, i.e., the area where orbit points can occur. The orbital was described probabilistically, allowing us to draw conclusions about the probability of rotor deflection (in this case, at critical speeds). If we use this deflection information for Condition Monitoring, we can estimate the probability of a detected state occurring, which allows us to quantify the risk of an incorrect diagnostic conclusion. When configuring a rotor monitoring system based on eddy current probes, knowledge of the estimated probability of rotor deflection enables us to better select the sensor required for the configured system.