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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript investigated the dynamic characteristics of two rotor configurations, first one is a centrally mounted unbalanced disk (Rotor05un) and another one is an off-center unbalanced disk (Rotor025un). The probability distribution of vibration under different working conditions was analyzed by numerical simulation and Monte Carlo simulation. And the authors need to answer the following questions and make manuscript revisions before agreeing to publish it.

Some specific comments are given below.

 

1. Please unify the writing format of the reference literature, and the novelty of the references is not prominent enough,

2. The process of Monte Carlo simulation and calculation is not given in the manuscript, and it is suggested that this relevant information should be supplemented.

3. In the description of Figure 25 in lines 387 to 391, "Rotor05un generally exhibits larger displacements compared to Rotor025un, particularly at higher quantiles." However, the first subgraph of (a) and the first subgraph of (b) are not what you described. Please give a reasonable explanation.

4. The introduction needs improvement by reviewing more related works. For example, Applied Mathematical Modelling, 128, p392-409, 2024. International Journal of Mechanical Sciences, 268, 109051, 2024. Therefore, literature survey is not sufficient to present the most updated status for further justification of the originality of the manuscript.

Comments on the Quality of English Language

English writting should be improved.

Author Response

We would like to thank the reviewers for their time and expertise. Their comments have been valuable in refining our manuscript.

This manuscript investigated the dynamic characteristics of two rotor configurations, first one is a centrally mounted unbalanced disk (Rotor05un) and another one is an off-center unbalanced disk (Rotor025un). The probability distribution of vibration under different working conditions was analyzed by numerical simulation and Monte Carlo simulation. And the authors need to answer the following questions and make manuscript revisions before agreeing to publish it.

Some specific comments are given below.

 

1. Please unify the writing format of the reference literature, and the novelty of the references is not prominent enough.

References updated.

2. The process of Monte Carlo simulation and calculation is not given in the manuscript, and it is suggested that this relevant information should be supplemented.

Information regarding Monte Carlo simulation supplemented in the chapter 3. Methodology.

3. In the description of Figure 25 in lines 387 to 391, "Rotor05un generally exhibits larger displacements compared to Rotor025un, particularly at higher quantiles." However, the first subgraph of (a) and the first subgraph of (b) are not what you described. Please give a reasonable explanation.

We appreciate the reviewer's careful observation regarding Figure 25. The first subgraphs of both a) and b) show negative displacements (negative direction to the coordina sysytem). In this context, a larger displacement should be understood as the one with greater absolute value. This observation indicates that the off-center disk in Rotor025un actually results in smaller absolute displacements in these positions.

4. The introduction needs improvement by reviewing more related works. For example, Applied Mathematical Modelling, 128, p392-409, 2024. International Journal of Mechanical Sciences, 268, 109051, 2024. Therefore, literature survey is not sufficient to present the most updated status for further justification of the originality of the manuscript.

Introduction expanded to include suggested, more recent and relevant studies.

Reviewer 2 Report

Comments and Suggestions for Authors

The study aims to investigate how disk position and unbalance affect the dynamic behavior of rotor systems to improve the design and efficiency of rotating machinery. The authors need to address the following recommendations to improve the quality of this article.

1.      The study appears to build on well-established concepts in rotor dynamics, such as the Jeffcott rotor model and Monte Carlo simulations. While these are robust methods, the paper could be enhanced by clearly stating what new insights or advancements it offers compared to previous studies. What is the unique contribution of this work that has not been addressed by prior research?

2.      The authors should highlight the novelty of the study in the abstract.

3.      The authors need to address the need for this study clearly, and how this study overcomes the challenges of the previous research?

4.      Please add the overall proposed methodology in section 3.

5.      The methodology section is too small. Enlarge it.

6.      The paper employs Monte Carlo simulations to account for uncertainties, which is a solid approach. However, the source and nature of these uncertainties could be better defined. Are these uncertainties based on manufacturing tolerances, operational conditions, or another factor?

7.      The paper could benefit from a more thorough explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

8.      The authors should add more explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

9.      The probabilistic results, such as the probability distributions of shaft orbital positions, are interesting but could be better connected to practical applications. How do these findings inform the design or operation of real-world rotary systems?

 

10. The authors can add a more detailed future work description and address the limitations of the current study.

Comments on the Quality of English Language

None. 

Author Response

We would like to thank the reviewers for their time and expertise. Their comments have been valuable in refining our manuscript.

Comments and Suggestions for Authors

The study aims to investigate how disk position and unbalance affect the dynamic behavior of rotor systems to improve the design and efficiency of rotating machinery. The authors need to address the following recommendations to improve the quality of this article.

1. The study appears to build on well-established concepts in rotor dynamics, such as the Jeffcott rotor model and Monte Carlo simulations. While these are robust methods, the paper could be enhanced by clearly stating what new insights or advancements it offers compared to previous studies. What is the unique contribution of this work that has not been addressed by prior research?

Contribution and the novelty additionaly mentioned in conclusion.

2. The authors should highlight the novelty of the study in the abstract.

Novelty mentioned in the abstract and in the conclusion.

3. The authors need to address the need for this study clearly, and how this study overcomes the challenges of the previous research?

Stated as a novelty in conclusion.

4. Please add the overall proposed methodology in section 3.

Methodology with description of the tools and methods used added.

5. The methodology section is too small. Enlarge it.

Methodology updated.

6. The paper employs Monte Carlo simulations to account for uncertainties, which is a solid approach. However, the source and nature of these uncertainties could be better defined. Are these uncertainties based on manufacturing tolerances, operational conditions, or another factor?

Additionaly mentioned in the chapter 3. Methodology .

7. The paper could benefit from a more thorough explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

Experimental device and its description referenced and mentioned in the article.

8. The authors should add more explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

Comparison of numerical and experimental results are shown in Fig. 3. 

9. The probabilistic results, such as the probability distributions of shaft orbital positions, are interesting but could be better connected to practical applications. How do these findings inform the design or operation of real-world rotary systems?

 Usefulness of probabilistic results mentioned in chapter 3. Methodology, chapter 4. Results (after Fig. 22) and quantiles mentioned with example (after Fig .25).

10. The authors can add a more detailed future work description and address the limitations of the current study.

Limitations of the study and comparison with other methods are mentioned in the last paragraph of the introduction.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

It can be accepted for publication.

Author Response

Thank you for your review.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have made the recommended changes in the manuscript.  However, for future revisions, it would be helpful if the authors could include detailed responses to each review point in the response to the reviewer’s file, rather than just a single line. This would make it easier for reviewers to track and verify how the recommended changes were addressed throughout the manuscript.

Author Response

We apologize for our oversight in not providing detailed responses to each review point. We fully recognize that this is a fundamental aspect of the revision process, and we regret any inconvenience this has caused. Our previous brief response was not meant to be dismissive or offensive in any way, and we regret if it came across as such.

We appreciate the time and effort our reviewers have volunteered, and we understand the importance of their work.

We prepared a comprehensive response addressing each point raised by Reviewer #2, clearly indicating where and how we have implemented the suggested changes in the manuscript.

Thank you for your patience.

1.The study appears to build on well-established concepts in rotor dynamics, such as the Jeffcott rotor model and Monte Carlo simulations. While these are robust methods, the paper could be enhanced by clearly stating what new insights or advancements it offers compared to previous studies. What is the unique contribution of this work that has not been addressed by prior research?

Novelty and and contribution to research mentioned in contribution (lines 482 – 506). The main points of contributions can be summarized in these 4 bulletpoints:

• Probabilistic method for visualizing and quantifying shaft orbital positions
• Cross-sectional analysis (revealing asymmetries and directional dependencies)
• Quantile-based comparison method for assessing disk positioning effects
• Visualization techniques including 3D probability distributions and contour plots

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 intuitive interpretation of complex data. This detailed orbital analysis offering 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 input random 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 in 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 in-formation 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.

2. The authors should highlight the novelty of the study in the abstract.

Novelty of this research mentioned in abstract aswell (lines 17 - 19): „Quantile analysis revealed approximate linear trends for Rotor025un, suggesting higher system stiffness and more predictable behavior near critical speeds.“ (22 - 26) „The study goes beyond traditional critical speed analysis to examine orbital patterns and point on orbit occurrence from a probabilistic perspective. (Additionaly mentioned in the abstract in the second round of reviews) 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, offering insights into the complex interaction behavior of chosen rotor configurations...“

3. The authors need to address the need for this study clearly, and how this study overcomes the challenges of the previous research?

The simulation incorporated random variables such as bearing damping, clearance, and unbalanced disk mass, with potential for expansion to more variables. This is additionaly stated in Conclusion (lines 493 – 497): „The input random 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.“

In other words, this basically leads to practical applications that include:

• Improved rotor system monitoring and design
• Probabilistic description of rotor deflection at critical speeds
• Enhanced risk quantification for diagnostic conclusions
• Better selection of sensors for rotor monitoring systems

4. Please add the overall proposed methodology in section 3.

Overall methodology stated and summarized in 5 points (lines 198 - 208) with additional information about solver used for the simulation:

1. Input variables (key design factors) were specified.
2. For each input variable, an interval of permissible values was defined.
3. 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.
4. 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.
5. The resulting set of values was statistically processed in Matlab in the form of histograms and quantiles.

 

5. The methodology section is too small. Enlarge it.

Methodology enlarged about additional information such as model assembly (lines 160 - 164): „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, nonlinearity, and realistic contact behavior. The aluminum shaft shackles acted as a shaft adapter between the bearings and the shaft.“

Also mentioned and refferenced previous article [19]. Monte Carlo simulation is also elaborated in a way that describe how i tis set up in the used software (lines 177 - 197): „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, 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). Investigation Strategy chosen to be Monte Carlo method and Design Type—Full Factorial. It 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 analysed 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.“

6. The paper employs Monte Carlo simulations to account for uncertainties, which is a solid approach. However, the source and nature of these uncertainties could be better defined. Are these uncertainties based on manufacturing tolerances, operational conditions, or another factor?

Uncertainties mentioned in the lines 193 – 195: „In this study, we identified key design factors affecting dynamics of the analysed rotor configurations such as: bearing damping, bearing clearance, and mass imbalance.“

Also additionaly in the conclusion (lines 493 - 497): „The input random 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“

7. The paper could benefit from a more thorough explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

Experimental device was thoroughly described in the previous publication so the refference [19] was added to the text. Short description of the model based of experimental device/rig was additionaly provided in Methodology (lines 160 - 164): „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, nonlinearity, and realistic contact behavior. The aluminum shaft shackles acted as a shaft adapter between the bearings and the shaft.“

8. The authors should add more explanation of the experimental setup, including how closely it replicates the conditions assumed in the numerical model.

Comparison of numerical and experimental results are shown in Fig. 3. with corrected refference (from  [16] to [19]).  Exact simmilarity was not quantified in this study.

9. The probabilistic results, such as the probability distributions of shaft orbital positions, are interesting but could be better connected to practical applications. How do these findings inform the design or operation of real-world rotary systems?

Practical applications and usefulness mentioned in the Methodology (lines 209 -212): „The probabilistic nature of results allows manufacturers or designers 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.“

Results: (lines 220 - 222) „Understanding the correlation between disk placement and system behavior enables informed decisions regarding mass distribution to achieve desired performance objectives.“ (lines 423 - 425) „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.“ & (lines 454 - 456): „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.“

This is also mentioned in the Conclusion (lines 498 - 506).

10. The authors can add a more detailed future work description and address the limitations of the current study.

Limitations of study and comparison with other methods (such as Machine learning) used by other authors in their studies is mentioned in the introduction (lines 103 - 112)