A Novel Overlapped Compensation Structure and Its Effectiveness Verification for Expansion Joints in Plate-Type PMEDS Vehicles

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the annotated pdf file for the scope of improvement of the paper before publishing. Thanks

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

Thank you sincerely for your time and thoughtful evaluation of our manuscript. We deeply appreciate your expertise and the constructive feedback you provided, which has helped us refine and strengthen our work. Below, we address each of your comments and suggestions point by point, incorporating your insights into the revised manuscript where applicable. Please find our detailed responses outlined sequentially.

 

 

Comments 1: Is there any relevant literature about factor affecting temporal variation of such expansion joints? (Page 2, end of first paragraph)

Response 1: As suggested, we have revised the manuscript to incorporate relevant references on expansion joints (S. Tan, C. Lin, and Q. Mei, "Study on the Operational Status of Rail Expansion Joints in Ballasted Track of High-Speed Railways," Journal of Railway Science and Engineering, vol. 18, no. 4, pp. 837–843, 2021.) Drawing on the cited literature's analysis of temporal variation characteristics in high-speed rail expansion joints, we extrapolate the temporal variation behavior of PMEDS expansion joints, thereby reinforcing the theoretical foundation of this study.

 

Comments 2: Why disrupted? please elaborate a bit here? (Page 2, second paragraph, seventh and eighth lines)

Response 2: The term "disrupt" was rigorously chosen to capture the nonlinear electromagnetic dynamics arising from air gaps (conductivity≈0) at expansion joints. Specifically, when the conductive plate’s continuity is interrupted, the induced eddy current fields experience vortex fragmentation due to abrupt spatial discontinuities in current paths, coupled with magnetic flux leakage caused by increased reluctance at the air gap. This disruption directly destabilizes the quasi-static equilibrium of eddy currents.

The revised manuscript here is modified to read: When PMs traverse these expansion joints during guideway operation, the air gaps at the joints (where conductivity approaches zero) disrupt induced eddy current fields in the conductive plates, resulting in stepwise changes in levitation force

 

Comments 3: Is it necessary to write sentences about descriptions of sections in a research paper? (First paragraph under the section heading of section 2)

Response 3: The inclusion of brief escriptions of the research structure at the beginning of each section in this paper aims to provide readers with a clear logical framework and guidance on the research trajectory. For interdisciplinary readers in particular, the section descriptions reduce barriers to comprehension, enabling them to more easily identify key substance.

 

Comments 4: Means no discontinuity? (Figure 4)

Response 4: Yes, "complete conductive plate" denotes the absence of expansion joints (no air gaps). This is an idealized model conceptualized to facilitate analytical investigation.

 

Comments 5: Please provide some references if the equations are from some text book. (Equation 2)

Response 5: As requested, we have added the reference (T Cao, H Shi, J Liu, X Wu and Z Deng, “Investigation of Electromagnetic Force Characteristics of Permanent Magnet Electrodynamic Suspension under Multi-Operation Conditions,” Transactions of China Electrotechnical Society, vol. 39, no. 17, pp. 5262-5277, Sept. 2024.) to explicitly address the governing equations for each subdomain in the PMEDS quasi-static system. The cited study validates the boundary condition assumptions and provides benchmarks for the coupling mechanisms between eddy current redistribution and force discontinuities, thereby reinforcing the mathematical consistency of our model.

 

Comments 6: Why is it a range? & Why varying length? (Table 1 & Figure 6)

Response 6: In practical PMEDS operation, the length of expansion joints inherently varies (as described in the text). Therefore, this parameter was analyzed as a range to systematically investigate how levitation force characteristics respond to its variation.

 

Comments 7: What is meant by during expansion joint? (The first paragraph below Equation 10)

Response 7: "During expansion joint" refers to the moment when Halbach array magnets traverse the expansion joint gap, inducing transient electromagnetic field discontinuities. This issue is explained in the revised manuscript.

 

Comments 8: Is the Eddy current distribution same for all the values of Le? (Figure 10)

Response 8: No, eddy current distributions vary with expansion joint length (Le). However, qualitatively, straight-joint completely truncate eddy currents for all Le, while overlap-joint maintain continuous redistribution via overlapping conductive paths.

 

Comments 9: please expand Le as length of expansion joint in the title of subsection. (3.2. Analysis and Optimization of Le)

Response 9: The subsection headings have been revised as requested (e.g., " Analysis and Optimization of length of expansion joint"), and all similar instances throughout the text have been standardized in the revised manuscript accordingly to ensure consistency.

 

Comments 10: Is there any simulation to show the thermal induced displacements with different Le? (Second line of the first paragraph below Figure 11)

Response 10: Thermally induced displacements due to conductive plate expansion/contraction (described in Section 1) inherently alter length of expansion joint (Le). While Le variations depend on material properties, nominal conductive plate length, and temperature gradients, simulating these multi-parametric engineering trade-offs would deviate from our core focus on electromagnetic discontinuity mechanisms. We instead emphasize generalized analytical relationships between Le and levitation force attenuation rate (LAR), prioritizing methodological insights over application-specific thermal simulation.

 

Comments 11: Then why don't plot pole pitch versus LAR? (Figure 12)

Response 11: Figure 12 focuses on determining the critical balanced overlap length (beyond which further increases yield diminishing returns). As for the relationship between LAR and PMs pole pitch, Figure 11 explicitly demonstrates that LAR decreases as pole pitch increases while other parameters remain constant. under a fixed length of expansion joint.

 

Comments 12: Is the minimum 120 mm? Please quantify. (The third line of the first paragraph below Figure 12)

Response 12: The 120 mm is not the minimum pole pitch. In the text, the PMs pole pitch varies from 120 mm to 240 mm to investigate the variation patterns of balanced overlap length under different pole pitches. The original description in this section was inaccurate and has been revised in the manuscript. The modified version reads: "Therefore, given a fixed PMs pole pitch, the expansion-joint design must ensure Lo is not less than the corresponding balanced overlap length, while accounting for manufacturing cost constraints."

 

Comments 13: Is it possible to show the skin depth at each speed? (The end of the first paragraph below Figure 13)

Response 13: In our article, the relationship between skin depth and speed has been analytically established through Equations (5) and (6). These equations demonstrate that the skin depth decreases with increasing speed. Since this trend can be directly derived from the theoretical framework, it is unnecessary to present the skin depth values at each speed.

 

Comments 14: Why base is also on the sideways with conductor plate in fig 14? (Figure 14)

Response 14: The PMEDS system with only the bottom conductive plate inherently constitutes a vertical undamped system. In practical applications, damping terms must necessarily be considered. The side conductive plates are specifically introduced to incorporate damping terms into the vertical dynamic analysis, thereby enabling the derivation of results that align more closely with practical scenarios.

 

Comments 15: Please correct to figure 15b (Equation 12)

Response 15: This error has been corrected in the revised manuscript.

 

Comments 16: Isn't it better to plot delta h vs time for better interpretation? (Figure 16.)

Response 16: As suggested, Figure 16 has been revised to display Δh as a function of time, and the corresponding descriptions in the text have been updated to reflect this modification.

 

Comments 17: Is it a drawing or actual image of the set up? (Figure 18)

Response 17: In Figure 18(b), which is an actual image of the setup, the caption has been revised to "Detailed image" in the revised manuscript.

 

Comments 18: It is not clear how the top plate is emulating different expansion joint configurations? (Figure 19)

Response 18: This section has been revised in the manuscript with the following expanded description: "The conductive plate above the magnetic wheel is rigidly mounted via the support plate. The outer circular holes on the insulating support plate are bolted to the upper force sensor, while the four central oblong slots are designed to accommodate different types of conductive plates. The conductive plates are bolted through their circular holes to the oblong slots on the support plate, enabling rapid replacement and allowing adjustment of the expansion joint length by sliding the bolts within the oblong slots. This configuration facilitates the simulation of various conductive plate arrangements under operational conditions."

 

Comments 19: does support plate exist in real world application? will it interfere with the levitation effect? why there are 4 grooves inside? (Figure 19)

Response 19: The support plate does not exist in practical applications. In the experimental setup, the support plate is made of Bakelite (a non-conductive material) to ensure it does not affect the levitation performance.

The four internal rectangular holes (not grooves) in the support plate are designed to allow adjustment of the expansion joint length during experiments by sliding the bolts inserted through them. This mechanism enables precise configuration of conductive plates with expansion joints to simulate varying operational conditions.

 

Comments 20: Did the author vary the angle for figure 19c configuration? (Figure 19)

Response 20: The angle variation in the Figure 19(c) configuration was not systematically investigated. It is worth noting that straight-joint can be equivalently regarded as angled joints with a 90° orientation. Magnetic wheel equivalence experiments indicate that angled-joint marginally outperform straight-joint, but their performance improvement remains limited compared to lap joints. Crucially, regardless of the angle adjustment in the Figure 19(c) configuration, eddy currents between the plates remain fully interrupted. Furthermore, angled-joint entails significantly higher manufacturing costs and exhibit lower economic practicality. For these reasons, this configuration was included solely as a comparative benchmark rather than a focus of in-depth study.

 

Comments 21: Is it good to decrease the levitation force or high force is good for the trains?

Response 21: For PMEDS trains, a higher levitation force is preferable. Greater levitation force directly correlates with improved load-bearing capacity and enhanced vertical vibration stiffness, both of which are essential for ensuring operational stability and system robustness under dynamic conditions.

 

Comments 22: What do the authors mean by minimizing force reduction? It is the LAR which is important for stability, not the levitation force itself, right? (Figure 21)

Response 22: In the magnetic wheel equivalent experiment, the PMs continuously passes over expansion joints, and the levitation force measured by the sensor corresponds to the minimum force experienced during joint traversal. When comparing the levitation force measured across expansion joints to that of the complete conductive plate, a greater reduction in levitation force directly correlates with a higher LAR. There is no conflict between LAR and levitation force itself for stability.

 

Comments 23: The force is found to vary with similar standard deviation in each design of junction. Why? If the LAR is not affected, then why don't we have straight joint with high levitation force itself? (Figure 21)

Response 23: In the magnetic wheel equivalent experiment, all tested expansion joint configurations exhibit comparable standard deviations in force variations. This uniformity arises from two factors:

1. The limited sampling frequency of the force sensor inherently restricts temporal resolution of force fluctuations.

2     As the magnet continuously traverses expansion joints during the test, the measured force essentially represents persistent oscillations around the minimum force value during joint traversal.

By comparing these measurements with the levitation force obtained from the complete conductive plate, the attenuation characteristics of different expansion joint types become evident.

 

Comments 24: earlier authors did not show angled-seam simulation results in figures *, 9 11, 12 or 13. Is it only done during experiments? (Figure 22)

Response 24: To emphasize the advantages of overlap-joint over straight-joint, the angled-joint configuration was intentionally omitted in the preceding analysis. Angled-joint was only introduced and analyzed as a comparative configuration in the magnetic wheel equivalent experiment.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper a novel overlapped compensation structure and Its effectiveness verification for expansion joint in plate-type vehicle is considered. A novel
overlapped compensation structure is proposed and its effectiveness is verified via simulation and experiment. The over-lapped compensation structure is modeled and the comparative time-varying levitation force is analyzed under different guide ways. The optimal structure specifications covering overlapped thickness
and length are studied. Dynamic analysis confirms the effect of the proposed overlapped compensation structure. The results are proved experimentally. It is  obtained that  the overlapped compensation structure can reduce levitation force attenuation by 20%. This result work is expected to be practically applied.

The paper has novelty in comparison to the previous published models. The applied model and method is good explained. The analytic results are compared with experimental ones. Good agreement is obtained. The results are widely discussed.

Number of figures is sufficient.

The Reference list is OK.

Author Response

Response to Reviewer:

Thank you for your positive evaluation of our manuscript and for acknowledging its novelty, methodology, and experimental validation. We sincerely appreciate your time and constructive feedback, which reinforces the significance of this work. Your support motivates us to further explore practical applications of the proposed structure. We are honored to contribute to this field and look forward to future opportunities for collaboration and innovation.

Reviewer 3 Report

Comments and Suggestions for Authors

actuators-3532631:

A Novel Overlapped Compensation Structure and Its Effectiveness Verification for Expansion Joint in Plate-Type PMEDS Vehicle

 

     I appreciate the elaborate effort on the comparison of straight-joint and overlap-joint.  But some points are ambiguous for me to understand the detail.  Please respond to the following questions and requests, and improve the manuscript for us to understand easily.

 

Abstract

 

1. Introduction

 

2. Structure and Principle
   • Structures of Expansion Joints

(Figure 2) The border of “Below overlap plate” and “Upper overlap plate” is difficult to discriminate.  Please draw their frame border in black color.

 

• Analytical Model of PMEDS with Overlapped Compensation Structure

(equation 1) Which region do you mean by “surface current region” and “air domain”?

Please cite a reference about this equation 1.

 

(Figure 4) Which region do you mean by “region I” and “region III”?

What do you mean by “\cent \tilde{n}”, “\cent \acute{o}”, and “\cent \grave{o}”?  What do you mean by “\pounds_1” and “\pounds_2”?  Are they typos or character corruptions?

 

(lines 1-2 under the figure 4) What do you mean by “conduction currents”?  Do you mean “displacement current”?

 

(equation 2) Please correct the partial differential operators “\partial_x”, “\partial_y”, and “\partial_z”. (“x”, “y”, and “z” are not subscripts.)

Please cite a reference about this equation 2.

 

(line 1 just after the equation 2) \mu 0 --> \mu_0 (“0” should be a subscript.)

 

(equation 4) What do you mean by “T_1” and “T_2”?

 

(equation 5) What do you mean by “\tau”?  Do you mean some pitch?

Please cite a reference about this equation 5.

 

(Figure 5) The contact face is insulated electrically between “Upper overlap plate” and “Below overlap plate”?

What do you mean by “\cent \tilde{n}”, “\cent \acute{o}”, and “\cent \grave{o}”?  What do you mean by “\pounds_1”, “\pounds_2”, “\pounds_3”, “\pounds_5”, and “\pounds_6”?  Are they typos or character corruptions?

 

3. Electromagnetic Characteristics and Optimization of Overlapped Compensation
   • Effectiveness Analysis of Overlapped Compensation Structure

(Table 1) The correspondence of the Values between the variable in Figure 5 and the parameter in this Table 1 is ambiguous.  Please write the variable name in such a way as “\ell_p: Magnet length”, “d: Conductive plate thickness”, “L_e: Length of expansion joint”, “L_o: Length of overlap”, for example, in this Table 1.

 

(Figure 6) How did you solve the equation 2 to obtain this figure result?  Did you solve the partial differential equations by ANSIS or COMSOL?

The sub-tick marks on the horizontal axis are not integer.  Please correct them.

When Time (ms) = 21, for example, how is the PMEDS configuration with straight-joint?  (Where is the straight-joint in Figure 4 at t = 21 ms?)

Why is the curve asymmetric weakly with respect to the minimum-levitation-force time even in case of “straight-joint”? 

 

(line 1 just after the equation 10) Please don’t put an indentation before “Where”.  (Because the sentence is not in the new paragraph.)

 

(Figure 8) Time (mm) --> Time (ms)

Why is the curve asymmetric strongly with respect to the minimum-levitation-force time in case of “Overlap-joint”? 

When Time (ms) = 22, for example, how is the PMEDS configuration with overlap-joint?  (Where is the overlap-joint in Figure 4 at t = 22 ms?)

Which direction is the PMEDS vehicle running?

How is the asymmetry if the “Upper overlap plate” is on the left-hand-side and the “Below overlap plate” is on the right-hand-side?

 

• Analysis and Optimization of Le

 

• Analysis and Optimization of Lo

(this sub-section number) 3.2 --> 3.3

 

• Analysis and Optimization of Tu

(this sub-section number) 3.3 --> 3.4

 

4. Dynamic Comparative Simulation
   • Vertical Dynamic Analysis of PMEDS Vehicle

 

• Comparison of Dynamic Performance

 

5. Magnetic Wheel Equivalence Experiment

(line 1 just after the figure 19) What do you mean by the “T” at the beginning of the first sentence?  Is it a typo?

 

(Table 2) The correspondence of the Values between the variable in Figure 5 (Figure 18) and the parameter in this Table 2 is ambiguous.  Please write the variable name in such a way as “d: Conductive plate thickness”, “L_e: Length of expansion joint”, “L_o: Length of overlap”, for example, in this Table 2.

 

6. Conclusions

 

References

Author Response

Dear Reviewer,

Thank you sincerely for your time and thoughtful evaluation of our manuscript. We deeply appreciate your expertise and the constructive feedback you provided, which has helped us refine and strengthen our work. Below, we address each of your comments and suggestions point by point, incorporating your insights into the revised manuscript where applicable. Please find our detailed responses outlined sequentially.

Regarding the symbol display issues in Figures 4 and 5: We confirmed that the original files (both Word and PDF) were error-free. However, the problem arose (PDF) after uploading to the journal platform, as verified by downloading the submitted files from the website. To resolve this, we have re-uploaded Figures 4 and 5 as separate high-resolution files in the "Supplementary Image" section of the revised manuscript.

 

 

Comments 1: (Figure 2) The border of “Below overlap plate” and “Upper overlap plate” is difficult to discriminate. Please draw their frame border in black color.

Response 1: As requested, we have revised Figure 2 by drawing black frame borders for both the "Below overlap plate" and "Upper overlap plate" to improve visual distinction.

 

Comments 2: Analytical Model of PMEDS with Overlapped Compensation Structure. (equation 1) Which region do you mean by “surface current region” and “air domain”? Please cite a reference about this equation 1.

Response 2: The "surface current region" refers to the domain where equivalent surface currents are distributed on permanent magnets (PMs), as visually depicted in Figure 3 of the manuscript.  The "air domain" denotes the non-conductive space surrounding the PMs.

Regarding Equation 1, we have added Reference (R. Zhou, G. Li, Q. Wang and J. He, “Torque Calculation of Permanent-Magnet Spherical Motor Based on Permanent-Magnet Surface Current and Lorentz Force,” IEEE Transactions on Magnetics, vol. 56, no. 5, May. 2020, Art no. 8200209.) to explicitly support the theoretical framework.

 

Comments 3: (Figure 4) Which region do you mean by “region I” and “region III”? What do you mean by "ñ"、"ó"、"ò"?  What do you mean by "£₁" and "£₂"? Are they typos or character corruptions?

Response 3: We sincerely apologize for the unintended character encoding errors in Figure 4 caused by formatting inconsistencies during manuscript preparation. The symbols "ñ," "ó," "ò," "£₁," and "£₂" were corrupted during file conversion and have now been corrected: "£₁" and "£₂" should read as T₁ (upper surface) and T₂ (lower surface) of the conductive plate, respectively. Figure 4 has been revised to clarify the spatial definitions: Region I denotes the upper air domain above the conductive plate, Region III represents the lower air domain beneath the plate, and Region II corresponds to the conductive plate itself. Key geometric parameters (permanent magnet length l_p, thickness h_p, levitation height h, and conductive plate thickness d) are now explicitly labeled to align with the analytical framework.

 

Comments 4: (lines 1-2 under the figure 4) What do you mean by “conduction currents”? Do you mean “displacement current”?

Response 4: "Conduction currents" was incorrectly used and has been revised to "displacement current", consistent with Maxwell’s framework for non-conductive regions. The revisions have been made in the first paragraph below Figure 4 in the revised manuscript.

 

Comments 5: (equation 2) Please correct the partial differential operators "∂x"、"∂y"、"∂z". (“x”, “y”, and “z” are not subscripts.) Please cite a reference about this equation 2.

Response 5: The partial differential operators in Equation 2 have been corrected to standard notation to eliminate subscript ambiguities. Additionally, Reference (T Cao, H Shi, J Liu, X Wu and Z Deng, “Investigation of Electromagnetic Force Characteristics of Permanent Magnet Electrodynamic Suspension under Multi-Operation Conditions,” Transactions of China Electrotechnical Society, vol. 39, no. 17, pp. 5262-5277, Sept. 2024.) has been added to validate the electromagnetic force formulation.

 

Comments 6: (line 1 just after the equation 2) "μ0"--> "μ₀" (“0” should be a subscript.)

Response 6: The formatting of μ_₀ at this location has been adjusted in the revised manuscript.

 

Comments 7: (equation 4) What do you mean by "T₁" and "T₂"?

Response 7: In Equation 4, "T₁" and "T₂" denote the upper and lower surfaces of the conductive plate, respectively, as labeled in Figure 4 of the revised manuscript.

 

Comments 8: (equation 5) What do you mean by "τ"? Do you mean some pitch? Please cite a reference about this equation 5.

Response 8: In Equation 5, "τ" represents the pole pitch of the PMs, which defines the spatial periodicity of the magnetic field arrangement. This parameter has been explicitly defined in the first paragraph below Equation 5 within the revised manuscript. As requested, we have cited the following reference to support the theoretical framework of Equation 5: (C. Huang, B. Kou, X. Zhao and X. Niu, “Improved Analytical Model for the Magnetic Drag of a Dual-Conductor Plate Parallel Electric Suspension and Guiding Mechanism,” IEEE Transactions on Industrial Electronics, vol. 70, no. 12, pp. 11994-12002, Dec. 2023.)

 

Comments 9: (Figure 5) The contact face is insulated electrically between “Upper overlap plate” and “Below overlap plate”? What do you mean by "ñ"、"ó"、"ò"、"£₁"、"£₂"、"£₃"、"£₅"、"£₆"? Are they typos or character corruptions?

Response 9: In Figure 5, the electrically insulated contact surface between the "Upper overlap plate" and "Below overlap plate" incorporates a minor gap in the model to emulate real-world operational conditions, as explicitly clarified in the revised first paragraph above Figure 5. Regarding the symbols "ñ," "ó," "ò," "£₁," "£₂," "£₃," "£₅," and "£₆," these were unintended artifacts from file encoding corruption. Shared symbols align with their definitions in the updated Figure 4, while the remaining symbols are now fully explained in the revised text preceding Figure 5. Both the insulation mechanism and symbol definitions have been rigorously addressed in the revised manuscript to ensure clarity and technical accuracy.

 

Comments 10: (Table 1) The correspondence of the Values between the variable in Figure 5 and the parameter in this Table 1 is ambiguous. Please write the variable name in such a way as “lₚ: Magnet length”, “d: Conductive plate thickness”, “Le: Length of expansion joint”, “Lo: Length of overlap”, for example, in this Table 1.

Response 10: Revised Table 1 adopts the suggested format [e.g., Magnet length (l_p)].

 

Comments 11: (Figure 6) How did you solve the equation 2 to obtain this figure result? Did you solve the partial differential equations by ANSIS or COMSOL? The sub-tick marks on the horizontal axis are not integer. Please correct them. When Time (ms) = 21, for example, how is the PMEDS configuration with straight-joint? (Where is the straight-joint in Figure 4 at t = 21 ms?) Why is the curve asymmetric weakly with respect to the minimum-levitation-force time even in case of “straight-joint”?

Response 11: Below is a point-by-point response to your queries regarding Figure 6:

1.Equation Solving Methodology:

The results in Figure 6 were obtained by numerically solving the partial differential equations in Equation 2 using COMSOL. It is now explicitly described in the revised manuscript.

2.Horizontal Axis Sub-Tick Marks:

This has been corrected in the updated Figure 6.

3.PMEDS Straight-Joint Configuration at 21 ms:

At Time = 21 ms (minimum levitation force), the straight-joint configuration aligns the symmetry axis of the magnet array with the geometric center of the expansion joint.

4.Weak Curve Asymmetry:

The slight asymmetry in the force-time curve arises from longitudinal edge effects of the straight-joint. Even in the "straight-joint" case, magnetic flux distortion occurs near the joint’s longitudinal ends due to abrupt permeability transitions.

 

Comments 12: (line 1 just after the equation 10) Please don’t put an indentation before “Where” (Because the sentence is not in the new paragraph.)

Response 12: The indent before "Where" post-Equation (10) has been removed. All similar formatting issues are now standardized in the revised manuscript.

 

Comments 13: (Figure 8) Time (mm) --> Time (ms) Why is the curve asymmetric strongly with respect to the minimum-levitation-force time in case of “Overlap-joint”? When Time (ms) = 22, for example, how is the PMEDS configuration with overlap-joint? (Where is the overlap-joint in Figure 4 at t = 22 ms?) Which direction is the PMEDS vehicle running? How is the asymmetry if the “Upper overlap plate” is on the left-hand-side and the “Below overlap plate” is on the right-hand-side?

Response 13: The "Time (mm)" label in Figure 8 has been corrected to "Time (ms)". For the overlap-joint case, the asymmetric levitation force curve relative to the minimum-force point (Time = 22 ms) arises from the Halbach array's motion in the +x-direction: the right-side overlap-joint exhibits a larger effective levitation height than the left-side joint, amplifying left-side force attenuation. Due to the Halbach array's strict symmetry, reversing the overlap configuration (upper plate left/lower plate right) would produce a mirror-image force curve.

The issues raised about Figure 8 have been addressed in the revised manuscript, with detailed explanations added in the two paragraphs above Figure 8.

 

Comments 14: Analysis and Optimization of Lo (this sub-section number) 3.2 --> 3.3 Analysis and Optimization of Tu (this sub-section number) 3.3 --> 3.4

Response 14: The sub-sections and all similar numbering errors have been thoroughly reviewed and corrected in the revised manuscript.

 

Comments 15: (line 1 just after the figure 19) What do you mean by the “T” at the beginning of the first sentence? Is it a typo?

Response 15: The letter "T" at the beginning of the first sentence following Figure 19 was an inadvertent typographical error and has been removed in the revised manuscript.

 

Comments 16: (Table 2) The correspondence of the Values between the variable in Figure 5 (Figure 18) and the parameter in this Table 2 is ambiguous. Please write the variable name in such a way as “d: Conductive plate thickness”, “Le: Length of expansion joint”, “Lo: Length of overlap”, for example, in this Table 2.

Response 16: Revised Table 2 adopts the suggested format [e.g., Conductive plate thickness (d)].

Reviewer 4 Report

Comments and Suggestions for Authors

The well-known problem of attenuation in the levitation force due to the eddy current truncation effect causing the permanent electrodynamic suspension (PMEDS) is addressed. An overlapped compensation structure comprising straight-joint pieces in conductive plate, where a Halbach array assists the PMEDS configuration with overlap-joint. Time-domain levitation force characteristics in straight-joint under varying lengths of expansion joint as well as dependence of levitation force attenuation rate on length of expansion joint, are given. Comparison of time-domain curves of the levitation force under different expansion is provided while the spatial distribution of the eddy currents is simulated. The several simulations involving levitation height response and vertical acceleration characteristics are also accompanied by magnetic wheel equivalence experiment, where time-domain measurements verify the data obtained via simulations.

It is a lengthy but complete work which deals with a real-world problem from many aspects (theory fitting models, sims, measurements); rarely we see such extensive studies in MDPI Actuators journal. The only element that is missing is a stronger novelty statement. The authors are clearly knowledgeable and should be able to devote a larger part of their work in explaining why their approach is beneficial and provide compare/contrast with other competing techniques.

In addition, in Figure 10, the different behavior in the spatial distribution of the two eddy currents (at straight and overlap joint) is not physically interpreted. The authors should explain the different nature of peaks as well as commenting on the possibility of using spherical scatterers isolated or in layers [1,2] to reduce that unwanted effect.    

[1] Single-series solution to the radiation of loop antenna in the presence of a conducting sphere, Progress In Electromagnetics Research, 2007.

[2] Eddy currents and conducting and magnetizable spherical inclusions fields in a nonmagnetic medium, Russian Journal of Nondestructive Testing, 2016.

Author Response

Dear Reviewer,

Thank you sincerely for your time and thoughtful evaluation of our manuscript. We deeply appreciate your expertise and the constructive feedback you provided, which has helped us refine and strengthen our work. Below, we address each of your comments and suggestions point by point, incorporating your insights into the revised manuscript where applicable. Please find our detailed responses outlined sequentially.

Regarding the concern about the weaker novelty statement: This work primarily focuses on structural innovation and analysis. Compared to traditional straight-joint conductive plates, we propose a novel overlap-joint to optimize the dynamic stability of the PMEDS system when traversing expansion joints. Additionally, in the eddy current analysis of Figure 10, we discussed the potential applications of spherical scatterers and cited the two references you recommended ([33,34]) to contextualize this exploration.

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

actuators-3532631R1:

A Novel Overlapped Compensation Structure and Its Effectiveness Verification for Expansion Joint in Plate-Type PMEDS Vehicle

 

     I appreciate the elaborate effort on the comparison of straight-joint and overlap-joint.  But only one point is ambiguous for me to understand the detail.  Please respond to the following question and requests, and improve the manuscript for me to understand easily.

 

Abstract

 

1. Introduction

 

2. Structure and Principle
   • Structures of Expansion Joints

 

• Analytical Model of PMEDS with Overlapped Compensation Structure

(equation 1) Which region do you mean by “surface current region” and “air domain”?

Please cite a reference about this equation 1.

Response 2: The "surface current region" refers to the domain where equivalent surface currents are distributed on permanent magnets (PMs), as visually depicted in Figure 3 of the manuscript. The "air domain" denotes the non-conductive space surrounding the PMs.

Regarding Equation 1, we have added Reference (R. Zhou, G. Li, Q. Wang and J. He, “Torque Calculation of Permanent-Magnet Spherical Motor Based on Permanent-Magnet Surface Current and Lorentz Force,” IEEE Transactions on Magnetics, vol. 56, no. 5, May. 2020, Art no. 8200209.) to explicitly support the theoretical framework.

(Further comment) The equation number (1) has been lost from the manuscript.  Please recover the equation number (1).

 

3. Electromagnetic Characteristics and Optimization of Overlapped Compensation
   • Effectiveness Analysis of Overlapped Compensation Structure

(Table 1) The correspondence of the Values between the variable in Figure 5 and the parameter in this Table 1 is ambiguous.  Please write the variable name in such a way as “\ell_p: Magnet length”, “d: Conductive plate thickness”, “L_e: Length of expansion joint”, “L_o: Length of overlap”, for example, in this Table 1.

Response 10: Revised Table 1 adopts the suggested format [e.g., Magnet length (l)].

(Further comment) In the revised Table 1, there appears a typo, “thicknes s (d) 12 mm”, which should be corrected as “thickness (d) 12 mm”.

 

(Figure 6) Why is the curve asymmetric weakly with respect to the minimum-levitation-force time even in case of “straight-joint”? 

Response 11:

4. Weak Curve Asymmetry:

The slight asymmetry in the force-time curve arises from longitudinal edge effects of the straight-joint. Even in the "straight-joint" case, magnetic flux distortion occurs near the joint’s longitudinal ends due to abrupt permeability transitions.

(Further comment) I’m afraid that I feel a theoretical jump in your logic about “4. Weak Curve Asymmetry”.  When the Halbach array moves closer to the straight-joint, abrupt permeability transition occurs.  But even when it moves away from the straight-joint, the abrupt permeability transition occurs.  Since the abrupt permeability transitions are symmetric, the force-time curve should be symmetric, I’m afraid.  Or do you mean the resistive-decay of the eddy current due to the permeability transition, when it moves away from the straight-joint?  And the weak curve asymmetry increases as the length of expansion joint (Le) increases?  Could you please explain the reason of the weak curve asymmetry more easily for me to understand?

 

• Analysis and Optimization of Le

 

• Analysis and Optimization of Lo

 

• Analysis and Optimization of Tu

 

4. Dynamic Comparative Simulation
   • Vertical Dynamic Analysis of PMEDS Vehicle

 

• Comparison of Dynamic Performance

 

5. Magnetic Wheel Equivalence Experiment

 

6. Conclusions

 

References

 

Author Response

Comments 2: Analytical Model of PMEDS with Overlapped Compensation Structure. (equation 1) Which region do you mean by “surface current region” and “air domain”? Please cite a reference about this equation 1.

Response 2: The "surface current region" refers to the domain where equivalent surface currents are distributed on permanent magnets (PMs), as visually depicted in Figure 3 of the manuscript.  The "air domain" denotes the non-conductive space surrounding the PMs.

Regarding Equation 1, we have added Reference (R. Zhou, G. Li, Q. Wang and J. He, “Torque Calculation of Permanent-Magnet Spherical Motor Based on Permanent-Magnet Surface Current and Lorentz Force,” IEEE Transactions on Magnetics, vol. 56, no. 5, May. 2020, Art no. 8200209.) to explicitly support the theoretical framework.

 

(Further comment) The equation number (1) has been lost from the manuscript. Please recover the equation number (1).

(Further response): Thank you. We've changed it in the revised manuscript.

 

 

Comments 10: (Table 1) The correspondence of the Values between the variable in Figure 5 and the parameter in this Table 1 is ambiguous. Please write the variable name in such a way as “lₚ: Magnet length”, “d: Conductive plate thickness”, “Le: Length of expansion joint”, “Lo: Length of overlap”, for example, in this Table 1.

Response 10: Revised Table 1 adopts the suggested format [e.g., Magnet length (l_p)].

 

(Further comment) In the revised Table 1, there appears a typo, “thicknes s (d) 12 mm”, which should be corrected as “thickness (d) 12 mm”.

(Further response): Thank you. We've changed it in the revised manuscript.

 

Comments 11: (Figure 6) Why is the curve asymmetric weakly with respect to the minimum-levitation-force time even in case of “straight-joint”?

Response 11:

4.Weak Curve Asymmetry:

The slight asymmetry in the force-time curve arises from longitudinal edge effects of the straight-joint. Even in the "straight-joint" case, magnetic flux distortion occurs near the joint’s longitudinal ends due to abrupt permeability transitions.

 

(Further comment) I’m afraid that I feel a theoretical jump in your logic about “4. Weak Curve Asymmetry”. When the Halbach array moves closer to the straight-joint, abrupt permeability transition occurs.  But even when it moves away from the straight-joint, the abrupt permeability transition occurs.  Since the abrupt permeability transitions are symmetric, the force-time curve should be symmetric, I’m afraid. Or do you mean the resistive-decay of the eddy current due to the permeability transition, when it moves away from the straight-joint? And the weak curve asymmetry increases as the length of expansion joint (Le) increases? Could you please explain the reason of the weak curve asymmetry more easily for me to understand?

(Further response): When the Halbach array approaches the straight-joint, abrupt magnetic field changes induce stronger eddy currents, akin to a sudden surge in friction during emergency braking. When moving away from the straight-joint, although the magnetic field variation is symmetric, resid-ual eddy currents require time to decay (similar to the gradual dissipation of residual heat after brake release). The difference in physical mechanisms between the "generation-decay" processes causes the force-time curve to exhibit a weakly asymmetric profile. As the Le increases, the magnetic field distortion zone expands, equivalent to prolonging the "brake pad contact time," which amplifies the asymmetric characteristics.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have adequately taken into account all the points made in the previous report.

Author Response

We sincerely thank the reviewers for their valuable time and constructive feedback during the peer-review process. Your insightful comments and suggestions significantly improved the quality of this manuscript. The rigorous critique helped us refine our methodology, clarify key arguments, and strengthen the overall presentation of our work. Once again, we deeply appreciate the expertise and dedication of the reviewers, which have been instrumental in advancing this research.

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

actuators-3532631R2:

A Novel Overlapped Compensation Structure and Its Effectiveness Verification for Expansion Joint in Plate-Type PMEDS Vehicle

 

     I appreciate the elaborate effort on the comparison of straight-joint and overlap-joint.  And my questionable points have been all cleared.  Now, I do hope that this manuscript would be published as soon as possible.