Review Reports - Load-Sensitive Tire–Road Friction Modeling and Dynamic Stability Analysis of Multi-Axle Trucks

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Authors investigate the important problem of integrating dynamic axle load transfer and load-sensitive friction characteristics into safe speed estimation for multi-axle trucks negotiating curves. However, it can not be accept in current paper. Here are some suggestions:

1. Line 42.” they neglect dynamic load transfer and the load sensitivity of tire–road friction, both of which play a crucial role in real driving conditions.”. Can the load-sensitive friction be simply understood as the coefficient of friction decreasing with the increase of the load? If not, please provide the definition of the load-sensitive friction, as well as the factors influencing its variation, and give a brief analysis thereof.

2. It is recommended to incorporate a descriptive schematic diagram in Section 2.2 to facilitate reader comprehension.

3. Figure 3: The labels and variables in Figure 3 are excessively small and suffer from low resolution.

4. Line 138. It is recommended to supplement the selection of the value 0.8 with relevant references to facilitate readers' consultation.

Author Response

Comment 1: Line 42.” they neglect dynamic load transfer and the load sensitivity of tire–road friction, both of which play a crucial role in real driving conditions.”. Can the load-sensitive friction be simply understood as the coefficient of friction decreasing with the increase of the load? If not, please provide the definition of the load-sensitive friction, as well as the factors influencing its variation, and give a brief analysis thereof.

Response 1: We appreciate the reviewer’s insightful comment. The load-sensitive friction used in our study refers to the experimentally observed nonlinear decrease in the tire–road friction coefficient with increasing normal load. This effect arises from the combined influence of tire–rubber viscoelasticity, contact pressure distribution, and thermo-mechanical coupling at the tire–road interface. To clarify this, the following explanation and references have been added in Section 2.3 (Line 148-160)

Comment 2: It is recommended to incorporate a descriptive schematic diagram in Section 2.2 to facilitate reader comprehension.

Response 2: Thank you for the suggestion. A schematic diagram has now been added as Figure 1 in Section 2.2, and the corresponding text has been revised to introduce the figure before the equations for improved readability.

Comment 3: Figure 3: The labels and variables in Figure 3 are excessively small and suffer from low resolution.

Response 3: We sincerely appreciate the reviewer’s helpful comment. In the revised manuscript, the font sizes of all labels and variables in Figure 3 have been increased to improve readability.

Comment 4: Line 138. It is recommended to supplement the selection of the value 0.8 with relevant references to facilitate readers' consultation.

Response 4: The rationale for selecting the rollover threshold LTRₘₐₓ = 0.8 has been clarified in Section 2.4 (Line 188-190). A supporting reference has been added, citing prior rollover-warning studies in which this threshold is commonly used. The added reference appears as Ref. [22] in the revised manuscript.
[22] Wang, R.; Xu, X.; Chen, S.; Guo, N.; Yu, Z. Vehicle rollover warning and control based on attitude detection and fuzzy PID. Appl. Sci. 2023, 13(7), 4339.

Reviewer 2 Report

Comments and Suggestions for Authors

see attached file

Comments for author File: Comments.pdf

Author Response

This study explicitly considers dynamic axle load transfer and load-sensitive tire-road friction characteristics in estimating the safe speed of multi-axle trucks negotiating curved road segments. It contains some interesting points those are valuable to readers working in this field, but several key issues should be addressed.

Comment 1: Section 1, what the necessity of using TruckSim Simulink environment for the implementation is could be further mentioned;

Response 1: We thank the reviewer for this helpful suggestion. In the revised manuscript, an additional sentence has been added at the end of the Introduction to clarify the necessity of using the TruckSim–Simulink co-simulation environment. Specifically, the following text has been included (Line 101-104):

“It is implemented within a TruckSim–Simulink co-simulation environment, which couples realistic multi-axle vehicle dynamics with the proposed friction and control logics, enabling physically consistent evaluation of dynamic load transfer and stability on curved roads.”

This brief addition explains that TruckSim provides a high-fidelity nonlinear multi-axle vehicle model, while Simulink allows integration of custom control and friction modules, thereby ensuring physically consistent evaluation under curved-road conditions.

Comment 2: for the books cited, the pages referred by this study could be given;

Response 2: We appreciate the reviewer’s suggestion. The cited books have now been supplemented with the corresponding page ranges and chapters referred to in this study. These pages describe the theoretical basis of tire–road interaction and nonlinear tire force generation, which are directly relevant to our friction-sensitivity modeling. The updated references are listed as [5]–[7] in the revised manuscript.

Comment 3: for Section 5. Conclusions, it should be more concise by showing the most important highlight, further, one paragraph is fine.

Response 3: We thank the reviewer for this valuable suggestion. The Conclusion section has been revised to be more concise and presented as a single paragraph summarizing the main findings and contributions of this study. The revised version emphasizes the essential outcomes—namely, the effectiveness of the proposed dynamic-load-sensitive framework for predicting rollover and skidding limits and its implications for real-time safety control of autonomous heavy trucks—while removing redundant descriptions.

Reviewer 3 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

General concept comments.

Comment 1 - Abstract: Authors should follow MDPI's established abstract writing style.

Response 1 - Thank you for the helpful recommendation. The abstract has been fully revised to follow MDPI’s structured format

Comment 2 - Keywords: We recommend that the keywords are specific to the article, yet reasonably common within the subject discipline. Pay attention to keywords: “Dynamic load transfer” (four occurrence in the text), “Safe speed estimation” (seven occurrences in the text), “Mult axle truck dynamics” (one occurrence in the text), “Autonomous heavy vehicles” (one occurrence in the text).

Response 2 - We agree with the reviewer. The keywords have been updated to reflect the core terminology used throughout the manuscript and to ensure relevance within the heavy-vehicle safety domain: Load-sensitive friction; Dynamic load transfer; Safe speed estimation; TruckSim–Simulink co-simulation; Static rollover threshold (SRT)

Comment 3 - Introduction: The problem addressed in the paper is clearly defined. The authors aimed to develop a more accurate method for evaluating safe speeds, one that accounts for changes in wheel loading and friction depending on load conditions. This issue is particularly pertinent to automotive industries, especially concerning autonomous vehicles and road safety. While many studies exist regarding truck safety, most previous works did not consider the impact of dynamic load variations or their effect on friction. Although the hypothesis about friction’s sensitivity to load is accepted without question, it is important to note that the relationship between friction and load varies across different surfaces and operational environments.

Response 3 - We appreciate the reviewer’s insightful comment. We agree that the friction–load relationship observed in experimental studies is not universally constant and may differ depending on pavement type and environmental conditions. To clarify this limitation, we have added a brief statement in the Introduction (Lines 75–76), noting that the load-induced change in friction can vary with surface and environment. This revision acknowledges that the μ–load model adopted in this study represents a generalized formulation rather than a surface-specific one.

In addition, Section 2.3 has been slightly expanded to include a more explicit physical explanation of load-sensitive friction and the factors contributing to its variation (Lines 148–160).

Comment 4 - Materials & Methods: The methodology appears comprehensive and technologically grounded. Using recognized software tools such as TruckSim and Simulink, quantitative assessments were carried out, leading to recommendations for improved speed control algorithms. Despite reconstructing realistic crash scenarios, there remains ambiguity surrounding the precise set of control parameters influencing system behavior (for example, inclination angles of the road or track width).

Response 4 - We thank the reviewer for the helpful comment. The physical and geometric parameters used in the simulation—including axle track width, CG height, axle load distribution ratio, and roll-stiffness ratio—were not arbitrarily selected, but were obtained from the accident vehicle dataset reported in Reference [11]. This has already been stated in Section 2.2, and Table 2 summarizes the complete set of parameters used in the model. To improve clarity, we have added an explicit sentence indicating that these values are directly derived from the referenced vehicle data. (Lines 266-269)

Comment 5 - Discussion: There is no mention of the accuracy of the proposed model nor any evaluation of its reliability and validity relative to existing methods for assessing traffic safety.

Response 5 - We appreciate the reviewer’s valuable comment. To the best of our knowledge, no previous study has explicitly formulated or applied a load-sensitive friction model to multi-axle truck stability or safe-speed estimation, so a direct comparison with existing approaches was not feasible.
Instead, we verified the model’s physical validity and numerical accuracy through internal consistency and empirical reproduction of experimental data. The proposed μ(N) formulation, calibrated with the results of Segel and Ervin [11], successfully reproduced the observed friction–load relationship, while the TruckSim–Simulink co-simulation results showed consistent variations in safe speed (≈7.5 km/h) and lateral offset (up to 0.52 m), supporting its validity. These clarifications have been added to the revised Discussion section (Lines 408-414)

Comment 6 - Conclusions: The results contribute significantly to our current understanding of truck safety on winding roads, emphasizing the necessity of integrating dynamic load and friction parameters. Most conclusions are supported by modeling outcomes but leave open questions regarding the applicability of the model to additional realworld scenarios and diverse vehicle types and road configurations.

Response 6 - We thank the reviewer for this valuable comment. We agree that clarifying the applicability of the proposed framework to a wider range of road and vehicle conditions is important. In the revised manuscript, the Conclusions section has been updated to explicitly acknowledge the current scope and to state the planned extensions of the model. The revised paragraph now specifies that the present framework assumes real-time availability of friction information and does not yet account for uncertainty in μ, stochastic road variation, or frequency-dependent tire behavior. It further states that future work will include online estimation of load and friction parameters, extension to different road geometries and operating conditions, and benchmarking against additional vehicle configurations.

Comment 7 - Lines 31-40: The authors claim that their results highlight the need to include dynamic load distribution and load-sensitive friction in evaluations of safe truck speeds. However, they do not provide citations from specific regulatory documents governing road design standards.

Response 7 - Thank you for the helpful comment. We have revised the Introduction to explicitly cite the latest AASHTO Green Book (7th ed., 2018) and FHWA design guidance, both of which specify fixed side-friction factors for curve-speed calculation and do not incorporate dynamic axle-load transfer or load-sensitive friction. This clarification has been added in Lines 56–67 of the revised manuscript, and Reference [2] has been updated from the 6th to the 7th edition accordingly.

Comment 8 - Lines 41-49: More detail could be added to describe the methods for quantifying dynamic wheel load and its effects on friction, specifying exactly what metrics were utilized for measurement purposes.

Response 8 - Thank you for the suggestion. We have revised Section 2.1 (Lines 123-128) to explicitly define the quantitative metrics used in evaluating stability—namely, the maximum LTR for rollover and the minimum wheel-level μ(Nw) derived from the load-sensitive friction model for skidding. This clarification now specifies how dynamic wheel loads are quantified and how they affect friction-based stability limits.

Comment 9 - Line 87: Why does friction reduction appear stronger for heavy-truck tires than passenger car tires? Additional explanations or references supporting this assertion are required.

Response 9 - Thank you for the comment. We have revised Section 3.3 to clarify why heavy-truck tires exhibit a stronger decrease in friction with increasing load. Segel and Ervin [10] provide the experimental evidence for the different μ–N trends, and an additional reference [24] has been included to explain that this behavior is consistent with established rubber-friction theory, where contact-area saturation and reduced adhesion under higher normal force accelerate friction loss. This mechanism is more pronounced in truck tires due to their higher inflation pressure and stiffer carcass construction. The corresponding explanation and citation have been added in Lines 340–348

Comment 10 - Line 210: Units of measure are missing for some table entries (such as air density or frontal area resistance), making interpretation difficult.

Response 10 - Thank you for the comment. The units for all dimensional quantities in Table 2 have now been checked and consistently included (e.g., air density in kg/m³, drag area in m², acceleration limit in m/s²). Dimensionless parameters are shown with an em dash (—), and this has been clarified in a table footnote. The revised version appears in Table 2

Comment 11 - Line 290: If possible, please elaborate further on the figure description, as the current title is insufficient for proper visualization comprehension.

Response 11 - Thank you for the helpful comment. We have reviewed the figure captions and expanded those that were too brief to ensure that each figure can be interpreted without referring back to the main text. In particular, the caption of Figure 6 has been updated to clarify how different values of the load-sensitive friction parameter c affect wheel-level friction loss and lateral deviation during ramp driving, allowing the results to be understood more clearly. The revised caption appears in the updated manuscript (Lines 316–321).

Comment 12 - Lines 335-340: Proposals for including load-sensitive friction models in realtime speed control algorithms require refinement and specification. How precise is the necessary input data, and what technical limitations must be taken into account?

Response 12 - Thank you for the constructive comment. We agree that the practical implementation of load-sensitive friction models in real-time speed control requires clarification of data requirements and technical constraints. Accordingly, the Conclusions section has been revised to explicitly state that the current framework assumes real-time availability of friction information obtained through external sensing or estimation, and that it does not yet address uncertainties in μ, stochastic variation in pavement conditions, or frequency-dependent tire behavior under oscillatory loading. The revised text also notes that future work will focus on online estimation of wheel loads and friction with sufficient accuracy for real-time control, as well as on identifying limitations related to sensor noise, estimation latency, and model uncertainty. These clarifications have been added in the revised manuscript (Lines 438–452).

Reviewer 4 Report

Comments and Suggestions for Authors

I think this approach simplifies the problem, where it is necessary not only to define the constant safe speed, but to determine the safe maneuvering in order to achieve the given task on sliding road. So the problem will belong rather to optimal control than to simple mechanics. Moreover, since the sliding level usually not known in advance the problem needs to take into account the possible observations from various sensors. So this class of problems belong rather to stochastic control.

Author Response

Comment: I think this approach simplifies the problem, where it is necessary not only to define the constant safe speed, but to determine the safe maneuvering in order to achieve the given task on sliding road. So the problem will belong rather to optimal control than to simple mechanics. Moreover, since the sliding level usually not known in advance the problem needs to take into account the possible observations from various sensors. So this class of problems belong rather to stochastic control.

Response: Thank you for the constructive comment. We agree that real-time maneuvering on low-friction or uncertain road surfaces ultimately requires optimal or stochastic control formulations that incorporate sensor-based friction estimation and uncertainty handling. However, the scope of the present study is limited to establishing a deterministic and physically consistent upper-level safe-speed boundary based on dynamic axle-load transfer and load-sensitive friction. In other words, the proposed model is intended to serve as a foundational speed-feasibility layer that can later be integrated into optimal or stochastic control architectures, rather than a complete closed-loop controller by itself. To clarify this point, we have revised the Conclusion to explicitly state the scope of the current work and to note that future studies will address controller-level implementation, online load/friction estimation, and stochastic uncertainties. (Lines 439-453)

Reviewer 5 Report

Comments and Suggestions for Authors

Dear Authors,

Your study presents a valuable modeling framework integrating load-sensitive friction, dynamic load transfer, and rollover analysis in multi-axle trucks. However, the paper requires major revision before acceptance. While the technical framework is strong, several sections lack sufficient background, justification, and broader contextualization. There are also methodological details that should be expanded for reproducibility and interpretability.

The Introduction should be expanded to emphasize the broader safety relevance of tire–road interaction modeling. Currently, it mainly focuses on truck dynamics and friction sensitivity. The societal implications of tire behavior under varying loads, especially its role in road accidents, should be highlighted. More or less, tire load sensitivity is a general traffic safety concern, not limited to trucks. Additionally, note that tire frequency-response modeling can enhance μ(N)-based models, particularly for transient or oscillatory load variations (for example doi: 10.17531/ein/163289). Together, these additions overview will position your work within a broader vehicle dynamics and safety framework, highlighting the real-world implications of tire–load interactions.

The co-simulation results are well-presented but should include quantitative validation against experimental or accident reconstruction data beyond reference [11]. Consider adding a comparison with empirical rollover speeds or measured lateral accelerations from controlled experiments or real-world datasets.

The selection of SRT levels (0.3 g, 0.5 g, 0.7 g) is reasonable but should include references or experimental justification for these exact thresholds.

Equations (6)–(9) are central to the study but would benefit from a more detailed derivation and parameter rationale.

The discussion could be enhanced by addressing frequency-domain implications and dynamic stiffness of tires, connecting to the suggested reference on frequency response. It would also strengthen the paper to outline how the proposed framework could be adapted for different road surfaces.

Author Response

Comment 1 - The Introduction should be expanded to emphasize the broader safety relevance of tire–road interaction modeling. Currently, it mainly focuses on truck dynamics and friction sensitivity. The societal implications of tire behavior under varying loads, especially its role in road accidents, should be highlighted. More or less, tire load sensitivity is a general traffic safety concern, not limited to trucks.

Response 1 - Thank you for the valuable comment. We agree that the Introduction originally focused too narrowly on heavy-truck dynamics. To address this, we have added a new contextual paragraph in the beginning of the Introduction (Lines 29–44), clarifying that load-sensitive friction and friction-related loss-of-control events affect all vehicle types, not only trucks. This revision expands the safety motivation before transitioning to the truck-specific discussion.

Comment 2 - Additionally, note that tire frequency-response modeling can enhance μ(N)-based models, particularly for transient or oscillatory load variations (for example doi: 10.17531/ein/163289). Together, these additions overview will position your work within a broader vehicle dynamics and safety framework, highlighting the real-world implications of tire–load interactions.

Response 2 - Thank you for the valuable suggestion. We agree that safe-speed estimation under unknown or time-varying friction conditions can lead to formulations that involve sensor-based estimation, stochastic modeling, and optimal control. To clarify the scope of the present work and to avoid the unintended impression that friction was treated as a fixed constant, we have revised the conclusion accordingly. The updated text now explicitly states that the framework assumes real-time friction information is supplied by an external sensing or estimation module, and that uncertainty in μ, stochastic road-condition variation, and optimal/stochastic control strategies are left for future extensions. (Lines 439-453)

Comment 3 - The co-simulation results are well-presented but should include quantitative validation against experimental or accident reconstruction data beyond reference [11]. Consider adding a comparison with empirical rollover speeds or measured lateral accelerations from controlled experiments or real-world datasets.

Response 3 - Thank you for this constructive suggestion. We fully agree that additional empirical validation would further strengthen the applicability of the proposed framework. However, publicly available datasets that simultaneously include axle-load variation and load-sensitive friction characteristics for multi-axle heavy trucks are extremely limited, and full-scale rollover experiments are beyond the practical scope of this work due to safety, equipment, and cost constraints. For this reason, the present study validates the method using a reconstructed real accident case, which is a commonly adopted approach in the literature when such data availability limitations exist. Expanding the validation to multiple empirical datasets remains an important direction for future research, and this has been clarified in the revised conclusion section. We sincerely appreciate the reviewer’s helpful insight.

Comment 4 - The selection of SRT levels (0.3 g, 0.5 g, 0.7 g) is reasonable but should include references or experimental justification for these exact thresholds.

Response 4 - Thank you for the insightful comment. We agree that the justification for the selected SRT levels should be explicitly supported by published data. Accordingly, we have revised the corresponding paragraph in Section 2.4 (Lines 196-209) to clarify that the three SRT values (0.3 g, 0.5 g, and 0.7 g) were chosen based on rollover threshold ranges reported in UMTRI datasets and subsequent experimental studies. These datasets show that fully loaded tank or container semitrailers typically fall within 0.25–0.45 g, medium-load multi-axle trucks around 0.45–0.55 g, and lightly loaded rigid trucks or SUVs up to 0.8–1.2 g. The revised text now states that the selected values represent the lower, mid, and upper portions of the practical stability spectrum observed in heavy-vehicle configurations.

Comment 5 - Equations (6)–(9) are central to the study but would benefit from a more detailed derivation and parameter rationale.

Response 5 - Thank you for the helpful comment. We agree that additional clarification was needed regarding the physical basis and rationale behind Equations (6)–(9). Accordingly, we have revised Section 2.3 (Load-Sensitive Friction and Skid Constraint) to (i) provide explicit experimental justification for the load-sensitive friction model, (ii) cite the foundational literature from which the empirical power-law form is derived, and (iii) explain the physical role of each equation in the force–balance formulation. This clarification now appears in the paragraph preceding Eq. (6), where supporting studies such as Segel & Ervin [10], Liu et al. [9], Wong [5], Rajamani [6], Pacejka [7], and Fortunato et al. [24] are cited to establish the empirical and theoretical basis of the adopted model. We have also added brief explanatory sentences after Eqs. (7)–(9) to clarify how the friction ellipse constraint, load-proportional lateral force allocation, and residual longitudinal capacity are sequentially used to determine the skid limit. (Lines 148-178)

Comment 6 - The discussion could be enhanced by addressing frequency-domain implications and dynamic stiffness of tires, connecting to the suggested reference on frequency response. It would also strengthen the paper to outline how the proposed framework could be adapted for different road surfaces.

Response 6 - We appreciate the reviewer’s suggestion regarding the relevance of frequency-response-based tire models. As advised, we have now explicitly acknowledged this limitation and indicated the extension path in the revised Conclusions section, where frequency-dependent tire behavior and oscillatory loading are stated as part of future research directions. (Lines 439-453)

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

n/a

Reviewer 5 Report

Comments and Suggestions for Authors

Dear Authors,

Thank you for your detailed revisions and thoughtful responses to the reviewers’ comments. The corrections have strengthened the manuscript, and I find it suitable for publication.

Best Regards.