Review Reports - Performance Analysis of Nonlinear Stiffness Suspension Based on Multi-Objective Optimization

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper studies quarter-vehicle ISD (inerter-spring-damper) suspensions in parallel and series layouts. A cubic spring nonlinearity controlled by a parameter ε is introduced, models for linear and nonlinear springs are formulated, and a multi-objective GA is used (design variables k1, b, c, ε) under random Class-C road excitation to reduce body acceleration (BA), suspension working space (SWS), and dynamic tire load (DTL). Reported best cases: parallel ISD with ε=0.15 and series ISD with ε=0.11; time- and frequency-domain results and an ε-sensitivity analysis are presented, and headline reductions relative to a passive baseline are given (e.g., parallel nonlinear: BA −6.07%, SWS −10.48%, DTL −3.94%)

The spring force is modeled as with ε stated as a dimensionless parameter in (0,1). As written, the cubic term has units of (N·m²), not N, hence the model is dimensionally inconsistent. Replace by a properly dimensioned cubic stiffness, e.g. (with in N·m³), or clearly define ε as carrying the necessary dimensions and adjust discussion accordingly.
In Sec. 2.1 the vehicle is called “semi-active parallel vehicle suspension,” but only a linear passive model is derived there. Align wording with the actual model presented in each subsection.
A transfer function is introduced (Eq. (3)), then appears inside time-domain equations of motion (Eq. (4)) as “,” which is not a valid time-domain ODE. Provide a consistent formulation (either state-space/ODE or Laplace-domain), or express the series branch through auxiliary states rather than in time-domain equations.
The text first sets ranges kg and N·s·m^-1, but Tab. 2 uses and . Unify these bounds and clarify which set was actually used for the reported optima.
The text announces “Parameters of the GA toolbox are listed in Table 2,” but Tab. 2 lists variable bounds, not GA hyper-parameters (population size, generations, crossover, mutation, etc.).
The configuration is repeatedly called “semi-active,” yet only passive components (inerter, damper, spring) are optimized offline; no online control law is implemented in the models of Sec. 2-4. Either justify the “semi-active” designation (e.g., variable-inertance/variable-damping hardware) or describe the setups as passive/mechatronic ISD networks to avoid confusion.
The introduction claims that suspensions “with nonlinear springs lack a natural frequency,” whereas, in general, they exhibit amplitude-dependent natural frequencies. Rephrase this passage cautiously to avoid a categorical statement.
The text says parallel results are in “Figures 3, 5, 6,” and series results in “Figures 4, 6, 8.” Given the captions, the parallel set should be 3 (BA), 5 (SWS), 7 (DTL); please correct the references and check all figure mentions throughout the paper.
Ensure each figure has a clean, self-contained caption that ends with a period.
Phrases like “Tables should be placed in the main text near to the first time they are cited” should be removed from Tabs. 2-5; ensure all template notes are deleted before submission.
5 states that simultaneous improvements occur for “approximately ” and that further increases reverse/saturate gains, yet the series optimum used elsewhere is ε=0.11. Reconcile the trend description with the reported optimum and the metrics in Tab. 5 (note also that DTL for nonlinear series (859 N) is higher than the linear series case (852 N)).
In Sec. 2, the spring stiffness is , but Eq. (5) uses ; unify notation and ensure all variables in equations are defined at first use with consistent symbols throughout.
The paper states “The authors do not have permission to share data,” which is unusual for simulations using published/assumed parameters. Consider adopting a statement appropriate for simulation studies, and, if possible, sharing code or random-seed settings to enhance reproducibility.
Numerous minor issues (e.g., “stragety” in Ref. 9) should be corrected. Ensure reference formatting is consistent and DOIs are valid/unbroken (e.g., extra space in Ref. 7’s DOI; broken “j.cnk i” in Ref. 13).
To situate the work within the most recent high-order waveguide modeling literature, it is recommended to cite DOI 10.2514/6.2024-3078. That paper introduces variable-kinematics plate/shell finite elements within the Carrera Unified Formulation and extracts dispersion via the Wave Finite Element (transfer-matrix) approach; validation spans thin/thick plates and shells in isotropic/orthotropic settings, making it directly pertinent to the present analysis of wave propagation in two-dimensional waveguides and to the chosen dispersion-computing framework. Including this reference would both acknowledge closely related methodology and provide readers with a state-of-the-art, efficiency-oriented alternative to full 3-D FE for comparable problems. It should be noted that the acceptance of the proposed addition to the current literature does not undermine the validity of this work.

Author Response

We sincerely appreciate your great efforts, constructive comments, and suggestions on our Paper. All comments and suggestions have been carefully considered to improve the quality of this manuscript. We hope this revision meets your requirements. Red fonts highlight modifications in detail in this revision.

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

The paper gives a clear and technically powerful study about optimizing vehicle suspension systems using nonlinear stiffness models and multi-objective genetic algorithms (MOGA). The topic looks interesting and fits well in the journal scope. The work is organized well and includes useful quantitative results. However, the paper needs some improvements in clarity, figure quality, and language before it can be accepted.

This study looks at how spring nonlinearity affects the performance of Inerter-Spring-Damper (ISD) suspension systems in vehicles. The authors developed both parallel and series quarter-vehicle ISD models, and they used a nonlinearity parameter (ε) to show how the spring stiffness changes. A MOGA is used to optimize the fundamental parameters (spring stiffness, damping, inertance, and ε) exposed by random road excitations. The results in both time and frequency domains show that tuning ε properly can make ride comfortable, suspension deflection, and tire load better compared to the linear ISD and conventional passive suspensions.

It would be good to add some recent studies (from 2023–2025) that used advanced optimization or intelligent control methods for suspension systems. That would make the novelty of the paper powerful.

Although the simulation results are solid, the authors should briefly explain why there is no experimental validation yet and how the simulation parameters were checked or compared with real-world data.

More information about the MOGA setup should be given for example, the population size, crossover and mutation rates, and how many generations were used.

It would also be nice to mention the convergence criteria or how the optimization stability was confirmed.

Figures 3–11 could be clearer, increase the resolution, make sure axis have labels and unit, and maybe use larger font for better readability.

It could also helpful to include a short statistical or sensitivity discussion about the optimization result.

The conclusion part should be shorter and avoid repeating the same number that are already shown on the result.

general language check (maybe by a native speaker or professional service) would improve readability and make the style more consistent.

in Page 2, Line 40, the sentence “It was against this backdrop that Professor Smith introduced the inerter…” can be simplified to “Professor Smith introduced the inerter, new element for vibration isolation.”

in Page 10, it would be helpful to define “primary resonance” frequency with actual number if possible.

You must also decrease the similarity rate in itenticate.

Comments on the Quality of English Language

The manuscript is technically clear and nicely organized, but the English language would benefit from a little bit more professional editing to enhance fluency and readability. Several sentences are lengthy and contain multiple clauses. Simplifying these sentence and improving the flow of transitions would make paper more engaging and easier to follow. Minor grammatical and stylistic inconsistencies (for example, article usage and plural forms) should also be corrected. In addition, some terminology, such as “degree of nonlinearity” and “nonlinearity degree,” should be used consistently all along the text. While the technical terminology is accurate, refining the phrasing and rhythm will elavate the overall presentation to the publication standard. A professional language editing service or a thorough revision by a fluent English speaker is recommended before final acceptance.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised manuscript version has been substantially improved, compared to the previous one. However, some aspects still need to be fixed.

Replace the use of inside time-domain ODEs with a consistent state-space representation (introduce an internal state for the inerter–damper series branch) or keep the full Laplace-domain treatment; the current hybrid form is not mathematically valid for time-domain simulation.
Use the same baseline values and precision in prose and in Tables 4 and 5 (e.g., either 888.80 N everywhere, or 888 N everywhere, but not mixed).
Minor language edits: Fix residual grammar/formatting issues (e.g., “an passive” → “a passive”; add missing spaces in headings like “4. Performance Analysis”, “6. Conclusion”.

Author Response

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Author Response File: Author Response.pdf