Non-Linear Forced Response of Vibrating Mechanical Systems: The Impact of Computational Parameters

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper investigates the influence of parameter settings on the accuracy and convergence of the Multi-Harmonic Balance Method (MHBM) in solving structural dynamics with dry friction nonlinearities and provides practical guidelines for users. Currently, MHBM is the most widely accepted method in academia for solving dry friction structural dynamics problems. However, its implementation in commercial software has been hindered by the large number of sensitive parameters that significantly affect computational results and convergence. To the best of the reviewer's knowledge, no prior studies have systematically addressed the guidelines for setting these parameters, nor have many researchers discussed this issue in detail. This work represents the first thorough investigation of these challenges, offering valuable insights for researchers in related fields.

The study adopts a reasonable model, employing the commonly used variable normal pressure hysteresis spring contact element to describe dry friction contact problems under coupled tangential-normal vibrations. The investigated parameters are comprehensive, covering contact stiffness, sampling points, harmonic numbers, and intermittent separation conditions, making the conclusions highly convincing.

Minor Issues to Address:

1. Recent literature citations are insufficient.Please supplement with the following references:
   • 1007/s11071-025-11352-4
   • 1016/j.jsv.2024.118410
   • 1007/s10483-023-3047-6
   • 1007/s11071-025-11065-8
   • 1016/j.cja.2023.11.026
2. Inconsistencies in Equations 9 and 10:
   • The symbols for normal force (Nand n) are not unified.
   • Equation 10 contains an expression for Tthat includes the sign function of T itself, which appears incorrect. Please clarify or revise.
3. Terminology inconsistency:
   • The terms "chapter" and "section" are used interchangeably. Please ensure consistency.
4. Tangential stiffness justification:
   • Provide the rationale for the chosen tangential stiffness values. How does this selection affect damping estimation?
5. Normal stiffness considerations:
   • Does the normal stiffness have physical significance? How does its value influence the results? What are the recommended values?
6. Generalizability to other contact models:
   • Would the conclusions remain valid if the contact model is altered (e.g., to a 3D contact model)?
7. Generalizability to higher degrees of freedom:
   • For systems with more contact DOFs, would the parameter selection criteria need adjustment?

Author Response

Revision of the paper “Non-linear Forced response of Vibrating Mechanical Systems: The Impact of Computational Parameters” by Enio Colonna, Teresa Berruti, Daniele Botto and Andrea Bessone

Author’s response to reviewer # 1

We sincerely thank the reviewer for her/his constructive comments and useful suggestions.

 Following the reviewer's suggestions, the paper is now clearer and quality improved. Please find below the authors response to the reviewer’s specific comments (reviewer’s comments in black, author’s response in blue, excerpts from the paper in green, new additions or corrections to the paper in red). The revised paper will present the changes in red, to better underline the modifications respect to the previous version.

 

1. Recent literature citations are insufficient. Please supplement with the following references:

1007/s11071-025-11352-4

1016/j.jsv.2024.118410

1007/s10483-02 3-3047-6

1007/s11071-025-11065-8

1016/j.cja.2023.11.026

 

Thank you for pointing this out of us, the suggested studies have been integrated into the paper literature.

 

2. The symbols for normal force (n and N) are not united. Equation 10 contains an expression for T(t) that includes the sign function of T(t) itself, which appears incorrect, please clarify or revise.

 

Thank you for pointing out the typo. We have now ensured consistency in the symbol for normal force.

 

Regarding the second point we agree that the equation you mentioned was written using notation that can be misunderstood, as it appears to include the sign of the unknown. The second of Equation 11 has been changed from “” to “”. The algorithm for tangential force calculation is a predictor-corrector scheme, so the computation is performed in two steps. The first step is the predictor, in which the tangential force  is computed assuming the stick state of the contact. The second step is the corrector one, in which the predicted force is compared with the Coulomb limit and it is recalculated as follows,

• If it is lower than the Coulomb limit the contact is in stick condition, so the predicted force is retained.
• If it is higher than the Coulomb limit the contact is in slip condition, so the predicted force is corrected and set equal to the Coulomb limit multiplied by the sign of the predicted term.

 

To make this clear in the manuscript the following addition has been integrated into section 2 at row 168:

 

“The computation of the tangential force is based on a predictor-corrector algorithm. This algorithm is discussed in detail in [22]. The main outcomes are summarized below. The tangential force is computed by the algorithm at each time instant, under the assumption that the contact is in a stick state (predictor step).

(10)

The next step is the corrector one, in which the force computed in Equation (10) is compared to the Coulomb limit. Then, its value is recalculated as follows:

(11)

3. The terms “chapter” and “section” are used interchangeably. Please ensure consistency.

 

We provide a uniqueness of the term, all the terms “chapter” are replaced in this paper with the term “section”.

 

 

4. Provide the rationale for the chosen tangential stiffness values. How does this selection affect the damping estimation?

 

Thank you for this request of clarification. In sections 3 and 4, the tangential stiffness values were chosen without considering the physical properties of a real surface, but they were taken from values found in the literature for systems with a single degree of freedom. The purpose of these two sections was as follows: section 3 attempted to demonstrate the differences in frequency response functions and solution methods with and without intermittent contact; section 4 focuses on the sensitivity analysis of the computational parameters.  On the contrary, in section 5 the value of the tangential stiffness is chosen considering the values adopted in literature, for real turbine applications.

To make clear this concept in the manuscript, the following addiction is integrated in Section 3, at row 437:

 

“The contact parameters adopted in this section and in the next one are chosen considering the typical order of magnitude of stiffness values used in literature for 1 GDL systems [26]. These values are not linked to physical properties of a real contact surface because the focus here is a sensitivity analysis which is independent from the method used for the stiffness values choice.”

 

About the damping, it is true, the value of the tangential stiffness affects the damping estimation. In fact, increasing the tangential stiffness value the contact reaches the slip state first, so there could be more friction damping. However, according to the authors, this aspect is not affecting the results obtained in this manuscript, in fact the trend shown for the solver convergence with the contact parameters is still valid for different values of stiffness values.

 

 

5. Does the normal stiffness have physical significance? How does its value influence the results? What are the recommended values?

 

We really appreciate this comment because it is pointing out an important subject. The normal stiffness has certainly a physical meaning, in fact it is modelling the compressive behavior between two contact surfaces and allows to estimate the normal contact forces starting from normal displacements. This point is addressed in the paper in section 1 at row 59: “Contact parameters, from the perspective of this paper, are input parameters for the non-linear solver. However, it is important to remember that they must model the physical properties of the contact, so they are closely related to the materials that make up the contact surfaces. These parameters can be derived from analytical models or from experimental tests on dedicated test rigs”. To better underline that the contact parameters, including normal stiffness, are strongly related to the physical properties of the contact surfaces, the previous expression is modified as follow:

 

“Contact parameters, from the perspective of this paper, are input parameters for the non-linear solver. However, it is important to highlight that they must model the physical properties of contact. These parameters can be derived from analytical models or from experimental tests on dedicated test rigs. For instance, the normal stiffness, in some cases, can be estimated using Hertz contact theory, that defines relationships between the indentation depth and the normal force for two contacting bodies. The single relationship depends on the elastic modulus, the surface geometry and the surface finish, and the ratio between the normal force and the indentation depth can be expressed as a normal stiffness, which in a contact model, is represented as a spring element.”

 

The value of normal stiffness significantly impacts the results, because it affects the contact states and transitions between them. Consequently, the shape of the nonlinear FRF curve changes. This can lead to more samples per period, which is necessary to correctly compute the Fourier coefficients. This argument is well discussed in section 4.3. To author’s knowledge there are not recommended values of normal stiffness, they depend on the specific blade model. The main suggestion for designers that manage non-linear tools is to use enough contact nodes, in order to better distribute the global normal stiffness, avoiding convergence issues.

 

 

6. Would the conclusions remain valid if the contact model is altered (e.g., to a 3D contact model)?

 

The authors believe that the conclusions remain valid also for a 3D contact model. This concept can be better understood considering that in a 3D contact model the contact algorithm is the same as the 2D contact model, the only difference is that in the 3D case two tangential components of the friction force must be computed, but each single tangential component is computed in the same way discussed in this paper. To better underline this aspect the following paragraph has been added in the conclusions, section 6, at row 1149:

 

“The results that come out from the sensitivity analysis performed in this paper are obtained considering a 2D contact model, which includes a variable normal force and a tangential friction force in one direction on the contact surface. These results can be easily extended to a 3D contact model where, instead of calculating the tangential force on the contact surface in a single direction, the tangential force is calculated in two perpendicular directions, but the calculation scheme remains the same.

 

7. For systems with more contact DOFs, would the parameter selection criteria need adjustment?

 

Thank you for this question, which emphasizes an important step in non-linear dynamic analysis of bladed disks. The parameter selection criterion is the same for a system with a single contact pair and for a system with a higher number of contact pairs. The number of the contact DOFs only influences the local value of the stiffness, so to avoid convergence issues is important, in real turbine applications, to have enough contact pairs. To make clear that the selection criteria does not need to be modified, the following paragraph has been added in the manuscript at the row 1017 in section 5:

 

“The values of the contact parameters shown in Table 10 are chosen considering numerical analysis carried out in literature on real bladed disks. In real turbine applications the number of contact pairs and the values of the corresponding contact parameter are usually chosen so as to obtain, when contact surfaces are stuck (without sliding), the same frequencies that would be obtained in Ansys in the same condition. To the author’s knowledge, it is recommended to choose the minimum values of contact parameters that guarantee the match of frequencies, to avoid convergence problems.”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

as in the attachment

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The entire manuscript could be subjected to a very slight English language polyshing.

Author Response

Revision of the paper “Non-linear Forced response of Vibrating Mechanical Systems: The Impact of Computational Parameters” by Enio Colonna, Teresa Berruti, Daniele Botto and Andrea Bessone

Author’s response to reviewer # 2

We sincerely thank the reviewer for her/his constructive comments and useful suggestions.

 

Following the reviewer's suggestions, the paper is now clearer and quality improved. Please find below the author’s response to the reviewer’s specific comments (reviewer’s comments in black, author’s response in blue, excerpts from the paper in green, new additions or corrections to the paper in red). The revised paper will present the changes in red, to better underline the modifications with respect to the previous version.

 

1. Generally, the Harmonic Balance Method assumes a solution in the form of a series of multiple harmonics. Why do the authors call it the “Multi Harmonic Balance Method”?

 

We agree with the reviewer that the term "Harmonic Balance Method" already implies a linear combination of harmonics of the fundamental frequency. The term “Multi” is redundant. Several examples in the literature refer to the multi harmonic balance method to emphasize that more than one harmonic is considered. The term "Multi Harmonic Balance Method" has been replaced with "Harmonic Balance Method" in the paper.

 

 

2. As should be understood, the carefully prepared numerical procedure covering the vast majority of the work is addressed to the simulations of vibrations of complex blade structures modeled using the finite element method. So why in section 5 was the proposed method applied again to a simple, academic mechanical system with 2 degrees of freedom and the equivalent parameters given in Table 10? How do the numerical values of these parameters compare to the corresponding parameters of the structural model of a real blade or entire blade rim? Since so much space in the paper was devoted to numerical testing by means of a trivially simple model, which in principle is correct, wouldn’t it be possible to illustrate the operation of such a laboriously developed approach using a more reliable model of at least 2-3 blades, proudly shown in Fig 19?

 

Thank you for bringing up this important point, which gives us the opportunity to clarify the purpose of this paper.

We agree with you that Figure 19 may cause misunderstanding, as it shows a portion of a bladed disk. This figure has been included mainly to provide an example of the typical context in which the considerations made in this paper can be exploited. The nonlinear forced response calculation was then performed on a system with only two degrees of freedom but with the values of the natural frequencies and the contact parameters close to the first natural frequencies and contact parameters of a real turbine blade. This is because the aim of this section (section 5) was to show how all the considerations made in the previous sections remain valid even with this new parameter values. It is true that the system has only two degrees of freedom, but in the authors' opinion, adding linear degrees of freedom would not have added value to the results concerning how to model the single contact behavior.

To better explain the meaning of Figure 19, the following paragraph has been added to section 5 at row 1010:

 

Figure 19 shows a portion of a bladed disk as an example of the typical context in which the considerations made in this paper can be exploited. For simplicity of calculation, the results shown below will still be on a system with 2 degrees of freedom but with system frequency values and contact parameter values typical of the blades of a real turbine.”

 

3. A degree of freedom is a very fundamental and precisely defined notion in mechanics and dynamics of mechanical systems. So what does the concept of “nonlinear DOFs” means on pages 3 and 28? A “degree of freedom” is simply a “degree of freedom”, regardless of the linear or nonlinear properties of the mechanical model used. If it is some kind of mental shortcut or jargon, it should be eliminated from the scientific article.

 

Thank you for clarifying this point. Indeed, there can be confusion regarding terminology.

The expression “nonlinear DOFs” on pages 3 and 28 refers to contact nodes. Since the concept of "nonlinear DOFs" can be misunderstood, we have substituted this expression with "contact nodes."

 

 

4. Starting sentences with “Where” is nor befitting good English.

 

The word "where" has been replaced with the following expressions:

• “The terms of the previous equations are defined below” on line 101 of Section 2.
• “The terms of previous equations are defined below” on line 174 of Section 2.
• “All system parameters are defined below” on line 419 of Section 3.

 

5. The abbreviation “GDL” should be defined for a wide range of uninitiated readers.

 

Thank you for pointing this out. We substituted "GDL" with "DOF" at row 382 in section 3.

 

6. There are for Co-authors declared, but only three of them “contributed equally to this paper”, see page 31. What about the fourth one

 

Thank you for pointing this out, there was a typo. “The three authors contributed equally to this paper” is substituted with “The four authors contributed equally to this paper”.

Author Response File: Author Response.pdf