Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the shimmy instability of the landing gear system by considering the influence of nonlinear damping has been studied using the concept of Hopf bifurcation and the central manifold theorem and the canonical method. Considering following comments is recommended and may improve your manuscript:

 

1)     In relation 11, two terms of main diagonal of matrix A are zero? Are these usual?

2)     Please explain more about how is the equation 47 obtained. According to figure 8, it seems that at V=8.98 and V=85.1, Hopf bifurcations are occurred. Please determine and explain in text about their types including subcritical and supercritical etc.  

3)     In figure 5, what is frequency at V=0?

4)     Please explain more figure 6 and explain its relationship with figure 8.

5)      It seems that figure 7 isn’t used in the text. Anyway, why is the relation of the figure 7 and 8. On the other hand, why for V>85.1, damping isn’t zero in figure 7? Is there any conflict between figures 7 and 8?

6)     Please explain more about novelty of the paper.

Comments on the Quality of English Language

words such as theo-rem are correct?

Author Response

Thank you very much for your careful review! I'm glad to get your approval of this paper and put forward a lot of valuable suggestions. According to your suggestions, I have revised the article. Each item is modified as follows:

1. In formula 11, the matrix A does have two main diagonal terms of zero, and I checked the established dynamic equation, which is a conventional model common in many shimmy papers.
2. Formula 47 is derived from formula 19, which can be obtained by calculating formula 18, and formulas 13 through 20 describe the derivation process in detail. The two velocities in FIG. 8 are supercritical Hopf bifurcation. In FIG. 8, the bifurcation type is analyzed by the first Lyaplov coefficient. The purpose of FIG. 8 is to analyze the eigenvalues to verify the accuracy of theoretical analysis.
3. According to formula 48, it can be seen that when the speed is very small, the frequency becomes sharply smaller. When the speed is 0, the frequency is 0. However, for dynamic research, especially the nose landing gear shimmy, the study of low speed below 2m/s is not of great significance.
4. Figure 6 studies the robustness of the landing gear design without considering the damper, and realizes the shimmy stability of the aircraft through the study of stability distance. At the same time, it is found that a critical stability distance can make the stability of the landing gear system change in the velocity domain. The detailed explanation is given in the formula derivation and theoretical explanation above Figure 6. On the one hand, if damping is added to the formula derivation in Figure 6, the formula will be complicated, which is not convenient for formula derivation, and is not enough to reflect the inherent stability of the landing gear system itself. The damping is added in Figure 8 to compare the time-domain simulation analysis in section 4.1, and to show that there are indeed virtual roots of eigenvalues at the two velocity points, and forks occur.
5. In Section 4.3, by analyzing the damping coefficient, it is theoretically calculated that when there are two characteristic roots that are virtual roots, that is, the damping coefficient is 19.8389 Nms, the aircraft shimmy will not occur at any speed. Figure 7 adopts the two-parameter analysis and the relationship between velocity and critical damping, and the results of the two analyses are consistent, which verifies its accuracy. As shown in Figure 7, when the speed is above 85.1m/s, it can be found that the required damping coefficient is less than 10 Nms, and all the analyses in Figure 8 are based on the damping coefficient of 10 Nms. Therefore, the eigenvalue of Figure 8 is negative when the speed is above 85.1m/s, and the system is stable.
6. In this paper, the center manifold method is integrated with the nose landing gear shimmy system for the first time, and the landing gear system is theoretically analyzed and derived. The first Lyapunov coefficient is used to validate the bifurcation type of the shimmy, and the concept of creeping distance is introduced, which provides guidance for the design of the landing gear shimmy. And the paper uses the combination of theoretical analysis and time domain simulation to ensure the accuracy of the paper from the aspects of frequency calculation and comparison and stability boundary.

Thank you very much for your careful review!

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the dynamic model of the landing gear system is built. The shimmy instability of the system is studied using the first Lyapunov coefficient of the system. The characteristics of Hopf bifurcation is also studied. The work is quite meaningful in this field. However, the writing of the manuscript needs to be improved. The paper is recommended to be published in Aerospace after answering following questions and finishing corresponding major revision in the manuscript.

 

(1)   In Line 11, it should be “.” Instead of “,”. Similar typo errors should be checked carefully in the manuscript, such as Line 287.

(2)   The schematic diagram of the model in Figure 1 and 2 is not clearly illustrated. A 3D model with coordinate XYZ should be given in the figures. What’s the two lines for the angle φ.

(3)   The symbol FKλ and MKφ in equation (1) and (2) is a bit confusing. It’s better to has symbols with single subscripts.

(4)   What’s the λ0, y0 and e in equations (3) and (4). The meaning of all symbols should be explained when it first appears. All other equations should also be checked in the manuscript.

(5)   How is equation (9) derived?

(6)   The English writing in the manuscript should be improved.

(7)   The paper is quite length and not friendly to read. Some deriving parts and unimportant formulas should be put the appendix. 

Comments on the Quality of English Language

The writing of the manuscript needs to be improved.

Author Response

Thank you very much for your careful review! I'm glad to get your approval of this paper and put forward a lot of valuable suggestions. According to your suggestions, I have revised the article. Each item is modified as follows:

1. The article does have some symbol errors and initial letter capitalization errors. After careful inspection and modification, the format errors are modified one by one.
2. The pictures in FIG. 1 and FIG. 2 are indeed not specific and detailed enough in describing the landing gear shimmy, so they have been redrawn and modified in combination with the composition of other shimmy papers.
3. Some numbering in the paper, especially with subscripts, have some problems in some formats, including the wrong position of subscripts and multiple subscripts, which have been processed.
4. The idea that the meaning of symbols should be explained at the time of their first appearance should indeed be emphasized, and in addition to the few symbols suggested by the reviewers, new symbols appearing in other equations in the manuscript are also explained.
5. Equation 9 is the basic equation of landing gear swing vibration mechanics. The tire, axle, piston rod and torque arm at the lower end of the landing gear are free from the outer cylinder, so there is no elastic force. The dynamic equation is established for rotation around its own axis, and the only forces are the tire torsional moment, the lateral force caused by the stability distance and the damping moment caused by the swing reducer. So we can express it as the formula 9.
6. I am sorry that the English expression of the paper has affected the reviewer's reading experience. After seeing this opinion, I have made certain adjustments to the full text, hoping that the paper can be significantly improved and smooth in the subsequent review.
7. Since this paper mainly focuses on the derivation of formulas, in order to ensure the accuracy and consistency of the paper, the derivation process of all formulas is introduced in detail, so the paper is too long, and the subsequent article layout arrangement can be handled by the editor.

Thank you very much for your careful review!

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thanks for your responses to questions. Please have a more attention to question 2. Is this sentence in your response to question 2 correct "The two velocities in FIG. 8 are supercritical Hopf bifurcation."?

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript has been carefully revised. The questions raised by the reviewer have been answered well. The paper can be published as it is now.