Review Reports

Round 1

Reviewer 1 Report

The manuscript titled "Analysis of Three-Dimensional Rigid-Body Impact with Friction" deals with a comparative study between three low-speed contact models through analytical and numerical studies. Both two and three dimensional impacts are modelled. The study has some merit (although not substantial) in comparing three models, deriving some analytical relationships (which I believe are somewhat novel), as well as demonstrating similarities between the two model and suggesting that Newton model should not be used (this last point perhaps needs more demonstration as to why). Some results are somewhat expected. For example, it is trivial that more energy is lost when the coefficient of restitution is low or the coefficient of friction is high. Nevertheless, the manuscript is still publishable, once some improvement is made as below (also in the attached file):

- Currently low-speed impacts and which model to use are also a point of debate in landing small solar system objects such as an asteroid, so I think the authors should mention that in their introduction as well.
- 3D impacts could have included more than just a rod, for example a cube etc., for more realistic applications.
- The paper is generally well written, but the text and figures need some improvement, which are noted in the commented pdf file.
- There are also some other points for clarification and some questions on the parameters selected and methodology, which are also noted in the attached pdf file.    

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for the constructive comments and especially for the annotated manuscript. We have made all of the correction that have been suggested, including space applications, and comments in the annotated manuscript.

We have the following replies. The reason why we chose a rod for both 2D and 3D impact is to compare the two and to show that, when orientation angles are selected in a certain way, the #D impact and 2D impact results become the same. Otherwise, the proposed formulation is applicable to any 3D object.

We have added a flowchart to the manuscript as recommended. But, because of the length of equations involved, we have opted to keep the verbal description of the analysis procedure. Unfortunately, the limit in quality of the figures arises when Matlab figures are converted to png files. Nevertheless, we ran the code on all of the figures and improved all the figures.

Otherwise, all of the reviewer's recommendations (general comments on manuscript and also editorial marks on manuscrit) have been incorporated.

Reviewer 2 Report

The topic of this paper is interesting, since 3D impacts with friction still remain a not-so-easy dynamical phenomenon to handle. The developments seem correct and logical.

a) Nevertheless the introduction is a kind of awful tutti-frutti on impact modelling, and it has to be rewritten. In fact  there are several ways to classify impact models within gross classes. One way is as follows (there is always some arbitrary choice in such classifications, but the following ones seem relatively reasonable):

A) Single impacts, i.e., a single collision occurs without any overlap with foregoing or next collisions, in 2D or 3D,

and

B) Multiple impacts, i.e., several collisions occur simultaneously in the system.

This is independent of the fact that the system may be a multibody system like a kinematic chain, or just an impact between two rigid bodies. This article deals with single impacts between two bodies. In any case (single or multiple), there are three main classes of models:

1) Algebraic (zero-order) models which assume instantaneous collisions between perfectly rigid bodies, and relate post- and pre-impact velocities through a so-called restitution rule.
  2) First-order dynamics following the Routh and the Darboux-Keller approaches, where the normal contact force impulse is used as a new time-scale.
  3) Second-order dynamics which rely on compliant models of lumped flexibilities and dampings (linear or nonlinear spring-dashpot rheological models), through assemblies of various basic components: elastic, visco-elastic, frictional, fractional elastic, visco-plastic, etc. 

One can further make the distinction between impacts with and without friction, since friction is indeed a very important effect. This paper deals with A) 1) with friction.

I am not asking the authors to make a comprehensive review of impact models in the introduction. However they have to improve it drastically. As it is it mixes almost everything (for instance the contribution of [16] is about multiple impacts, while Keller, Stronge and others use the Darboux-Keller approach and should be grouped together), and it misses some fundamental contributions. Authors can have a look at the monograph: B. Brogliato, Nonsmooth Mechanics, Models, Dynamics and Control, Springer International Publishing Switzerland, 3rd Edition, 2016, where a relatively good and complete review of impact modelling is made. Especially chapters 2, 4, 6 in that book should be consulted and authors can easily extract some essential citations from the many that can be found in that book.

b) Be careful with the so-called Stronge coefficient. In fact, as recalled in the above book (section 4.3.6) the energetic CoR was introduced long before Stronge (and thus it is not recent as stated line 177).

c) It is also important to cleary understand what is the so-called Darboux-Keller approach, which is one of the most used in the Impact Mechanics literature, see the above book section 4.3.5.

d) the contents look correct as far as I could see. It would be good that the authors state more clearly in the introduction what is the real contribution compared with previous works, since this is a topic which has already witnessed many studies since many years.

e) Authors compare the three basic CoRs. How do you compare with previous analysis by Stronge and others after him (reference [20], results by Batlle, etc), or to the analysis made in section 4.3.5.3 in the above book ?

f) The expressions in (23) (24) look like the Thompson and Tait's formula, see details in the above book.

 

 

 

 

Author Response

We agree with the reviewer's comments and we have rewritten the introduction section entirely to incorporate the reviewer's suggestions for classifying impact and describing our contribution. We also added two new references. We also revised the abstract in view of the reviewer's comments. In addition, we have added comments to the manuscript to put the contribution of our paper in better perspective.

Round 2

Reviewer 2 Report

Authors have done a correct revision of their work. However, I still believe that the acknowledgement of previous works on the topic is missing. Especially in section 2, the 2D impact, which has long been studied in the Applied Mechanics literature, is "reviewed" in an 8-page section. Why not, but it would be very useful for readers to know what kind of improvements authors are providing here, compared to the several studies that have been published on the topic. In this respect the material in the introduction is not sufficient. I think it is mandatory to add a paragraph, perhaps at the end of section 2, which recapitulates the novelties proposed in section 2 for the 2D case analysis. For instance section 4.3.5 in the monograph [10]  proposes an analysis of the 2D case, and furnishes a small bibliography. This could serve as a base to write such a paragraph.

 

About the 3D case: if I trust reference [10, Proposition 4.9], Darboux and Stronge proved an important result about direction and orientation of sliding when sliding resumes before the end of the collision. Are authors going beyond this ? Do their results confirm this, or not  ?

 

In summary, I am not questioning the interest of the topic, but I want the authors to write clearly and shortly what  they bring to the field.

 

 

Author Response

In response to the reviewer's comments, we have made the following revisions:

1. We have added more references to the manuscript. We disagree with the approach of many authors who provide a very length reference list. Rather, we list pertinent and representative works, including the very comprehensive review listed in Sec. 4.3.9 in Brogliato. If published, our article would be another contribution that would be included in a further edition of that review.
2. We have added a paragraph to the end of Section 2, summarizing the contributions of this section, new material in the section, and how this section lays the foundation for the study of three-dimensional impact.  
3. We have revised the next-to-last paragraph of the Introduction Section to put into better perspective our contribution that the approximation we introduce permits use of the algebraic approach to solve the impact equations. If you look at the literature, the first- or higher-order approximations are used for solving specific cases of impact and are limited in scope. Our approach, while approximate, is valid for all types of three-dimensional impacts (except multiple impacts, which will be our next extension of the research). We have nothing against the fine work done by previous researches, Darbaux, Stronge, Battle, or others. All we are doing is introducing a new approach to the solution of the problem, one that estimates the change in direction of the sliding when sliding ends and restarts in a different direction. Our contribution is to provide an alternative approach, one that relies on the algebraic equations and does not change the general nature of equations to be solved.