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Review of “Mathematics of stable tensegrity structures”

Ajay Harish, Vijay Nardurlikar, Shubham Deshpande, Stephanie Andress,
Alexander Popp

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Ajay Harish, Vijay Nardurlikar, Shubham Deshpande, Stephanie Andress, Alexander Popp. Review of “Mathematics of stable tensegrity structures”. 2023. hal-04082168

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Submitted on 24 May 2023

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Open Review

oai:hal-04082168v1

Reviewed Article

DOI 10.46298/jtcam.7337

History

Review Sep 13, 2021

Rebuttal Jun 1, 2022

Review 2 Jul 19, 2022

Rebuttal 2 Aug 24, 2022

Licence

CC BY 4.0

©The Authors

# Review of “Mathematics of stable tensegrity structures”

Ajay HARISH¹, Vijay NARDURLIKAR¹, Shubham DESHPANDE², Stephanie ANDRESS³, and Alexander POPP⁴,E

¹ School of Mechanical, Aerospace and Civil Engineering (MACE), University of Manchester, UK

² Indian Institute of Technology, Dharwad, India

³ Department of Mechanical Engineering, Purdue University, USA

⁴ Institute for Mathematics and Computer-Based Simulation, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

E Editor

## Review of version 1

Permalink: oai:arXiv.org:2101.09616v1

### Reviewer 1 (Anonymous)

**Reviewer** The authors present the mathematics of the stability of tensegrity structures, which is interesting and merits research.

**Authors** Thank you very much for the appreciation.

**Reviewer** Figures and tables should be reviewed for clarity (text size, addition of subcaptions, figure labels, axis labels, and unit rounding).

**Authors** Thank you very much. The figures have been modified and new images have been added for Figures 1, 2, 3, 9, 10, 13.

**Reviewer** The manuscript flow from the mathematical stability discussion to form finding to the 10-segrity is lacking and is not clear as to the contributions of the author.

**Authors** The flow of the document has been improved with further explanations, nomenclature listing, uniformity in variable names. The primary contribution from the author is the Eq. 33 and also the entire form-finding section. The mathematical discussion provided in the Sec. 4 is necessary to arrive at the Eq. 33.

**Reviewer** The proposition of a 10-string 3 bar tensegrity structure for engineering is circumventing the greater discussions needed to bring tensegrity structures into reality which are connection design and contact mechanics of strings and bars.

**Authors** We completely agree that more needs to be done in the direction of contact mechanics of bars and strings. However, this work tries to:

(a) eliminate the concept that tensegrities need to be pre-stressed

(b) bring together the mechanics of stability into the form-finding process through a simple equation (Eq. 33 in the paper) that relates the minimum number of strings required to have stable structures.

(c) As shown in Fig. 14, this work facilitates building of tensegrity structures that are stable and can be scaled. Recent works [35, 36, refs from manuscript v3] discuss new modelling techniques that facilitate considering joins and contact in such lattice structures. We believe that this is one such direction to be considered in future works.
**Reviewer** Additionally, with the 10th string, the structure will have an asymmetrical center of rigidity, which for engineering design adds complexity

**Authors** Yes, we agree that in this case, the 10th string here leads to an unsymmetric structure. However, this is not necessarily true that the resulting structures will be unsymmetric. Symmetry has been used as a condition for design in many of the string-deficient designs that have been presented till-date. Thus, symmetry is not necessarily a favorable aspect for these structures. The complexity brought in by the asymmetrical center of rigidity definitely warrants additional work but would be much and not in the direction of this paper.

**Reviewer 2 (Anonymous)**

**Reviewer** The present contribution deals with a mathematical framework for form finding of stable tensegrity structures, excluding undesired swinging modes. The general concept and the presented results seem to be original and of interest.

But from my point of view the present version of the manuscript requires several modifications before publication. There also exist some issues in the examples chapter, which require further clarification. In the following, my comments are separated into rather major and rather minor ones.

**Reviewer** The structure of the Introduction is not very well. It starts with about three pages of literature review, which may be structured more clearly. This is followed by subsection 1.1., but subsection 1.2. does not exist, which prohibits the existence of 1.1.

**Authors** Thank you very much. The Introduction section has been further divided into areas of Literature review. The literature review is further sub-divided into multiple areas of interest to the tensegrity community.

**Reviewer** The theory part covered in sections 2 to 5 involves numerous different variables, but several of them are not introduced, e.g., epsilon and B on page 6. In addition to naming the variables, some meaning or mathematical/physical interpretations could improve comprehensibility.

**Authors** This is a great point. A comprehensive variable list has been added to the start of the paper. In addition, we have modified to ensure consistent use of variables across the paper.

**Reviewer** Furthermore, some variables are not unique, e.g. R is a list on page 5 and some (rotation) operator on page 6, or A in Eqs. (5) and (16).

**Authors** As in point 2.1, this has been addressed.

**Reviewer** Furthermore, the notation is not introduced and changes several times, i.e. which quantities are written in bold (points are sometimes bold and sometimes not), what is the notation for a matrix, etc.

**Authors** The notations have been clarified with a dedicated nomenclature at the start of the manuscript.

**Reviewer** All in all, the theory sections a hard to read since the reader has to deal too much with such technical issues first. From my point of view, this may be severely improved.

**Authors** Thank you very much. We definitely agree that the concept of spaces and linear algebra is not always simple. In this paper, we have tried to bring a balance between a mathematical and engineering-oriented paper. We have done this by reducing the lemma and theorems to a form that most computational engineers might be able to comprehend more easily. However, it is expected that the reader needs to have a background in at least linear algebra. Most researchers in the area of computational and theoretical mechanics should be well versed in the above. Keeping the size of the manuscript in mind, it will be hard to introduce from the basics. However, we are very happy to improve if the reviewers can guide on which specific parts need to be improved.

**Reviewer** The finite element modeling in section 6.3 is not comprehensible. From the information given, it is not possible to reproduce the presented results, since necessary information is missing. A list of questions is given in the following:

**Authors** Thank you very much. Additional information has been added as iterated below.
**Reviewer** What are the considered cross sections of rods and strings?

**Authors** Thank you very much for pointing this. The cross-sectional dimensions of the strings and the rods have been added to Sec. 6.3.1.

**Reviewer** How many beam elements are used to discretize a rod/string?

**Authors** 20 - rods
14 to 15 - string

**Reviewer** What type of element technology is used to prevent shear locking?

**Authors** The FEM simulations use B32 elements from Abaqus standard. These are 3-noded quadratic beam elements. Standard Abaqus elements have been used here, since (a) The work is not motivated towards developing new beam element technologies (b) It is easier to reproduce if the elements used are already available for everyone.

**Reviewer** Which type of degrees of freedom does the element possess?

**Authors** The elements have displacement and rotational degrees of freedom. A citation has been added to the Abaqus manual where more information can be found.

**Reviewer** Are the strings pre-stretched? If yes, but which amount?

**Authors** This is the primary difference in this work. Till date, tensegrities have been thought of as pre-stressed structures. It has been iterated time and again that these structures are only stable when prestressed. However, it is important to note that this is a wrong notion that has been crept in since all the tensegrity designs so far are based on pre-stretched elastic strings. We want to emphasize here that the structures are not necessarily prestressed. This has been highlighted in the sections 1.3 and 3.1.

**Reviewer** How is the zero stiffness of strings under compression considered in a modal analysis? This would lead to a permanent change of systems.

**Authors** We completely agree. We also agree that the zero-stiffness in compression needs to be replaced by a better string-element. However, this merits itself to be addressed as a separate work. There are several string elements in the literature already. But the goal here was to: (a) use elements directly available in tools like Abaqus to ensure reproducibility (b) Engage the community to discuss about the notion of the necessity of "pre-stress" in tensegrity structures.

However, we thank the reviewer for bringing this discussion into light to make our paper better. This has now been added into the sections 6.3.1 and 6.3.2.

The strings cannot have any stiffness in compression. The whole concept of stability of a tensegrity is based on the idea that the strings are always in tension and rods in compression. This is why the notion of pre-stressed gained a lot of traction since the inception of tensegrities.

Once the strings are in compression, they have zero stiffness and they can deform and maintain a deformed configuration without any external work. This also means that they are in continuous equilibrium over a finite range of motion. This is also what can be physically seen that once a string is slack, they can occupy several positions.

The positions for the strings shown in higher modes would be only one representation of the several finite shapes that the string can attain. However, the primary noteworthy point here is that such weaker configurations are possible easily in the lower-energy modes in the existing 9-string rather than the 10-string structures.

**Reviewer** How are the joints modeled? The plotted mode shapes give the impression that rotational degrees of freedom are coupled, i.e. bending is transferred from string to string/rod. Is this realistic?

**Authors** The joints are not considered as a part of this work. The work uses beam elements. It is assumed that these line elements share one common node at the joints. Thus, all the degrees of freedom at these nodes remain the same while no additional constraints are added.

Bringing in contact mechanics is definitely important but much beyond the scope of the current work. The current work is not aimed at investigating the contact forces that help transfer between the strings to rods. Mention of this has been added to the section on conclusions and
future works. There has been new literature on modelling techniques and this has been added into the conclusions section. Such new techniques could pave way for modeling joints etc.

**Reviewer** It would be nice to see the applicability of the proposed form finding strategy to a second problem involving more members, e.g. the dodecahedron from Figure 4.

**Authors** While this is a great suggestion. The entire class of polygonal tensegrities can be classified as Voronoi tensegrities. It would warrant a complete discussion, including literature survey and a full-fidelity analysis to discuss Voronoi-tensegrities. There are two particular directions for future works (a) tensegrity towers (b) Voronoi tensegrities. We believe both are important topics that deserve their own manuscripts with detailed discussion. Thus, we believe that this is beyond the mandate of the present manuscript. We would request that this be allowed to be presented in full as a separate manuscript going ahead.

**Reviewer** Section 7 “Conclusions and future work” does not discuss any future research directions. What is your aim for research future? Which ingredients are missing or need to be improved?

**Authors** Further improvements have been made to the conclusions and future works

**Reviewer** Minor comments

**Reviewer** Abstract: what is both stable?

**Authors** Thank you very much. This has been corrected in the latest version (v3).

**Reviewer** Page one last paragraph: I would replace the epsilons.

**Authors** Thank you very much. This has been replaced by the subset symbol in the latest version (v3).

**Reviewer** Page 5, first sentence: n rods, b beads, m cables. Below: 3N tuples

**Authors** Thank you very much. This has been corrected in the latest version (v3).

**Reviewer** Page 5, 2.1, first paragraph: ... tensegrity has...

**Authors** Thank you very much. This has been corrected in the latest version (v3)

**Reviewer** Page 6, first sentence: ... motion about the point... about?

**Authors** This has been corrected to mention the motion of the point P

**Reviewer** Page 7, last sentence: ... translations about x-, ... about?

**Authors** Thank you very much. This has been corrected in the latest manuscript (v3)

**Reviewer** Page 11/12: condition (A/B)? (consistently bold or not), but A is already used twice.

**Authors** The notation has been corrected in the latest version (v3)

**Reviewer** Page 12, below Eq. (35): rows

**Authors** Thank you very much. This has been corrected in the latest version (v3).

**Reviewer** Page 13, below Eq. (38): Eq. (38) (brackets missing)

**Authors** Thank you very much. This has been corrected in the latest version (v3). We have modified all references using the cleverref package to bring about uniformity.

**Reviewer** Page 13: ... the coefficients form... (without s)

**Authors** This has been corrected in the latest version (v3).

**Reviewer** Figure 9: additional Theta?

**Authors** Thank you very much. The plot has been replaced with a new one that is clearer and also the extra theta has been removed.

**Reviewer** General comment: Color scheme in Figures hard to read in black/white printed version. But this is just a matter of taste.
**Authors** Thank you very much. This is a great point. The plot has been replaced with a new one that is more clear and should offer better insight even when printed in black and white

**Reviewer** Page 18, before last paragraph: ...red red curve..., definitely lead...

**Authors** Thank you very much. This has been edited and corrected in the current version (v3).

## Review of version 2

Permalink: oai:arXiv.org:2101.09616v3

### Reviewer 2 (Anonymous)

**Reviewer** The scope and the contents of the paper are appropriate for JTCAM. The present contribution deals with a mathematical framework for form finding of stable tensegrity structures, excluding undesired swinging modes without pre-stretching the cables.

The general concept and the presented results seem to be original and of interest. During revision of the draft, the authors have improved the manuscript in an appropriate way. The overall quality of the manuscript is quite high. Still some minor issues remain, thus I recommend to accept the manuscript with minor revision.

Some comments are listed below:

**Reviewer** You state that you added a comprehensive variable list to the start of the paper. Where is it?

**Authors** Sincere apologies and thanks for pointing this out. This has been rectified in the latest manuscript that is now available on arXiv. The issue was with how our overleaf document was transferred to arXiv. Our overleaf document showed the nomenclature (as shown in the screenshots below). However, it seems like this was not available when compiled on arxiv but also did not show any error. The nomenclature was created using the nomencl package. This required us to submit additional files by generating them from overleaf. This has been done and corrected now on arXiv as well.

**Reviewer** Reply to the previous major comment " How are the joints modeled? The plotted mode shapes give the impression that rotational degrees of freedom are coupled, i.e. bending is transferred from string to string/rod. Is this realistic?": With this comment I was not intending to recommend using contact constraints at the nodes. I was just questioning whether it is reasonable to couple the rotations of string and rod at the nodes. I didn't mean additional constraints. But if these elements share a common point and all degrees of freedom are equal (like it is done in the present work), bending is transferred from rod to string. I rather expected a hinged connection, that is coupling only the three displacement degrees of freedom, while all three rotations are free.

**Authors** I agree with the reviewer that sharing the node will transfer the bending loads from the rod to the string as well. This work uses beam models and considers a common node at the ends. Joining the strings through a common node, instead of joint constraint, is expected to allow the bending mode of deformations to be transferred to the strings as well. This has been addressed in this work through the usage of no-compression constitutive model for the strings. Any bending deformation will necessarily introduce compression but however such compressive loads cannot be sustained by the strings and thus non-physical and will not be permitted. However, a string element as introduced in the work of Crussells-Girona [13, ref from manuscript v4] would be more appropriate alongside usage of appropriate joints and will be considered in the future work.

This has also been added to the final section (Conclusions and future) in the manuscript.

**Reviewer** Last sentence: did you mean “from ground up”?

**Authors** Yes, thank you very much for spotting this. This has been corrected.

## Editor's assessment (Alexander Popp)

The given contribution proposes a new mathematical framework for form finding of stable tensegrity structures with a special focus on engineering applicability. Two independent reviews
by expert referees have been written for the first version of the submitted manuscript, one of
which was immediately very positive and only provided minor comments for improvement, i.e. a
request to the authors to add comments on connection design and on the increased complexity of
the newly identified three-rod ten-string structure. The other review in the first round has asked
the authors more closely and critically about their work and consequently demanded a major
revision. Important remarks of this referee addressed the incomplete description of the finite
element modeling in the results section as well as the lack of a second, independent form finding
example. Within a revised version of the manuscript, the referees' concerns were mostly resolved,
and after a second revision both referees recommended publication.

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