the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Dynamic Analysis of the Tensegrity Structure of a Rotary Airborne Wind Energy Machine
Abstract. The dynamic behavior of the tensegrity structure (helix) of a Rotary Wind Energy (RAWE) machine was investigated by combining experimental and numerical techniques. Taking advantage of the slenderness of the helix, a dynamic model for the evolution of its center line and the torsional deformation was developed by using Cosserat theory. The constitutive relations for the axial, bending and torsional stiffness, which are a fundamental component of the model, were obtained experimentally by carrying out laboratory tests. Three scenarios of increasing complexity were then studied with the numerical tool. Firstly, a stationary solution of the model, i.e. with fixed ends and no rotation, was found numerically and used to verify the correct implementation of a numerical code based on finite elements. The stability analysis of this solution, which corresponds to the state of the structure just after deployment but before operation, showed that the natural periods of longitudinal, lateral, and torsional modes of the RAWE structure under consideration are around 0.03 s, 0.2 s and 0.4 s, respectively. Secondly, the dynamic in nominal operation was investigated by keeping fixed both end tips and implementing a controller that adjusts the torque at the ground to reach a target angular velocity of 120 rpm. Key characteristic variables like the tension and the response times of the helix were obtained. Thirdly, the dynamic of the helix when the lower end is fixed and the upper end is driven in a circular motion of frequency f_{1} was studied experimentally and numerically. The helix's tension in the experiment increased for f_{1} above certain threshold and the structure collapsed at f_{1} ≈ 5 Hz. Simulation analysis revealed a resonance of the structure at a frequency higher to the one observed in the experiment (around 13 Hz).
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RC1: 'Comment on wes2023170', Anonymous Referee #1, 09 Jan 2024
This paper highlights the potential nonlinearities that can be observed in a tensegrity structure of a rotary airborne wind energy system. In particular, the hysteresis of the torsional stiffness and the discrepancy in the numerical vs experimental resonance frequencies are valuable additions to the literature and could be useful for further studies.
I only have two minor comments:
1. I understand the justification for only considering aerodynamic drag in equation 14. However, it would be useful to briefly comment on whether the lift generated by the spinning rotor is significant. I assume it’s not.
2. The hysteresis observed in fig. 5 is interesting. As the authors have noted in section 5.3, this could be a contributing factor to the resonance observed in experiments that were not predicted by numerical simulations. I think that the unmodelled hysteresis is the main reason for this discrepancy, so some discussion on how this can be studied further can be useful. There are some ODEbased methods that can model hysteresis in a dynamical system (e.g, the GomanKhabrov model. This is more for unsteady aerodynamics, but the ODEbased approach can be extended to other applications). The authors can consider adding a brief discussion on how this hysteresis can be modelled in a future study.
Lastly, I have a few suggestions regarding grammar/spelling in the attached pdf.

AC1: 'Reply on RC1', Gonzalo SanchezArriaga, 16 Jan 2024
Comment 1. Equation 14 is our model for the aerodynamic force acting on the helix (and not in the rotor). Only the aerodynamic drag is considered, as such a force component may dominate the lift. The spinning helix is made of rigid bars and tethers and its function is not to generate any lift but to transmit the torque from the rotor to the ground station. The lift generated by the rotor is indeed important, but the rotor is not part of our model, which is focussed on the helix.
Comment 2. The numerical simulations predicted a resonance, but at a different frequency to the one observed in the experiments. We fully agree that the unmodelled hysteresis can contribute to this discrepancy. Following the advice of the Reviewer, we added a few sentences about this important point.
Comment 3. We really thank the Reviewer for providing this revision. Most of his/her suggestions were incorporated to the manuscript.
Citation: https://doi.org/10.5194/wes2023170AC1

AC1: 'Reply on RC1', Gonzalo SanchezArriaga, 16 Jan 2024

RC2: 'Comment on wes2023170', Anonymous Referee #2, 23 Jan 2024
This paper presents a model of a rotary AWE system. The structure model is ‘decoupled’ from the airborne rotor with the setting up of boundary conditions. Interesting new insights are obtained from the combined numerical and experimental studies. The investigation of the three scenarios following an increased complexity makes very good sense. The work includes clear novel contributions. The conclusion section is well written.
The following minor comments are for the authors’ reference.
 In Figure 5, the initial point and the ending point are not identical for each tension applied. What causes this difference? Should it be a closed loop (in theory)? Also, when the tension is 233N (green colour), the pattern seems to be quite different from the rest. Is there a reason for this isolation?
 In Section 5.3, it is said the frequency $f_1$ was verified. Is this forcing frequency of the eccentric arm introduced in the experimental study only, not covered in the modelling somewhere?
 If yes to the previous comment, what is the model used for bifurcation analysis (Figure 8)?
 The use of mathematical notations is confusing in places. Better give a table to list key variables, parameters, reference frames, operators, etc. It would be helpful to specify the use of bold letters and plain letters especially when the same letters are recycled, e.g., the two $v_A$ terms in Equation (14), the time t and the bold letter of t. If possible, keep a consistency in partial derivative representations in PDEs and derivatives in ODEs.
 In several places, the term ‘dynamic’ should be ‘dynamics’ for dynamic systems.
 In Line 71, it should be ‘r(L_0,t)’ not ‘r(L,t)’, isn’t it?
 In Lines 266267, ‘Fig. 1’ should be ‘Fig. 2.’
 What is the $i_E$ used in the boundary conditions (72) and (76)? It was introduced after Equation (14) but not explained there.
 In Figure 6, add the text to the y axis of the bottom figure.
 Some materials in Section 3 can perhaps be moved to appendix to keep a smooth reading.
 There are some typos and gramma mistakes in writing which can be easily removed.
Citation: https://doi.org/10.5194/wes2023170RC2 
AC2: 'Reply on RC2', Gonzalo SanchezArriaga, 02 Feb 2024
We are glad to know that the Reviewer found that our work provides interesting insights and has novel contributions. We thank him/her for providing the minor comments that we address in the lines below.
Comment 1. In our opinion, the results of Fig. 5 are a consequence of the structure of the helix, which is made of bars under compression connected by tethers and knots. As the torsional torque M_{t} increases, in steps to give a torsional angle of 10º, the tension T_{e} also increases and exhibiting hysteresis. Clearly, the behaviour of the structure depends on the history. We think that it is because the exact position of the knots and the contact points between bars are not fixed but certain sliding occur and also friction plays a role. The exact positions of such a contacts points depend on the internal tension imposed for zero torsion (for this reason the curve for T_{e0 }= 233N is so different) and also on the history. Due to the presence of knots and the sliding of the bars, each time the helix passes from being relaxed to be tensioned it acquire a slightly different configuration that translate into different mechanical properties.
Comment 2. We are not sure to fully understand the question. In the experiments, frequency f_{1} was imposed by the eccentric arm and it was measured and controlled.
In our model of the helix, frequency f_{1 }appears as a boundary condition in Eq. (76). In our numerical analysis such a frequency was varied to mimic the conditions of the experiment.
Comment 3. The model of the helix used in the bifurcation analysis of Fig. 8 is the model explained in Sec. 2 together with the boundary condition of Eq. (76). In the bifurcation analysis, we fixed a value of the frequency f_{1} and integrated the equations of motion numerically by using the initial condition explained in Sec. 5.1. The integration was long enough to let the system converge to a solution, which is periodic. We then computed the maximum value of the tension for a given oscillations.
Comments 4. We revised all the equations carefully and we did not find any error. However, we agree that some key aspects can be highlighted to help the reader to understand the notation. We added several sentences in the revised manuscript to help its readability and clarity. We thank the Reviewer for this comment.
Comment 5. We made a search to the word “dynamic” and corrected it.
Comment 6. Yes, the typo was corrected in Line 71.
Comment 7. The Reviewer is right. We corrected it in Lines 266267.
Comment 8. Vectors i_{E}, j_{E} and K_{E} are the unit vectors of frame S_{E}. We added a sentence in Sec 2.1 and another in Sec. 5.1 to remind their meaning to the readers.
Comment 9. We cannot add a text to the y axis because Fig. 6 presents two different quantities, which are explained in the legend.
Comment 10. We agree that such idea improves the readability of the manuscript. Part of the material of Sec. 3 was moved to a new Appendix.
Comment 11. We revised the manuscript a fix some typos.
Citation: https://doi.org/10.5194/wes2023170AC2

RC3: 'Comment on wes2023170', Anonymous Referee #3, 26 Mar 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023170/wes2023170RC3supplement.pdf

AC3: 'Reply on RC3', Gonzalo SanchezArriaga, 02 Apr 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023170/wes2023170AC3supplement.pdf

AC3: 'Reply on RC3', Gonzalo SanchezArriaga, 02 Apr 2024
Status: closed

RC1: 'Comment on wes2023170', Anonymous Referee #1, 09 Jan 2024
This paper highlights the potential nonlinearities that can be observed in a tensegrity structure of a rotary airborne wind energy system. In particular, the hysteresis of the torsional stiffness and the discrepancy in the numerical vs experimental resonance frequencies are valuable additions to the literature and could be useful for further studies.
I only have two minor comments:
1. I understand the justification for only considering aerodynamic drag in equation 14. However, it would be useful to briefly comment on whether the lift generated by the spinning rotor is significant. I assume it’s not.
2. The hysteresis observed in fig. 5 is interesting. As the authors have noted in section 5.3, this could be a contributing factor to the resonance observed in experiments that were not predicted by numerical simulations. I think that the unmodelled hysteresis is the main reason for this discrepancy, so some discussion on how this can be studied further can be useful. There are some ODEbased methods that can model hysteresis in a dynamical system (e.g, the GomanKhabrov model. This is more for unsteady aerodynamics, but the ODEbased approach can be extended to other applications). The authors can consider adding a brief discussion on how this hysteresis can be modelled in a future study.
Lastly, I have a few suggestions regarding grammar/spelling in the attached pdf.

AC1: 'Reply on RC1', Gonzalo SanchezArriaga, 16 Jan 2024
Comment 1. Equation 14 is our model for the aerodynamic force acting on the helix (and not in the rotor). Only the aerodynamic drag is considered, as such a force component may dominate the lift. The spinning helix is made of rigid bars and tethers and its function is not to generate any lift but to transmit the torque from the rotor to the ground station. The lift generated by the rotor is indeed important, but the rotor is not part of our model, which is focussed on the helix.
Comment 2. The numerical simulations predicted a resonance, but at a different frequency to the one observed in the experiments. We fully agree that the unmodelled hysteresis can contribute to this discrepancy. Following the advice of the Reviewer, we added a few sentences about this important point.
Comment 3. We really thank the Reviewer for providing this revision. Most of his/her suggestions were incorporated to the manuscript.
Citation: https://doi.org/10.5194/wes2023170AC1

AC1: 'Reply on RC1', Gonzalo SanchezArriaga, 16 Jan 2024

RC2: 'Comment on wes2023170', Anonymous Referee #2, 23 Jan 2024
This paper presents a model of a rotary AWE system. The structure model is ‘decoupled’ from the airborne rotor with the setting up of boundary conditions. Interesting new insights are obtained from the combined numerical and experimental studies. The investigation of the three scenarios following an increased complexity makes very good sense. The work includes clear novel contributions. The conclusion section is well written.
The following minor comments are for the authors’ reference.
 In Figure 5, the initial point and the ending point are not identical for each tension applied. What causes this difference? Should it be a closed loop (in theory)? Also, when the tension is 233N (green colour), the pattern seems to be quite different from the rest. Is there a reason for this isolation?
 In Section 5.3, it is said the frequency $f_1$ was verified. Is this forcing frequency of the eccentric arm introduced in the experimental study only, not covered in the modelling somewhere?
 If yes to the previous comment, what is the model used for bifurcation analysis (Figure 8)?
 The use of mathematical notations is confusing in places. Better give a table to list key variables, parameters, reference frames, operators, etc. It would be helpful to specify the use of bold letters and plain letters especially when the same letters are recycled, e.g., the two $v_A$ terms in Equation (14), the time t and the bold letter of t. If possible, keep a consistency in partial derivative representations in PDEs and derivatives in ODEs.
 In several places, the term ‘dynamic’ should be ‘dynamics’ for dynamic systems.
 In Line 71, it should be ‘r(L_0,t)’ not ‘r(L,t)’, isn’t it?
 In Lines 266267, ‘Fig. 1’ should be ‘Fig. 2.’
 What is the $i_E$ used in the boundary conditions (72) and (76)? It was introduced after Equation (14) but not explained there.
 In Figure 6, add the text to the y axis of the bottom figure.
 Some materials in Section 3 can perhaps be moved to appendix to keep a smooth reading.
 There are some typos and gramma mistakes in writing which can be easily removed.
Citation: https://doi.org/10.5194/wes2023170RC2 
AC2: 'Reply on RC2', Gonzalo SanchezArriaga, 02 Feb 2024
We are glad to know that the Reviewer found that our work provides interesting insights and has novel contributions. We thank him/her for providing the minor comments that we address in the lines below.
Comment 1. In our opinion, the results of Fig. 5 are a consequence of the structure of the helix, which is made of bars under compression connected by tethers and knots. As the torsional torque M_{t} increases, in steps to give a torsional angle of 10º, the tension T_{e} also increases and exhibiting hysteresis. Clearly, the behaviour of the structure depends on the history. We think that it is because the exact position of the knots and the contact points between bars are not fixed but certain sliding occur and also friction plays a role. The exact positions of such a contacts points depend on the internal tension imposed for zero torsion (for this reason the curve for T_{e0 }= 233N is so different) and also on the history. Due to the presence of knots and the sliding of the bars, each time the helix passes from being relaxed to be tensioned it acquire a slightly different configuration that translate into different mechanical properties.
Comment 2. We are not sure to fully understand the question. In the experiments, frequency f_{1} was imposed by the eccentric arm and it was measured and controlled.
In our model of the helix, frequency f_{1 }appears as a boundary condition in Eq. (76). In our numerical analysis such a frequency was varied to mimic the conditions of the experiment.
Comment 3. The model of the helix used in the bifurcation analysis of Fig. 8 is the model explained in Sec. 2 together with the boundary condition of Eq. (76). In the bifurcation analysis, we fixed a value of the frequency f_{1} and integrated the equations of motion numerically by using the initial condition explained in Sec. 5.1. The integration was long enough to let the system converge to a solution, which is periodic. We then computed the maximum value of the tension for a given oscillations.
Comments 4. We revised all the equations carefully and we did not find any error. However, we agree that some key aspects can be highlighted to help the reader to understand the notation. We added several sentences in the revised manuscript to help its readability and clarity. We thank the Reviewer for this comment.
Comment 5. We made a search to the word “dynamic” and corrected it.
Comment 6. Yes, the typo was corrected in Line 71.
Comment 7. The Reviewer is right. We corrected it in Lines 266267.
Comment 8. Vectors i_{E}, j_{E} and K_{E} are the unit vectors of frame S_{E}. We added a sentence in Sec 2.1 and another in Sec. 5.1 to remind their meaning to the readers.
Comment 9. We cannot add a text to the y axis because Fig. 6 presents two different quantities, which are explained in the legend.
Comment 10. We agree that such idea improves the readability of the manuscript. Part of the material of Sec. 3 was moved to a new Appendix.
Comment 11. We revised the manuscript a fix some typos.
Citation: https://doi.org/10.5194/wes2023170AC2

RC3: 'Comment on wes2023170', Anonymous Referee #3, 26 Mar 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023170/wes2023170RC3supplement.pdf

AC3: 'Reply on RC3', Gonzalo SanchezArriaga, 02 Apr 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023170/wes2023170AC3supplement.pdf

AC3: 'Reply on RC3', Gonzalo SanchezArriaga, 02 Apr 2024
Model code and software
LAgrangian Kite SimulAtors (LAKSA) G. Sanchez Arriaga and A. Pastor Rodríguez https://github.com/apastor3/laksa
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