The dynamic behavior of the tensegrity structure (helix) of a rotary airborne 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, 0.2, and 0.4 s, respectively. Secondly, the dynamics in nominal operation was investigated by keeping both end tips fixed 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 dynamics of the helix when the lower end is fixed and the upper end is driven in a circular motion of frequency

The increase in wind power density with altitude

An interesting subfamily of AWE systems involves concepts based on rotating kites or rotors that use the well-known phenomenon of auto-rotation for harvesting wind energy

Flight testing activities with the RAWE machine showed that the tensegrity structure exhibits a rich dynamics that involves longitudinal, lateral, and torsional waves. The tensegrity structure has a helix-like shape that acquires a high torsional and bending stiffness when a traction load is applied. Finding the operational limits of the helix is crucial for the reliability of the RAWE machine. According to experimental tests, the structure can collapse if the torque is above a certain threshold

An important property of the tensegrity structures of RAWE machines is that they are slender; i.e., their unstretched length (

Figure

Sketch and frames of reference of the rotary AWE machine.

Following

At every point of the center line we also define a local frame

Momentum and angular momentum equations of the structure with total mass

The model is completed by the constituent relations that depend on the elastic properties of the material and on the shape and the dimension of the cross section of the structure. Writing the twist vector as

The set of equations is simplified if some considerations based on the geometry of the structure and the forces and torques that are expected to act on it are taken into account. We start by assuming that

After substituting these results in Eqs. (

For convenience, we introduce the normalized variables

Regarding the boundary conditions, we do not impose the exact conditions of a real rotary machine, i.e., coupling the dynamics of the structure and the rotor. Such a complete analysis is beyond the scope of this work that is focused on the helix. We assume that the position vector of the central line at

Since our work does not couple the helix with a rotor, the torque at

Equations (

Following a Galerkin method, we write Eqs. (

Integrations by parts of Eqs. (

After constructing the vector

Several tests were carried out to verify the correct implementation of Eq. (

Three laboratory tests were carried out to find the values of

Panels

As shown in the top panel of Fig.

Experimental results of the axial test.

A classical bending test was carried out to determine the

Experimental results of the bending test.

Difficulties were found for the experimental determination of the torsional stiffness. As shown in the bottom panel of Fig.

In principle, a simple model for the relation between the torsion torque and angle is

Experimental results of the torsion test.

This section studies three different dynamic scenarios of the helix. For all of them we used the parameters shown in Table

Common parameters used in the analysis.

We consider a helix with the properties of Tables

Parameters used in the analysis of solutions with fixed ends.

The analysis focuses on stationary solutions of Eq. (

The solution of System (

The linear stability of the stationary solution was then investigated in the framework of Eq. (

Stability results of the stationary solution.

n/a: not applicable

We now investigate the dynamic response of the helix when the torque at the upper end varies as

Figure

Panels

The real operation of a RAWE machine is more complex than the scenario described in Sect.

The experiment, which was carried out inside the laboratory, used a structure with the characteristics in Tables

Parameters used in the analysis of solutions with a mobile end.

As shown in Fig.

Evolution of the tension (left) and the forcing frequency

The same scenario was also studied by using the simulator. Besides the parameters in Tables

The result of the analysis is the bifurcation diagram with

Therefore, we conclude that the numerical tool is able to capture some interesting phenomena correctly, and it can be used to predict them qualitatively. However, if the interest is in getting quantitative values, then it should be used carefully. This conclusion is reasonable because the simulator relies on a set of simplifying hypotheses and also needs to be fed with inputs that have a certain level of uncertainty. As shown by the results of the experimental characterization of the structure in Sect.

Maximum values of the tension at

The results of this work show that the proposed model based on Cosserat theory is appropriate for getting insight into the dynamics of the helical structure of a RAWE machine. It offers some advantages as compared to previous models based on springs and disks or a large set of masses linked with springs and dampers. For instance, unlike previous models, our model captures the three interesting motions of the helix, thus improving the fidelity. It provides a couple of nonlinear partial differential equations that can be solved with well-known numerical methods like the finite-element method used in this work. Such a compact formulation, which includes stretching, bending, and torsional effects, highlights the role of the different terms in the dynamics of the structure and their coupling. It has also been used here to estimate the characteristic times for the propagation of longitudinal, lateral, and torsional waves in the helix. Moreover, the airborne rotor and the ground generator enter the model as boundary conditions that include the position and velocity of the end sections of the structures and the external torque applied to them. The model was completed with the axial, bending, and torsional stiffness of a real AWE machine obtained by conducting three dedicated experiments in the laboratory. Since axial, bending, and torsional stiffness is essential information for RAWE machine simulators, such experimental work fills an important gap in the field. The relation between the bending and the torsional stiffness and the tension was measured for the tensegrity structure. It was also shown that the torsional stiffness depends on both the tension without torsion and the history of the structure as the tension is increased. A hysteric behavior was measured, which poses a challenge for the modeling of the helix.

The analysis of the stationary solution with no torsion and fixed ends, which was found independently by using a shooting algorithm, allowed us to verify the correct implementation of the finite-element code. A linear stability analysis was performed to identify longitudinal, lateral, and torsional modes and their natural periods. The quickest mode of the RAWE machine of SomeAwe corresponds to a longitudinal mode with a period of 0.03 s, whereas lateral and torsional modes exhibit natural oscillations with 0.2 and 0.4 s. Therefore, the natural frequencies of the helix predicted by the simulator are 33.3, 5, and 2.5 Hz. Simple estimations based on classical results for beams are in agreement with these results and constitute a second test for the correct implementation of the code.

The simulation tool provided interesting information about the nominal operation of the RAWE machine like the transient of the angular velocities of the cross sections of the helix, the tension, and the torques. A numerical analysis was carried out by keeping the two end points of the structure fixed. It assumed that the rotor imposes a time-dependent external torque that approaches 25 N m, which is a typical value for the RAWE machine under consideration. It was shown that a simple proportional-derivative controller for the torque at the ground generator can stabilize the angular spinning velocity of the structure at the target value (120 rpm). For this configuration, the tension is almost constant in time and throughout the structure. The evolution of the variables and their values are well aligned with the data collected in field testing by SomeAWE Labs.

In a second numerical analysis, the lower end of the helix was kept fixed, and the upper end was mobile. Although an auxiliary kite is used to anchor the rotor, in a real RAWE machine the upper-end point is not fixed due to wind velocity fluctuations and the dynamic coupling of the structure and the rotor. This scenario was mimicked by imposing a circular periodic motion to the upper end of the structure with a forcing frequency

By assuming that the first derivative is continuous across the elements, we find the following relation between

The code presented in this work was added as an independent module to the open-source software LAKSA (

The data that support the findings of this study are available on request from author Christof Beaupoil (christof.beaupoil@gmail.com).

The model of the RAWE machine was developed by GSA and ACV. The implementation of the code, the numerical analysis, and the preparation of the manuscript were performed by GSA. DU and CB performed the experiments to characterize the RAWE machine in static and dynamic conditions. All the authors contributed to the discussion of the results and the final version of the paper.

Gonzalo Sánchez-Arriaga, Álvaro Cerrillo-Vacas, and Daniel Unterweger declare that they have no conflict of interest. Christof Beaupoil is the founder and owner of SomeAWE Labs.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work was carried out under the framework of the GreenKite-2 project (grant no. PID2019-110146RB-I00) funded by MCIN/AEI/10.13039/501100011033.

This paper was edited by Alessandro Croce and reviewed by three anonymous referees.