## Abstract

Most studies regarding models of tensegrity systems miss the possibility of large static deformations or provide elaborate and lengthy solutions to determine the system dynamics. Contrarily, this work presents a straightforward methodology to find the dynamic characteristics of a guyed tensegrity beam structure, allowing the application of vibration control strategies in conditions of large deformations. The methodology is based on a low-order, adaptive, nonlinear finite element model with pre-stressed components. The method is applied to numerical and experimental models of a class 2 tensegrity structure with a high length-to-width aspect ratio. Image processing and accelerometer data are combined to extract the experimental natural frequencies of the structure, which are compared to numerical results. Prony’s method is applied to estimate damping, and a numerical control strategy is employed using the dynamical model of the structure.

## 1 Introduction

Tensegrity structures became popular in 1948 by Snelson as an art form. Fuller and Applewhite recognized their engineering value and created the term tensegrity as a contraction of “tensional” and “integrity” [1]. Tensegrity structures contain compressive discontinuous parts (struts) and continuous tensile parts (cables) connected with ball joints (pin-joint) [2]. The rigidity of tensegrity systems results from a state of self-stressed equilibrium between cables under tension and compressed rigid bodies [3]. Generally, the self-weight of the cables can be disregarded [4]. However, recent articles have explored factors such as friction and contact between struts [5,6]. Tensegrities are considered class one if the bars do not touch each other. Otherwise, the class is given by the maximum number of struts sharing a common node. A fundamental aspect of tensegrities is the uniaxial stress property of the components: cables and bars must be under tension and compression, respectively [7]. This property contributes to an optimized choice of materials and geometry, focusing on resistance to traction in the cables and compression (and buckling) in the bars [3].

Tension structures, such as cable nets, membrane structures, and tensegrity domes, offer significant advantages over conventional structures, such as steel structures [8]. Due to their design, tensegrities can serve as the foundation of lightweight and strong mechanical structures using less material [9] when designed efficiently [10]. Among various traditional approaches, the tensegrity concept is one of the most promising for active and deployable structures [11–13]. For deployment, the disjointed struts provide a crucial advantage of the tensegrity concept, enabling a compact package [14]. In addition, they may integrate structural and control systems since the elastic components can carry both sensing and actuating functions [15]. Also, a small amount of energy is sufficient to control the shape of tensegrity structures, which is advantageous for active control [16]. Tensegrity structures also exhibit excellent shape change capability and shock resistance. These features can be beneficial in mechanisms, robots, and space exploration rovers [17–20]. The concept and features of tensegrities are valuable in high technology and aerospace structural applications [21].

Recently, tensegrities have received significant attention from scientists and engineers in fields such as architecture, civil engineering [22], biology [23], and in the construction of dampers [24], shape-shifting structures, such as twisting wings [25] and ocean wave energy harvesting mechanisms [26]. Many form-finding methods and topology designs have been developed recently [27–33] along with the introduction of new tensegrity families [34,35]. The kinematic and dynamic behaviors of tensegrities have been investigated in works such as those in Refs. [5,6,9,11,14,36,37]. Particular concerns in these studies include the magnitudes of structural displacements, which can be large, even if the deformation of individual members are small [38]. Slack cables also represent a relevant problem, especially when such elements transition from tensioned to slack states in tensegrity mechanisms, generating roughness in the system movements [39,40]. The literature shows a large number of works related to the nonlinear analysis of tensegrity systems. Kebiche et al. [38] developed a calculation method for tensegrity systems taking into account geometrical nonlinearities. They used a total Lagrangian formulation to determine the tangent matrix and the internal stress vector for a four-strut tensegrity system subjected to traction, compression, bending, and torsion loading. They also studied a multi-cell beam under traction. The results reveal nonlinear behavior due to flexibility, with rigidity increasing with the external load and self-stress level, except in compression loads. The mechanical behavior was observed to be similar to anisotropic materials, where the displacements response depends on the orientation of the loads. Tran and Lee [41] also presented a numerical method for large deflections, including both geometric and material nonlinearities. They utilized total Lagrangian and updated Lagrangian formulation to treat the geometrical nonlinearity, while material nonlinearity was handled through the elastoplastic stress–strain relationship. Their proposed method calculates responses of the quadruplex unit module, double layer quadruplex grid, and five-quadruplex module beam under external loads. The results indicate that the stiffness of tensegrity structures increases with the self-stress level. In the quadruplex unit module, the stretching stiffness dominates over bending. The bending strength capacity of the double layer quadruplex tensegrity grid is not significantly affected by the self-stress level. The updated Lagrangian formulation is recommended for the large deflection analysis of tensegrity structures. Zhang et al. [42] developed an efficient numerical method capable of capturing mechanical responses of tensegrity structures with very large and highly nonlinear deformations under different conditions. This method is applicable for all types of tensegrities subjected to either external or internal applied actuation.

The literature is limited in terms of experimental results on high aspect ratio tensegrity beams. This work extends the available collection to assist the community in validating new static and dynamic models. Additionally, considering the dynamical analysis works investigated, there is a knowledge gap regarding vibration control of tensegrity structures with fast and large geometry changes. Most kinematic and kinetic models of tensegrities found in the literature employ lengthy and time consuming adaptive or relaxation techniques to evolve from the different conditions of large displacement movements. Implementing control strategies based on these techniques can be challenging. In contrast, the present work proposes simpler manners to build a dynamic model of tensegrity structures that can be assembled in real-time, yielding sufficient parameters to allow the vibration control of the structure at any stage of its present large displacement movement. The proposed methodology is based on the superimposition of a low-order, adaptive, nonlinear finite element model with pre-stressed components onto the nodes of the statically deformed structure. The node positions of the structure, then, are determined by a polynomial function obtained previously from static analysis data, undergoing the workspace of possible large deformations. To verify and validate the proposed analysis technique, the numerical model and a physical prototype of a planar tensegrity guyed beam are utilized. The dynamic characteristics under different operational conditions are also investigated.

Furet and Wenger [43] also studied the kinematics and actuation of a planar tensegrity manipulator with two levels (2-*X*) and observed that friction can be relevant to the dynamics of the mechanism, which should be added in future reports. Even though the results from a two-level tower can be extrapolated to a certain extent, the behavior might change significantly in higher aspect ratios. The prototype presented in this work features six levels (6-*X*), and the damping ratios are estimated. There is limited data available in the literature regarding these kinds of analyses on tensegrity guyed beams. The tensegrity structure presented in this paper counts on comparisons to the studies presented in Refs. [44,45], which focused on a solid, long, guyed beam under large displacements. However, the present model was observed to behave similarly. Furthermore, the dynamic model developed in this work is used to apply the vibration control procedure shown in Ref. [45] for long beams, which has been verified experimentally [46]. An adaptation of the dead-band controller shown in Ref. [47] for tensegrity systems is another viable option.

## 2 General Modeling Methodology

### 2.1 Static Analysis of Tensegrity Structures.

*i*th node coordinates are described by a position vector

**n**. A node matrix

_{i}**N**containing all

*b*nodes of the structure is defined (Eqs. (1) and (2)). The connectivity

**c**of the

_{k}*k*th member joining nodes

*i*and

*j*is given by Eq. (3), where

**e**is a vector with a length of

_{i}*b*, filled with zeros and containing 1 in the

*i*th position. The connectivity matrix is thus assembled (Eq. (4)).

**m**represent the

_{k}*h*members of the structure in the members matrix

**M**, which is obtained from the node and connectivity matrices (Eq. (5)).

**K**of a member

_{k}*k*can be defined (Eqs. (6) and (7)) as

*s*

_{k}=

*f*

_{k}/

*l*

_{k}is the force density of a member and

*K*

_{B}=

*E*

_{k}

*A*

_{k}/

*l*

_{k}is the linear elastic stiffness for bars. Therefore, the stiffness matrix

**K**is composed of the pre-stress

_{k}**K**and material $K\varphi $ components. Terms

_{Σ}*f*

_{k}and

*l*

_{k}represent, respectively, the normal force and length of the

*k*th member.

**c**

_{k}c_{k}^{T}is

*b*×

*b*and

**L**is 2 × 2 (in a planar system), the product

_{k}**c**

_{k}c_{k}^{T}⊗

**L**is 2

_{k}*b*× 2

*b*, the same dimensions as the global stiffness matrix

**K**, which is assembled by the simple sum of all element stiffness matrices

_{G}**K**. Finally, for a given load vector

_{k}**f**, the displacements

**u**are calculated as

**f**, while temperature loads can be introduced directly into the pre-stress component

**K**as the effect of thermal expansion.

_{Σ}### 2.2 Nonlinear Static Analysis.

The method described in the previous section is appropriate for small displacements, but can be adapted to assess behavior under large deformations. The total load, which causes large displacements, must be applied in small increments to satisfy the small deformations assumption in all steps, resembling the strategy used in Ref. [49]. The intensity of the total force **f** is divided by the number of steps *p*, transforming one nonlinear analysis into *p* linear analyses [50,51]. Material properties *E* and *ρ*, cross-sectional areas *A*, and incidences are assumed to be constant during all steps of the analysis.

**N**and internal stresses

*σ*have to be updated at every step according to the new displacements, as shown in Fig. 1. The internal loads affect the pre-stress component

**K**of the stiffness matrix

_{Σ}**K**. The force density

_{k}*s*

_{k}is related to the internal stresses

*σ*of the deformed members of the tensegrity structure by the relation in Eq. (9).

The Euler’s incremental loads procedure is not recommended for solving models containing a high number of elements due to its relatively lower efficiency when compared to more sophisticated methodologies. However, models with few elements can be calculated in an adequate simulation time. Additionally, the procedure is relatively simple to implement and combine with the methodology that assumes small displacements.

*lf*and starting

*lo*lengths of a determined cable are used to represent the external load in a dimensionless measure

*c*=

*lf*/

*lo*. The polynomials are expressed in matrix form (Eq. (10)), where

**X**and

**Y**represent the coefficient matrices, and

**r**is the position vector of node

_{n}*n*in

*x*(

**i**) and

*y*(

**j**) coordinates. Furthermore, the image of the polynomials for 0 <

*c*< 1 can be interpreted as the workspace of the mechanism joints, which adds to the work developed in Ref. [52].

### 2.3 Modal Analysis.

*q*is assumed to be equally distributed within the

*w*bars, and first-order shape functions are used to calculate each element mass matrix

**H**(Eq. (11)).

_{k}**H**is the superposition of the element mass matrices

_{G}**H**. The natural frequencies

_{k}*ω*and modes of vibration

**d**are extracted from the solutions of the eigenvalue problem (Eq. (12)).

### 2.4 Damping Parameters.

**D**is the damping matrix and can be represented as proportional to mass and stiffness (Eq. (14)). The proportional damping assumption has to be taken carefully in tensegrity systems [53,54]. Numerical results (Fig. 9) show that the first two natural frequencies are sufficiently separated and the dynamic study focuses on them. Additionally, the experimental procedures considered impulse excitation instead of harmonic. These criteria do not guarantee that a proportional damping model will be accurate, but suggest that the assumption can lead to reasonable results. This hypothesis is assumed subject to a second inspection in the results section.

*α*(geometry related) and

*β*(material and joints associated). The cost function (

*J*) of the optimization problem that has to be minimized is presented in Eq. (17):

*d*corresponds to the vibration mode,

*D*is the total number of modes, and

*ξ*

_{num}and $\xi exp$ are the numerical and experimental damping ratios, respectively.

To calculate the cost function *J*, the number of numerical damping ratios must match the number of experimental damping ratios. Therefore, a model order reduction is performed to achieve a second-order state-space model. Minimizing *J* is equivalent to finding *α* and *β* that generate numerical damping ratios *ξ*_{num} as close as possible to the experimental damping ratios $\xi exp$.

### 2.5 Control.

**H**, stiffness

**K**, and damping

**D**matrices of the tensegrity beam model are used in the implementation of a control technique. The numerical

*H*

_{∞}control strategy presented in Ref. [45] is replicated, with a single motor acting on the structure. The external load vector

**f**(

*t*) in Eq. (13) can be expressed as a combination of disturbance and actuator terms (Eq. (18)), where

**B**and

**b**stand for disturbance matrix and force input vector, while

**w**(

*t*) and

*u*(

*t*) are the disturbance and actuator forces, respectively. Transient environmental conditions, such as wind loads, can be incorporated into the disturbance term.

*hinfsyn*matlab function is used. Their filter gains must be adjusted to minimize the

*H*

_{∞}norm, thereby increasing the system’s robustness.

## 3 Guyed Tensegrity Beam Model

### 3.1 Numerical Procedures.

A planar tensegrity beam with six sections, as shown in Fig. 2, is studied in this paper. Thick lines represent bars and thin lines stand for the tendons. The maximum number of rigid bodies connected by the same node is two. Therefore, it is a class 2 tensegrity. A cable pulls the tip toward a fixed point located close to the base and causes large deformations in the structure, following the studies presented by Holland et al. [56] and Kurka et al. [45], but with a guyed tensegrity instead of a long beam. The large deformations require a nonlinear static study. Additionally, the high aspect ratio suggests low frequencies and motivates a vibration analysis.

The numerical model was implemented under the following assumptions:

The structure is composed of 30 elements (12 bars and 18 cables) and 14 nodes.

Bars and cables are linearly elastic and one-dimensional elements.

The total mass of the structure is attributed to the bars.

Damping is included in the joints and assumed to be proportional to mass and stiffness.

Gravity is included. The self-weight is inserted as a load.

Rubber bands are applied as cables because their natural length is shorter than the starting distance between nodes. Therefore, they begin the experiment under tension and bear a larger displacement before becoming slack. Additionally, friction in the joints was not considered in the static analysis because it mostly depends on relative motion.

The load changes its direction during the analysis. As the top node moves while the base does not, the guying cable varies its orientation. Therefore, the force has to be adjusted in each step of the incremental loads procedure. Despite the relatively low efficiency of the incremental loads technique, in this study, all 30 members can be represented by a single element each, leading to a very low computational cost in each iteration. To conveniently access the static analysis results, the node positions are associated with the relative length *c* = *lf*/*lo* of the pulling cable through a polynomial fit. Base nodes remain fixed, as their coordinates do not vary under different levels of *c*. The node coordinates matrix (which can be quickly built from a given pulling cable relative length *c*) provides enough information to assemble the mass and stiffness global matrices.

*λ*

_{d}for each

*d*mode are obtained from experimental responses using Prony’s method (as described in Ref. [55]) and using Eq. (19) as a fitting function.

*P*

_{d}is the amplitude, and

*D*is the number of modes.

### 3.2 Experimental Procedures.

The prototype is built with 150 mm 3D-printed bars and rubber bands as cables, forming a class 2 tensegrity. Each bar is composed of two strips that entwine the strips of its pair (Fig. 4), minimizing displacements in depth and increasing the critical buckling load, which could be problematic in planar tensegrity systems [57]. Also, one of their ends is slightly arched to allow acute angles. The use of 3D printing technology facilitates this unique design. The bi-dimensional beam is hung upside down against a grid paper and guyed by a nylon cable, as shown in Fig. 3. The prototype is bi-dimensional and offers more stability when hung upside down, which reduces out-of-plane motion. Although most applications suggest a stand-up position, it is important to validate the numerical model under reduced error conditions. Deformed configurations are photographed, and image pixel positions are calibrated to yield metric deformation measurements. The camera is placed as far as possible from the prototype, and the images are zoomed in to minimize perspective errors. Tolerance for these uncertainties is indicated in the results section.

Four deformed configurations and the natural layout under self-weight are studied, and the final length of the pulling cable is replicated in the numerical model for all configurations. Vibration analyses are performed in all positions. An accelerometer is attached to the last bar (highlighted in Fig. 3) to extract the natural frequencies. The accelerometer is used to generate a frequency response function using the impact hammer test. Additionally, the experiment is recorded by a 30 frames per second (fps) camera, and image processing software is used to track the node positions over time. Finally, a fast Fourier transform algorithm is applied to convert those position-time data sets into frequency responses, providing more experimental data to validate the method. Utilizing image processing to the vibration study is convenient in this experiment because this structure exhibits low natural frequencies, staying within the camera’s fps limits. Furthermore, the mass of the accelerometer may interfere with the results. However, the image processing approach is impracticable for higher frequencies, leaving a range of interest to rely on the accelerometer data.

### 3.3 Properties of the Cables.

Rubber bands are employed as cables due to their low Young’s modulus and extensive elastic range, enabling small input loads to induce large displacements without plastic deformation. Furthermore, as the relaxed length of the bands is shorter than the distance between nodes, they are initially pre-stressed, reducing the occurrence of slack cables in the analysis. An experiment is conducted to find the Young’s modulus of the rubber. A hook scale pulls a 0.083 m long band, and the normal force is acquired for every 0.01 m or 0.005 m displacement, yielding the results shown in Table 1.

l (m) | f (N) | $\u03f5$ | σ (Pa) |
---|---|---|---|

0.083 | 0.000 | 0 | 0 |

0.090 | 0.883 | 0.08 | 3.92 × 10^{5} |

0.095 | 1.275 | 0.14 | 5.67 × 10^{5} |

0.100 | 1.668 | 0.20 | 7.41 × 10^{5} |

0.105 | 1.962 | 0.27 | 8.72 × 10^{5} |

0.110 | 2.256 | 0.33 | 1.00 × 10^{6} |

0.120 | 2.747 | 0.45 | 1.22 × 10^{6} |

0.130 | 2.943 | 0.57 | 1.31 × 10^{6} |

0.140 | 3.335 | 0.69 | 1.48 × 10^{6} |

0.150 | 3.630 | 0.81 | 1.61 × 10^{6} |

l (m) | f (N) | $\u03f5$ | σ (Pa) |
---|---|---|---|

0.083 | 0.000 | 0 | 0 |

0.090 | 0.883 | 0.08 | 3.92 × 10^{5} |

0.095 | 1.275 | 0.14 | 5.67 × 10^{5} |

0.100 | 1.668 | 0.20 | 7.41 × 10^{5} |

0.105 | 1.962 | 0.27 | 8.72 × 10^{5} |

0.110 | 2.256 | 0.33 | 1.00 × 10^{6} |

0.120 | 2.747 | 0.45 | 1.22 × 10^{6} |

0.130 | 2.943 | 0.57 | 1.31 × 10^{6} |

0.140 | 3.335 | 0.69 | 1.48 × 10^{6} |

0.150 | 3.630 | 0.81 | 1.61 × 10^{6} |

Since the rubber material exhibits a nonlinear stress–strain behavior, we use the Young’s modulus from the stress–strain relation observed on the prototype, which never exceeds 30%. Therefore, the best-fitting Young’s modulus for the prototype of 3.00 MPa is obtained using the first six experimental stress–strain test data points, as shown in Fig. 5.

### 3.4 Relevance of the Force Density Term.

*s*

_{k}compared to the classical bar stiffness

*EA*/

*l*. Since its contribution to the global stiffness might not be significant [58], this term could be removed to simplify the methodology. This subsection focuses on showing the relevance of such term and why it should not be removed in this study. For a horizontal element of length

*l*, cross-sectional area

*A*, and Young’s modulus

*E*under the action of a regular force

*f*, the members’ matrix is defined by Eq. (21).

**L**and

_{1}**K**matrices are built (Eqs. (22) and (23)) following the methodology described in Section 2.1.

_{G}*K*

_{B}and force density

*s*

_{k}contribute directly to the stiffness matrix. The first one is assumed to be constant, but the second one is not, as the normal force

*f*changes depending on the loads. The tension

*σ*does not exceed 3MPa in this study, therefore we may plot the comparison between

*s*

_{k}and

*K*

_{B}versus tension

*σ*(Fig. 6). Since

*s*

_{k}reaches up to $12%$ of stiffness

*K*

_{B}, the contribution of the force density must be considered in the stiffness matrix of the present prototype.

## 4 Results and Discussion

### 4.1 Nonlinear Static Analysis.

The prototype’s response and the simulation results are represented by dashed and solid lines, respectively (Figs. 7(a)–7(e)). Thick lines represent the struts, and thin lines stand for the cables. The circles show the 10 mm radius tolerance of the experiment regarding node positions. The total weight of the structure (290 g) is equally distributed among the 12 bars, and the cables are considered massless. The Young’s modulus of the bars is considered to be 2 GPa, and they are assumed to have a Hookean behavior. The final lengths of the pulling cable are 0.61 m, 0.49 m, 0.40 m, 0.28 m, and 0.16 m for configurations 0, 1, 2, 3, and 4, respectively.

Because the structure is hung upside down, gravity points upwards in the graphs as shown in Figs. 7(a)–7(e). The first graph contains data from the experiment and simulation without any loads in the pulling cable (under self-weight only). Therefore, the tip of the beam holds its own weight, and the base supports the weight of the whole structure, causing more significant deformation of the sections close to the support. All the other four graphs contain the guying cable, each with a different final length.

Despite showing a slightly nonlinear behavior (Fig. 5), which brings errors to the graphs in Figs. 7(a)–7(e), the rubber bands are helpful in reducing the number of discontinuities (transitions between slack to tensioned states) in the analysis. Additionally, despite the geometric constraints of the bars that force the experiment to remain two-dimensional, they exhibit slight bending, allowing for a small displacement in depth. Using the tip position as a reference, the modeling error relative to the beam length is calculated (Table 2). Most of the experimental data points align with the simulations, and the maximum error in the tip position is 3.51% in configuration 1, indicating the robustness of the model.

Config. | Experimental | Numerical | Error (%) | ||
---|---|---|---|---|---|

x (m) | y (m) | x (m) | y (m) | ||

0 | 0.112 | 0.535 | 0.101 | 0.524 | 2.97 |

1 | 0.252 | 0.474 | 0.235 | 0.468 | 3.51 |

2 | 0.332 | 0.400 | 0.319 | 0.400 | 2.49 |

3 | 0.394 | 0.290 | 0.389 | 0.290 | 1.02 |

4 | 0.411 | 0.168 | 0.420 | 0.169 | 1.64 |

Config. | Experimental | Numerical | Error (%) | ||
---|---|---|---|---|---|

x (m) | y (m) | x (m) | y (m) | ||

0 | 0.112 | 0.535 | 0.101 | 0.524 | 2.97 |

1 | 0.252 | 0.474 | 0.235 | 0.468 | 3.51 |

2 | 0.332 | 0.400 | 0.319 | 0.400 | 2.49 |

3 | 0.394 | 0.290 | 0.389 | 0.290 | 1.02 |

4 | 0.411 | 0.168 | 0.420 | 0.169 | 1.64 |

### 4.2 Polynomial Interpolation of Nodal Positions.

The trajectory of each node is mapped (Fig. 8) for 150 load increments from configuration 1 to configuration 4.

**X**and

**Y**are defined in Eqs. (24) and (25) for generating

*x*(

**i**) and

*y*(

**j**) coordinates in millimeters.

^{5}less than running a nonlinear static analysis of this model. This fast transformation allows this technique to be implemented in real-time vibration control strategies.

### 4.3 Modal Analysis.

The first set of graphs (Figs. 15, 16, 17, 18, and 19) contains the numerical results displaying the two first vibration modes and their natural frequencies. The experimental results acquired through image processing and the accelerometer (impact hammer test) are presented in Figs. 20–25 and Figs. 26–30, respectively (these figures are available in the Appendix). In Figs. 20–25, the thin lines represent the frequency spectrum of different points in the same capture or different captures, while the thick line in each graph represents the mean spectral value. Some of the frequency responses show the first two natural frequencies in the same spectrum (positions 0 and 2), but two analyses are required to extract each frequency in position 1. In Figs. 26–30, each thin line represents the spectrum of a different external input, and the thick line in each graph is the mean spectral value. Finally, the results for natural frequencies are compiled and compared in Table 3.

Configuration | Mode | Numerical (Hz) | Image processing (Hz) | Accelerometer (Hz) | Error (%) |
---|---|---|---|---|---|

0 | 1st | 2.80 | 2.36 | 2.87 | 2.5 |

2nd | 4.45 | 4.38 | – | 1.6 | |

1 | 1st | 2.89 | 2.93 | – | 1.4 |

2nd | 4.41 | 4.40 | – | 0.2 | |

2 | 1st | 2.99 | 3.20 | 3.50 | 7.0 |

2nd | 4.56 | 4.37 | – | 4.2 | |

3 | 1st | 3.19 | 3.67 | 3.25 | 1.9 |

2nd | 4.96 | – | – | – | |

4 | 1st | 3.40 | 3.44 | 3.38 | 0.6 |

2nd | 5.82 | – | 6.12 | 5.2 |

Configuration | Mode | Numerical (Hz) | Image processing (Hz) | Accelerometer (Hz) | Error (%) |
---|---|---|---|---|---|

0 | 1st | 2.80 | 2.36 | 2.87 | 2.5 |

2nd | 4.45 | 4.38 | – | 1.6 | |

1 | 1st | 2.89 | 2.93 | – | 1.4 |

2nd | 4.41 | 4.40 | – | 0.2 | |

2 | 1st | 2.99 | 3.20 | 3.50 | 7.0 |

2nd | 4.56 | 4.37 | – | 4.2 | |

3 | 1st | 3.19 | 3.67 | 3.25 | 1.9 |

2nd | 4.96 | – | – | – | |

4 | 1st | 3.40 | 3.44 | 3.38 | 0.6 |

2nd | 5.82 | – | 6.12 | 5.2 |

The *Error* column compares the experimental and numerical results and shows the lowest experimental error. Configurations 0, 1, 3, and 4 report a good agreement between numerical and experimental results, but there is a considerable error in the first mode of configuration 2. The resolutions of the spectrum obtained through image processing are approximately 0.4 Hz, which fit in the 7$%$ error that has garnered attention. These discrepancies primarily stem from the experimental data capturing damped frequencies and the uncertainties associated with material properties and geometry. Furthermore, the peaks shown in the spectra are considerably wide, encompassing the natural frequencies obtained through simulations within their width. The second frequency of configuration 4 could not be extracted through image processing, but was acquired by the accelerometer and concurred with the numerically predicted result. Due to the hardware limitations of the uniaxial accelerometer, some vibration modes where the displacement is mostly orthogonal to the sensor could not be extracted. This limitation is noticeable in the second vibration mode of configurations 0, 1, 2, and 3 and the first mode of configuration 2. Moreover, a high damping ratio may affect the accurate extraction of higher mode natural frequencies through experimental methods. In this study, this issue is present in the second mode of configuration 3, where the damping factor is more significant (Table 4).

Modal parameters | ||
---|---|---|

Configuration | Natural frequency (Hz) | Damping ratio (%) |

0 | 2.72 | 32.5 |

1 | 2.95 | 32.3 |

2 | 3.09 | 17.4 |

3 | 3.33 | 49.4 |

4 | 3.76 | 18.8 |

Modal parameters | ||
---|---|---|

Configuration | Natural frequency (Hz) | Damping ratio (%) |

0 | 2.72 | 32.5 |

1 | 2.95 | 32.3 |

2 | 3.09 | 17.4 |

3 | 3.33 | 49.4 |

4 | 3.76 | 18.8 |

The increase in stiffness in a more tensioned structure is gradual because some cables of the tensegrity become slack under larger deformations, reducing the growth rate of the global stiffness. Contrarily, as the geometry of the tensegrity changes, its effective length decreases, leading to higher natural frequencies in these deformed configurations. In other words, shortening the guying cable causes a few tendons to become slack, but it also places more stress on the taut tendons and shortens the beam, causing an overall increase in the global stiffness. This pattern is more evident in the numerical results for all intermediate configurations (Fig. 9), where higher loads lead to higher natural frequencies.

The vertical axis compares the final length *l*_{f} of the pulling cable in each configuration to the starting length *l*_{0} (configuration 0), therefore keeping the ratio between 0 (maximum load) and 1 (no load). The axis orientation is reversed to follow the pattern used in the work presented in Ref. [45]. Also, the best-fitting experimental results for the first two modes are compared to numerical outputs (lines) in Fig. 10.

### 4.4 Damping.

Free damped responses for configurations 0–4 (Fig. 11) are obtained from the video recordings used in Sec. 4.3 to determine the natural frequencies of the tensegrity.

Prony’s method is applied for all five configurations. The first mode natural frequencies and damping ratios of each configuration are presented in Table 4. Based on previous studies [45], the damping ratio in intermediate configurations can be reliably estimated from five states using a low-order polynomial interpolation. The steepest descent algorithm is applied to solve the optimization problem, and the proportional damping parameters are listed in Table 5.

Parameters | ||||
---|---|---|---|---|

Config. | α · 10^{3} (1/s) | β · 10^{3} (s) | Natural frequency (Hz) | Damping ratio (%) |

0 | 4.0 | 36.7 | 2.83 | 32.5 |

1 | 1.5 | 35.1 | 2.95 | 32.3 |

2 | 7.3 | 18.6 | 3.05 | 17.4 |

3 | 10.8 | 51.8 | 3.29 | 49.4 |

4 | 6.39 | 37.8 | 3.48 | 18.8 |

Parameters | ||||
---|---|---|---|---|

Config. | α · 10^{3} (1/s) | β · 10^{3} (s) | Natural frequency (Hz) | Damping ratio (%) |

0 | 4.0 | 36.7 | 2.83 | 32.5 |

1 | 1.5 | 35.1 | 2.95 | 32.3 |

2 | 7.3 | 18.6 | 3.05 | 17.4 |

3 | 10.8 | 51.8 | 3.29 | 49.4 |

4 | 6.39 | 37.8 | 3.48 | 18.8 |

### 4.5 Control.

To evaluate the proposed procedures, the dynamical characteristics of configuration 2 are used as inputs for the *H*_{∞} control strategy. This configuration is selected because it presents the lowest damping factor and the structure is in an intermediate shape between configurations 0 and 4. The filters replicate those used in Ref. [45] but with adjusted gains to satisfy the different structural dynamics. The open- and closed-loop systems are compared in Fig. 12 in terms of singular values. The frequency peak at 2.88 Hz is significantly reduced by the controller. Additionally, an impulse force (Fig. 13) and a 2 N root-mean-square (RMS) random vibration (Fig. 14) are loaded on the tensegrity beam tip node in both directions. The infinite norm is 0.224 and 0.078 for the system without and with control, respectively. The controlled response to an impulse input shows a higher vibration suppression compared to the random input response. This suggests that this control technique is less efficient under random inputs for this model. Still, the vibration levels of the controlled response to random inputs are approximately 36% lower than the uncontrolled response. The control results regarding configurations 0, 1, 3, and 4 are equivalent in behavior to configuration 2.

## 5 Conclusions

A lightweight, long, and flexible 2D tensegrity cable guyed beam was both modeled and built. The model accounted for self-weight and was verified in static experiments with five different loads. The proposed methodology offers reliable results through a more straightforward process compared to current techniques. It involves combining Euler’s incremental loads method for solving nonlinear problems in finite element analysis with a procedure designed to calculate linear statics of pre-stressed tensegrity structures. The incremental loads procedure is convenient to implement, but also less efficient than other methodologies. However, the structure under study requires modeling very few elements, which keeps the computational cost of the simulation low and highlights the advantages of simpler implementation. Most of the experimental data aligned with the simulation outputs, indicating the accuracy of the proposed methodology. The image processing technique provides closer results to the numerical outcomes than the accelerometer data for low frequencies, which are not far from the minimum range of the instrument (1 Hz). However, image processing is limited by the camera’s fps, narrowing the acquisition range. Advantages such as high accuracy, ease of use, low cost, quick data acquisition, and absence of instrumentation attached to the structure justify its use in low-frequency experiments. Damping ratios were estimated through an optimization technique comparing image processing and numerical data using Prony’s method. This study estimates the damping ratio for five configurations of the structure, with intermediate points reliably predicted by a low-order polynomial interpolation. The numerical natural frequencies obtained through Prony’s method showed good agreement with experimental results, with a maximum difference of 7.45%. Finally, a numerical *H*_{∞} control strategy is applied to suppress vibrations in the tensegrity beam using the presented dynamical model. Future research steps involve assessing the influence of environmental factors on the behavior of the structure, such as wind and temperature loads, and performing experiments to verify the numerical control results and optimize the controller.

## Acknowledgment

The authors thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil’s research support foundation, for sponsoring this work.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*b*=number of nodes

*c*=final to initial length ratio

*f*=normal force (N)

*l*=member length (m)

*q*=structure’s mass (kg)

*s*=force density (N/m)

*t*=time (s)

*u*=actuator force (N)

*w*=number of struts

*x*=position in

*x*direction (m)*y*=position in

*y*direction (m)**b**=force input vector

**c**=connectivity vector

**e**=auxiliary connectivity vector

**f**=external loads vector (N)

**m**=member vector (m)

**n**=position vector (m)

**r**=position vector in the static analysis (m)

**u**=displacements vector (m)

**w**=disturbance forces (N)

**x**=position vector in the dynamical analysis (m)

**z**=order reduction auxiliary vector

*A*=cross-sectional area (m

^{2})*D*=number of vibration modes

*E*=Young’s modulus (Pa)

*J*=cost function

*P*=amplitude

**B**=disturbance input vector

**C**=connectivity matrix

**D**=damping matrix (N s/m)

**G**=order reduction auxiliary matrix

**H**=mass matrix (kg)

**K**=stiffness matrix (Pa)

**L**=stiffness matrix auxiliary component (Pa)

**M**=members matrix (m)

**N**=node matrix (m)

**Q**=order reduction auxiliary matrix

**X**=*x*coefficients matrix (m)**Y**=*y*coefficients matrix (m)*ft*=fitting function