Natural frequencies describe the pre-stress in tensegrity structures

Computers and Structures 138 (2014) 162–171

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Natural frequencies describe the pre-stress in tensegrity structures Nasseradeen Ashwear, Anders Eriksson ⇑ KTH Mechanics, Royal Institute of Technology, Osquars backe 18, SE-100 44 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 7 June 2013 Accepted 13 January 2014 Available online 11 February 2014 Keywords: Tensegrity Pre-stress Resonance spectrum Health monitoring Buckling

a b s t r a c t This paper investigates the effect of pre-stress level on the natural frequencies of tensegrity structures. This has been established by using Euler–Bernoulli beam elements which include the effect of the axial force on the transversal stiffness. The axial-bending coupling emphasizes the non-linear dependence of the natural frequencies on the pre-stress state. Pre-stress is seen as either synchronous, considering a variable final pre-stress design or as tuning, when increasing pre-stress is followed in a planned construction sequence. It is shown that for a certain tensegrity structure, increasing the level of pre-stress may cause the natural frequencies to rise or fall. This effect is related to whether the structural behavior can be seen as compression or tension dominant. Vanishing of the lowest natural frequency of the system is shown to be related to the critical buckling load of one or several compressed components. Modes of vibration show that when the force in the compressed components approaches any type of critical buckling load, this results in lower vibration frequencies. The methods in this study can be used to plan the tuning of the considered tensegrity structure towards the design level of pre-stress, and as health monitoring tools. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Over the last four decades, tensegrity structures first invented by Fuller [1] have been extensively studied. They are efficient in some applications, such as certain space applications where the flexibility of shape is important. On the other hand tensegrity structures in some applications may be considered as inefficient [2]. They are used in a wide diversity of fields like aerospace [3], large space structures [4], civil engineering [5] and robotics [6]. Tensegrity structures are classified as free standing pre-stressed pin-jointed cable-strut systems in which contacts are allowed between the struts [7]. They are said to be class one if the bars do not touch each other and class two if two bars are connected at joints [8]. A fundamental aspect of tensegrities is the stress unilateral property of the components: cables and bars must be under tension and compression, respectively. Another important aspect of a tensegrity structure is its pre-stress state, which is an initial internal equilibrium state without external loads, and which to a high degree decides the response properties, e.g., the stiffness to external loading. The design of a tensegrity thereby has to include these aspects in addition to the definition of a specific geometry. In literature, focus lies primarily on the form finding and self-stress state design, [5,9], but only few studies are devoted to the dynamic response and behavior of the tensegrity. Normally, no

⇑ Corresponding author. Tel.: +46 87907950. E-mail address: [email protected] (A. Eriksson). http://dx.doi.org/10.1016/j.compstruc.2014.01.020 0045-7949/Ó 2014 Elsevier Ltd. All rights reserved.

analytical solution exists for the natural frequencies of tensegrity structures, and investigations must rely on numerical simulations. Furuya [10] investigated the vibrational properties of a tensegrity mast and shows that the natural frequencies increase as the level of pre-stress increases. Sultan et al. [11] conclude that the model dynamic range generally increases with pre-stress. Moussa et al. [12] demonstrate that the lowest natural frequency of the unloaded single and multi-module continuous strut tensegrity systems is a function of the pre-stress forces in the components of these systems. Tan and Pellegrino [13] tested the non-linear vibration of a cablestiffened pantographic deployable structure and concluded that it is possible to vary the stiffness of the structure by increasing the prestress level up to the point where the joints start behaving as fully clamped. Ali et al. [14] also discuss the importance of the level of self-stress as a design parameter. Their study indicates that the fundamental frequency of a tensegrity footbridge is not directly influenced by the self-stress level, which must be interpreted as limited variations in frequencies within a reasonable range of pre-stress. Their parametric studies show that the natural frequencies are affected by other design parameters such as the cross sectional area of the components: stiffness and vibration properties are connected. Many of the cited studies state that the natural frequencies of the tensegrity structures increase when the level of pre-stress is increased without addressing the issue to what extent this is true. It is therefore interesting to examine how stiffness, measured from vibration properties, is affected by the pre-stress level, which in a tensegrity introduces both compression in the bars and tension in the cables. According to Greschik [15], a combined set of bars

N. Ashwear, A. Eriksson / Computers and Structures 138 (2014) 162–171

and cables may have a dominant strut vibration effect or a dominant cable vibration effect depending on the material and geometry of that set; the structural response is obviously a combination of these aspects. From the mechanical viewpoint, tensegrities are inherently non-linear structures, with response to loading and excitation strongly dependent on the pre-stressing. With a specific designed pre-stressing introduced, by a shortening or lengthening of one or a few components from their unloaded lengths, the response can, however, be seen as linearized around this ideal state, and this is the common simulation approach. The present work primarily focusses on states during assembly, or after a possible fault in the structure. This means that the conditions deviate from the ideal, and simulations must be performed non-linearly. The nonlinear dependence of the resonance spectrum on pre-tension is a key feature of the present work. This with a particular interest in the lower part of the spectrum, where several frequencies are often very closely situated, from a measurement viewpoint. One of the failure causes of tensegrity structures is the buckling of one or several compressed components. Hanaor [16] states that it is desirable to design double layer tensegrity grids for bar buckling as the governing failure mechanism. Murtha-Smith [17] illustrates that a loss of members due to buckling of one or more bars in a critical truss area is more serious. Another failure mode is the loss of pre-tensioning level, due to relaxation of the control components, over shorter or longer time periods. Both these failure modes are affecting the resonance spectrum of the structure, and are detectable from accurate resonance measurements. In linearized simulations, tensegrities are commonly seen as trusses, and most of the formulations implemented in literature use truss elements for all members in numerical simulations. This gives only vibration modes where several joints are included, as only axial effects are considered. The above-mentioned non-linearity in the response is, however, primarily related to a coupling between axial and bending behavior of the components, where it is well known that, e.g., a string is tuned to its right resonance by the introduction of an axial force. Similarly, the resonance is affected by a compressive force, and the resonance frequency lowered with increasing force magnitude, until the buckling load, where transversal stiffness disappears and infinite non-periodic displacements are obtained. The consideration of the axial force also in compressed components is another new feature of the present work. The present work seeks the resonance frequencies of the structure as a function of the pre-stressing. In particular, it is shown that frequencies going down will correspond to an instability of the structure. These instabilities can be of different types, local or global, but will regardless of failure mode be manifested by a tangent stiffness matrix of the structure approaching singularity. The resonance modes, expressed as relative displacements in all degrees of freedom, will be indicated by the corresponding eigenvalues approaching zero with increasing pre-stress, and are more or less localized to one or a few components. A frequency approaching zero will most typically indicate that one or several compressed members are approaching buckling. In the considered context, this buckling of a member during the assembly stage will not lead to a dramatic collapse, but primarily manifest itself in a softening and possibly an incapacity of the structure to reach the planned pre-stressing state. The present work studies the improvement of current models for resonance frequency simulation of tensegrities by introducing the bending behavior of all components, and by a one-way coupling between the axial force and the stiffness. From this, both local and global vibration modes are obtained. The resonance frequencies are seen as non-linearly dependent on the pre-stressing level in the structure, thereby giving a basis for diagnosis of structural conditions from measured frequencies. The new aspects of tensegrity simulations are shown for simple, plane structures

163

but the basic methods are easily used also for more complex structures, as further discussed below. With respect to applications, the pre-stressing of the structure is seen in two different contexts below. The first context is a design consideration, when the frequencies are evaluated for a final geometry and topology as a function of the pre-stress level. We denote this as synchronous pre-stressing of all unstressed lengths of the components. The calculated resonance frequency spectrum can thereby be seen as a signature for the correctly assembled and pre-stressed structure. The second context is a production viewpoint. In a tuning prestressing, it is assumed that a design of geometry and pre-stress is chosen, thence all unstressed lengths are known. By shortening or lengthening the designated control components, the designed configuration and pre-stress is gradually reached. The frequencies are thereby functions of the level to which the structure is yet prestressed on its route to the designed state. These functions thus start at a state where most of the structure is assembled from unstressed components, and the designed pre-stress state is successively obtained while length-adjusting the control components to (and perhaps, beyond) the design lengths. The corresponding geometry is in general (slightly) deviating from the desired measures. The spectrum of resonance frequencies can then be seen as a progress indicator of the control process. The two contexts are thereby seen as tools in the light of the well-known damage identification process for aerospace, civil and mechanical engineering infrastructure known as structural health monitoring (‘SHM’). A wide range of highly-effective non-destructive assessment tools are normally available for such monitoring [18]. Different indicators are used in structural health monitoring, but vibration health monitoring methods (‘VHM’) use the natural frequencies as indicators of the change in one or more structural properties [19]. Vibration of civil engineering and aerospace structures can thereby be used as:   

An indication of design requirement satisfaction; a quality control tool in the manufacturing process; a damage detection tool during the service life of the structure.

Although many authors have focused on the design and form-finding of the tensegrity structures, vibration health monitoring as design or quality control tools has not been reported in the literature. However, VHM techniques were in [20] applied as a damage detection tool. Panigrahi et al. [20] experimentally applied this method for a single module tensegrity structure, and on a tensegrity grid structure as a damage detection tool without any analytical formulation. They conclude that this technique is convenient and cost effective. In the health-monitoring context, similar methods as the ones discussed in the present paper can be useful tools, when a measured resonance frequency spectrum can be inversely related to the current pre-tensioning state. In this context, where essentially the resonance spectrum is triggered by external excitation, the most visible modes will, however, be the transversal, more or less localized, cable vibration modes, corresponding to bending of the components. This fact emphasizes the assumption in the current work that bending behavior of the components, and an axial-bending coupling are highly useful. This, in turn, focusses also on the representation of joints between components in the structural model, and this is a key issue in the present work.

2. Formulation and method Tensegrity structures have some members which are under compression (bars) and some which are under tension (cables).

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They are essentially an assembly of components connected at the joints. These joints are computationally seen as perfect hinges where no bending moments are transmitted between elements. The analytical solution of the natural frequencies of one prestressed single simply supported plane beam can be found in [21, chap. 4]. The solution of the vibration equation leads to the natural frequencies of a beam with length L as a function of the axial prestress force N o , positive when tension:

x2n ¼

" !# np4 EI N o L2 1þ 2 2 m L n p EI

ð1Þ

c ¼ cosðhÞ ¼ ðx2  x1 þ u2  u1 Þ=Ls ; and s ¼ sinðhÞ ¼ ðy2  y1 þ v 2  v 1 Þ=Ls

where m is the mass per unit length of the beam, E is Young’s modulus, I is the relevant moment of inertia and n is any positive integer. Eq. (1) can be written as a function of the nth critical load of buckling Ncr;n as

x2n ¼

 np4 EI  No 1 m L Ncr;n

ð2Þ

where N cr;n ¼ n2 p2 EI=L2 . Eq. (2) shows that the first beam natural frequency vanishes for No equal to N cr;1 , i.e., at the lowest compressive buckling force of the beam. 2.1. A uniform pre-stressed simply supported beam element In order to include the coupling between transversal and axial stiffness of each component, a non-linear Euler–Bernoulli beam element was used [22, chap. 14]. When using this beam element for the tensegrity structures, rotational degrees of freedom at the member ends are moment released. In this study, material non-linearity has been ignored, while the geometric non-linearity has been considered in the tangent stiffness matrix:

T ¼ K E þ K G K

ð3Þ

T; K  E and K  G are, respectively, the element’s tangent, elastic where K and geometric stiffness matrices in the local coordinates. The geometric stiffness matrix was formulated to include the effect of the axial force on the transversal stiffness of the beam element, [22, chap. 14], but other non-linear effects were neglected. In the considered 2-D case, each element in the structure was thereby characterized by:

2

EA L

6 6 0 6 6 6 0 E ¼6 K 6 EA 6 L 6 6 0 4 0

0

0

EA L

0

12EI L3 6EI L2

6EI L2 4EI L

0

12EI L3 6EI L2

0

EA L

0

0

12EI L3 6EI L2

0

0

12EI 6EI L3 L2 6EI 2EI L L2

0

3

3 2 0 0 0 0 0 0 7 6EI 6 0 6 L 0 6 L 7 7 6 5 10 5 10 7 L2 7 7 6 2EI 7 L 2L2 L L2 7 6 7 0 0 L 10 15 10 30 7  G ¼ No 6 7; K 7 6 L 60 0 0 0 0 0 7 0 7 7 7 6 7 6 0 6 L 0 6 L 7 6EI 7 4 5 10 5 10 5 L2 5 L L2 2L2 4EI 0 10 0 L 30 10 15 L 0

ð4Þ where N o is the elastic forces evaluated from the change in distance between the end points, but neglecting the bowing effect, and A is the cross-sectional area. This stiffness matrix is here used for all elements, with their relevant I and A values. Similarly, the linear beam element mass matrix was formulated as:

2

6 6 6 qAL 6  6 M¼ 420 6 6 6 4

140

0

0

0

156

22L

22L 0

2

54

0 70 0 0

70

0

0

54

4L 0

0 140

13L 0

13L

0

156

13L 3L2

0

22L

with q the material density.

The formulation of the mass matrix equation (5), can be found in [23, chap. 3], where herein the rotational inertia has been neglected. In addition, this consistent mass matrix is used for both cables and bars. The transformation from local to global degrees of freedom was conventionally seen as a transformation matrix TðhÞ relating global to local coordinates through the current orientation angle h for an element. The sine and cosine values of the orientation angle for the element are including the effects of the displacements, so that:

0

3

13L 7 7 7 3L2 7 7 0 7 7 7 22L 5 4L2

ð5Þ

where Ls is the deformed length of the element:

Ls ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2  x1 þ u2  u1 Þ2 þ ðy2  y1 þ v 2  v 1 Þ2

ð6Þ

for xi ; yi the reference coordinates of the two end nodes i ¼ 1; 2, and ui ; v i the corresponding displacements. It is worth noting that in the tensegrity case, the coordinates xi ; yi denote the reference joint points in the designed configuration and ui ; v i the current position of the end points in relation to these points, but that ui ¼ v i ¼ 0 is normally not a stress-free equilibrium position, due to the desired pre-stress. The structural tangent stiffness and mass matrices are conventionally assembled according to the symbolic expressions:

KT ¼

X

 T T T LT ; LT K

M¼

X

 T LT LT MT

ð7Þ

where L is the element connectivity matrix, defined from topology, and the sums extend over all elements. The elastic and geometric stiffness matrices are evaluated while the structure is in equilibrium expressed by displacement coordinates u; v for all nodes. This equilibrium (with external loads possibly zero) is achieved iteratively using the Newton–Raphson method, as briefly explained below. It is noted that the state with pre-stress but without external load must also be solved for. With the tangent stiffness and mass matrices assembled, the problem can be linearized, and the small free undamped vibrations of a structure around the evaluated equilibrium state are obtained from the generalized eigenproblem:

x2 M/ þ K T / ¼ 0 2

ð8Þ

where x is an eigenvalue and / the corresponding eigenvector. The spectral decomposition thereby gives n natural frequencies xi and the related eigenvectors /i of the structure at the considered equilibrium state. It is noted that the vibration modes /i should be seen as small vibrations around a specific state, considering the corresponding internal forces. The number of frequencies is equal to the number of degrees of freedom of the structural model, with only a number of the lowest ones of primary interest. It is noted that the tensegrity structures normally contain a high degree of symmetry. This will mean that the resonance solutions will normally contain sets of closely situated frequencies, and possibly eigenspaces of higher dimensions [24]. Calculated modes in these cases will be arbitrary combinations of orthogonal vibration vectors in this space. The multiplicity of eigensolutions demands some care in interpreting obtained modes, but will otherwise cause only minor problems in the numerical treatment. The eigensolution extraction was in the present work performed by built-in functions of Matlab,1 which are highly accurate even for cases with dense spectra, and when the extraction is less well-conditioned, due to the tangential stiffness matrix approaching singularity. For accuracy and efficiency in tensegrity problems of more practically relevant sizes, subspace iteration methods, [25], would be the preferred choice; in those, a suitable number of the lower eigensolutions are conveniently obtained. 1

The MathWorks, Inc., Natick, USA

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where Ls is a function of the relevant coordinates and displacements for the element, Eq. (6). The evaluation of the internal forces is thereby based on a global displacement vector U i at iterative stage i. Assembly of element forces at stage i to structural forces F iin thereby defines residual forces DF i ¼ F iin  F ex and a correction to displacements are symbolically found from DU i ¼ U iþ1  1

U i ¼ ðK iT Þ DF i where the support conditions are considered. The iterative sequence normally converges in very few iterations (when jjDFjj < ) giving a displacement state U for a specific prestress state. 2.4. Pre-stress setting Varying the level of pre-stress for a tensegrity structure can be achieved in different ways. As noted in the Introduction, we in this study used two settings for level of pre-stress. The first setting was followed in all considered numerical examples while the second one was seen as a particular case in the third numerical example (tuning process).

Fig. 1. X-frame tensegrity beam T10C6.

2.2. Member end moment releases One of the characteristics of the tensegrity structure is that there is no bending moment effect at its joints where components are connected. Moment free connections between beam elements was achieved by introducing individual rotation degrees of freedom at joints where several elements meet, cf. the example in Fig. 1. These rotational degrees of freedom were released at joints during the construction of the structure stiffness matrix. Thereby each component can rotate independently, but the translations of the components are completely coupled. It is here worth noting that the beam element used in this study includes the effect of the axial force on the bending stiffness of the component. When a member is divided into several beam elements, the rotational degrees of freedom must be coupled at the internal nodes to keep the continuity of the component. For larger problems, model reduction techniques can be used to condense out the internal unloaded degrees of freedom, without significant loss of accuracy in the important results. 2.3. Equilibrium iterations The main calculation for the studied problem was seen as a successive unstressed length change of one or several of the components (depending on the context explained below), and finding the corresponding equilibrium state. This equilibrium is achieved iteratively using the Newton– Raphson method for the residual force equation [26]: DF ¼ F in  F ex ¼ 0 where F ex are the acting global external forces (vanishing P when an unloaded structure is considered) and F in ¼ Lf are the assembled forces, coming from the global internal forces for each element f ¼ N o ½c  s 0 c s 0T with N o for an element, from:

No ¼ EAðLs  Lo Þ=Lo

ð9Þ

2.4.1. Synchronous pre-stressing The pre-stress in this setting was seen as a level scalar w multiplied by the initial self-stress force vector Q calculated from  in which Ls is the vector of design lengths for each Q ¼ diagðLs Þq  is the self-stress force density vector evaluated component and q by the Force Density Method (FDM) [9,27], and if necessary chosen from a higher-order self-stress space by a suitable method. Each element Q i in Q represents the force in component N i ¼ wQ i . With N i evaluated, then the cut-length Lo for each component can be calculated from Lo ¼ EALs =ðN i þ EAÞ. By varying w; Lo varies accordingly. This is thereby seen in a design context where the final geometry defining Ls and level of pre-stress can be decided based on the application demands. The level of pre-stress w and the final geometry of the tensegrity structure are decided first, and then the unstressed length Lo for each component is calculated. This prestressing approach thereby changes all Lo in each level of prestress. 2.4.2. Tuning pre-stressing In this context we chose a pre-stress level value w, and the unstressed length Lo for each component was calculated as above. We assumed that each component was cut at exactly the unstressed length corresponding to this decided level of pre-stress, except the components that were seen as controls. These controls were assumed to be cut longer and then, by successively shortening their length, the level of pre-stress was varied. This method is seeing the problem in a production context and we use it to simulate the tuning of the tensegrity structure towards its designed state. We note that the geometry of the assembled structure will deviate slightly from the designed one, except exactly at the designed prestress. The changes in the pre-stress level thereby is achieved by stepwise modification of only Lo for the control components. Obviously, the same viewpoint is also relevant in a health-monitoring context, when components are checked for pre-stress relaxation. 2.5. Algorithm The Steps of the formulation of one sequence of pre-stress states are as follows: Step 0: Choose a pre-stress pattern and a pre-stress level based on the self stress states available. A choice is, if necessary, made in the self-stress space. The FDM is used in this study, as explained in [9,27].

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Step 1: For the considered context (synchronous for design purposes and tuning for production and health-evaluation purposes) introduce unstressed lengths Lo for all members. Step 2: Iterate for equilibrium by employing the Newton–Raphson method as above. Step 3: Evaluate the structural tangent stiffness and mass matrices of the converged state. Step 4: Carry out the spectral decomposition analysis and find at least an extensive subset of the natural frequencies and their corresponding modes. Step 5: Check if the frequency analysis indicates instability. In that case break the loop and conclude results. If not, then proceed to Step 6. Step 6: Modify the unstressed lengths for all components or just the controls based on the context explained above. Repeat Steps 1–5. For the development and implementation of the above algorithm, we used the Matlab software.

2.6. Basic test The method of investigation was first applied on a pre-stressed simply supported beam. The results were validated by the analytical solution from [21, chap. 4]. The aim was to verify the accuracy of this formulation and its dependence on the discretization of the components. The beam was assumed to have the properties: massive circular section with R = 0.015 m, beam length Lo ¼ 1 m, Young’s modulus E = 210 GPa, density q ¼ 7500 kg/m3 leading to a mass distribution m = 5.301 kg/m. Consequently, the Euler critical load of buckling N cr ¼ 82:408 kN. The lowest natural frequency for the axially unloaded beam is 62.342 Hz. The beam was first discretized by one beam element, Fig. 2, and then into two and six elements. For each case, the axial compressive force was increased step-wise until the lowest natural frequency was very close to zero. With higher compressive force, the natural frequencies became complex and at least one of the eigenvalues of the tangent stiffness matrix is less than zero. When using a single beam element, the natural frequency approached zero ð0:005 HzÞ when the axial load was N ¼ 100:197 kN, which is in error by 21.59% compared to Euler’s load of buckling. With a two beam element discretization, the natural frequency approached zero ð0:005 HzÞ when the axial load reached 83:028 kN with a relative error equal to 0:75%. For a six beam element discretization, the natural frequency approached zero ð0:008 HzÞ when the axial load reached 82:417 kN with a relative error equal to 0:01%. The lowest natural frequency for different discretizations of the axially loaded simply supported beam is shown in Fig. 3, clearly indicating the buckling compressive force. In addition, the first three mode shapes from the six element discretization case are depicted in Fig. 4, reproducing the well-known mode shapes of the simply supported beam. The vibration modes were evaluated at a compressive force of 82:417 kN.

Fig. 2. Plane beam element and its degrees of freedom.

In comparison to tensegrity systems below, it is noted that the pre-stressed length of this beam Ls is slightly below Lo ¼ 1 m due to the compression force.

3. Numerical examples We investigated the natural frequencies for a set of plane tensegrity structures with the method outlined above. The investigated structures were a class 1 planar tensegrity T4C2, Fig. 5, a class 2 planar tensegrity T2C4 (the inverted version of T4C2), and an X-frame tensegrity beam made of 3 connected T4C2 modules, T10C6, Fig. 1, where Tx and Cx are the notations used for the numbers of tensile and compressive components, respectively. For instance T4C2 is a structure that has four tensile components and two compressive components. All the structures were seen as simply supported, thus externally statically determinate. The material and geometry of the components were selected so that buckling of the compressed components occurred before yielding of the cables, using an infinite yielding stress. All examples gave, as noted above, sets of very closely situated resonance frequencies, related to the internal symmetries of the structures, which are typical for tensegrities. This clustering of frequencies gave no numerical problems with the algorithm used. In the presentation below, these sets will be described as coinciding, even if a small difference in frequencies were indicated by the numerical solution. These small differences typically were related to the spatial mass distribution of the structural models, reflecting the assumed displacement restraints introduced.

3.1. Example 1, an X-frame tensegrity T4C2 The single X-frame cell is the basis for X-module tensegrity structures, and has a single self-stress state. The tensegrity T4C2 with design size 1  1 m2 shown in Fig. 5 has two compressed bars and four cables. The investigated structure was assumed to have cable diameter Dc ¼ 0:015 m, bar diameter Db ¼ 0:035 m. The material for both cables and bars was defined by elastic modulus E ¼ 210 GPa and density q ¼ 7500 kg=m3 . Based on the element numbering scheme shown in Fig. 5, the initial self-stress force vec-

Fig. 3. Relationship between the axial load and the lowest natural frequency for different discretizations of the pre-stressed simply supported beam.

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(a) First mode of vibration

(b) Second mode of vibration

(c) Third mode of vibration

Fig. 4. First three mode shapes evaluated for the simply supported beam, discretized by six elements, at compressive force N ¼ 82:417 kN.

12

17 18

3

16

13

20

19

Bars Cables

4

14

5

11 15

2

6

buckling in the two first cases with vanishing frequency, but also give minor effects on the global displacements. It is noted that with this setting, the critical load of buckling is also slightly varying with the level of desired pre-stress as the analytical Euler load is a function of unstressed length Lo , which changes with the level of pre-stress in the synchronous context. It is further noted that the errors in calculated buckling load are identical to the ones in the simply supported beam above. The third comment is that, due to symmetry, two natural frequencies approach zero together, reflecting the symmetry of the two compressed components.

3.2. Example 2, an X-frame tensegrity T2C4 2

9 3

4 5

1

10

1

7 8 6

Fig. 5. X-frame tensegrity T4C2 meshed with single beam element for each component.

pffiffiffi pffiffiffi T tor of this structure is Q ¼ ½1 1 1 1  2  2 . Each member was considered to act as pin-ended. The pre-stress in this example was synchronously varied by multiplying the initial self-stress force vector Q by a level parameter w, giving the unstressed lengths Lo , of each component. Consequently, the final geometry was the same for any level of pre-stress considered, but the cut-lengths varied. To investigate the accuracy of this formulation in relation to the element discretization, the structure was first simulated with one element, and then with six elements, for each component. Using one beam element per component, the two lowest natural frequencies approached zero (0:004 Hz) when the pre-stress level w ¼ 65:569 kN, and the axial compression pre-stress force in the bars was 92:729 kN, which can be compared with the Euler critical load of buckling evaluated for the corresponding unstressed length Lo ¼ 1:4149 m which is 76:262 kN, an error by 21:59%. When six elements were used for each component the two lowest natural frequencies approached zero (0:004 Hz) when the prestress level w ¼ 53:943 kN and the axial force in the bars was 76:287 kN and the Euler critical load of buckling evaluated for the corresponding unstressed length Lo ¼ 1:4147 m was 76:279 kN with a relative error 0:01%. The relationship between the level of pre-stress and the lowest natural frequencies for this case of synchronous pre-stressing is shown in Fig. 6. It was observed that the higher frequencies of vibration increased with increasing level of pre-stress, Fig. 6. Another observation was that one eigenvalue of the tangent stiffness matrix crosses zero at the critical situation, indicating structural instability. The calculated first four mode shapes are shown in Fig. 7, where the two first should be seen as arbitrary vectors in the twodimensional eigenspace corresponding to the vanishing frequency. The modes essentially represent local bending deformations, i.e.

The inverted X-frame tensegrity structure was investigated. This structure has four compressed bars and two cables. The same material as above was used. The diameters were modified to Dc ¼ 0:019 m; Db ¼ 0:03 m in order to ensure that buckling occurs before yielding in the cables. The self-stress vector of this structure pffiffiffi pffiffiffi T is Q ¼ ½1  1  1  1 2 2 , with the numbering scheme in Fig. 5. Each component was divided into six beam elements. The pre-stress was varied as a synchronous pre-stressing. The four lowest natural frequencies approached zero (0.003 Hz) when the pre-stress level w ¼ 82:252 kN, and the axial compression force in all bars was 82:252 kN which again is 0:01% from the Euler load of buckling for the corresponding unstressed length Lo ¼ 1:001 m. The relationship between the pre-stress level and the lowest natural frequencies is shown in Fig. 8. The calculated first four mode shapes are shown in Fig. 9.

Fig. 6. Relationship between the natural frequency and level of pre-stress for Xframe tensegrity T4C2; each curve represents several coinciding frequencies, cf. legend.

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Fig. 7. X-frame tensegrity T4C2: mode shapes at buckling.

Fig. 8. Relationship between the lowest natural frequencies and the level of prestress for X-frame tensegrity T2C4; each curve represents several coinciding frequencies, cf. legend. Fig. 10. Lowest natural frequencies as functions of pre-stress level of the tensegrity beam T10C6; each curve represents several coinciding frequencies, cf. legend.

3.3. Example 3, a three module tensegrity beam T10C6 Three 2-D basic Snelson’s X-frame modules were connected to compose the two-dimensional tensegrity grid beam in Fig. 1, with 6 bars and 10 cables. For this structure the number of degrees of freedom is 288 with 96 elements, with six elements for each component, and moments released at the physical joints. The material for cables and bars was chosen as above, the diameter of the bars was Db ¼ 0:035 m and for the cables Dc ¼ 0:015 m. The self-stress space is here 3-dimensional and the self-stress vector was chosen as pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi T Q ¼ ½1 1 1 1 1 1 1 2 2 1  2  2  2  2  2  2 , with the numbering scheme as in Fig. 1, following the method in [9]. For a synchronous pre-stressing, the first six natural frequencies of the structure approach zero (0.004 Hz) for a pre-stress level

w ¼ 53:942 kN and the axial compression force in the bars is 76:286 kN, with an error 0:01%. The relationship between the pre-stress level and the lowest natural frequencies for this structure is shown in Fig. 10. The first four mode plots show that all the bars buckled at this axial compression force, Fig. 11. 3.4. Example 4, tuning pre-stressing of T10C6 The structure in Example 3 was tuned towards the design configuration with a value of pre-stress level decided to be 45 kN. The chosen components for tuning (controllers) in this example were the horizontal cables 8 and 9, Fig. 1. Their cut-lengths were

Fig. 9. X-frame tensegrity T2C4: mode shapes at buckling.

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Fig. 11. First four mode shapes for the tensegrity beam T10C6 at buckling.

Fig. 13. Variation of the lowest six natural frequency of tensegrity beam T10C6 during the tuning process towards the designed level of pre-stress. At point B, the structure became symmetric: the designed state. Fig. 12. Tensegrity beam T10C6 at rest stage, before tuning to 1  3 m2 (Relative displacements enlarged for visibility).

assumed to be longer than their designed unstressed length Lo , starting from a length where they can just barely be joined with their corresponding component ends. The investigation of this tensegrity shows that at the designed pre-stress level, the six lowest natural frequencies at the design configuration were 14.38 Hz, Fig. 10. The structure is expected to have these natural frequencies if the lengths of the controls components are shortened to their designed values. Starting from a very low level of pre-stress, where the lowest natural frequency of the asymmetric structure, Fig. 12, was 24.08 Hz, the six lowest frequencies followed an increasing tuning pre-stress and they

became practically identical when the designed configuration (1  3 m2 ) was reached, i.e., the structure in this state became symmetric and six of its lowest natural frequency are coinciding, point B in Fig. 13. After tuning the structure to the designed configuration, where its six lowest natural frequencies coincided with the designed natural frequency value (14.38 Hz), the final size of the tensegrity structure was exactly the same as the target design 1  3 m2 . Some of these tuning steps, before reaching the design configuration are shown in Fig. 14, with exaggerated deviations from nominal measures.

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4. Discussion

Fig. 14. Tuning process of tensegrity beam T10C6 towards the target design. (Relative displacements enlarged for visibility).

If the unstressed lengths of the tuning components are further decreased below their designed lengths, the six lowest natural frequencies are no longer coinciding, as the asymmetry of the structure is again present. Two natural frequencies go to zero when the two bars in the middle module buckle as the first and second vibration mode shapes shown in Fig. 15. At this critical state the lowest natural frequency of the structure approached zero (0.005 Hz) at a controller unstressed length of 0.996 m with a compressive force in the buckled bars (in the middle module) of 75:932 kN. This is approximately equal to their critical load of buckling 75:924 kN.

The method of formulation presented in this paper aims at a full investigation of natural frequencies in tensegrity structures, including transversal vibrations. In addition to a general tool for representing pre-stress level with resonance, it can thereby relate the natural frequency of the studied tensegrity structures with the nearness to buckling in one or more of its compressed components, thus providing a useful tool for anticipating the collapse of the tensegrity structure. Seeing the natural frequencies as non-linear functions of the introduced pre-stress, the viewpoint is also relevant for tuning of the structure towards the target design, geometry and pre-stress level. The studies found in literature [10–13] state in one sense or another, that increasing the pre-stress level will lead to increase of stiffness and thereby the natural frequency of the tensegrity structure without addressing to what level we can increase the level of pre-stress. Because tensegrity structures are composed of both compressed and tensile components, there are two criteria which have to be considered when modifying the level of pre-stress: yielding of cables and buckling of bars. In most of the studies found in literature the element used for formulating the tensegrity structure is the truss element which cannot represent the transversal bending and stiffness of the components. It will become more important to include the transversal stiffness whenever the structure has compressed components, this is also exactly what was included in [28]. The observations from this study are also similar to the study made by Greschik [15], where he mentions that pre-stress concurrently increases cable and decreases bar vibration frequencies. The formulation was first tested on a pre-stressed simply supported beam. Results compared well with the analytical solution Eq. (1) [21, chap. 4] but only if the member was divided into several elements. For example in Fig. 3, the 6 element discretization of the pre-stressed simply supported beam is almost identical with the exact solution, Eq. (1), and the difference between the axial compression force and the critical load of buckling of this beam was 0:01% at x1  0. However, we found that the computational

Fig. 15. First four mode shapes for tensegrity beam T10C6 tuned beyond the designed pres-stress.

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time can be reduced by using several beam elements for compressive components and a single beam element for tensile components with approximately the same results obtained. The simple example also showed that the present beam element formulation is capable to correctly consider compressive pre-stress. The calculations verified the need for consideration of the transversal behavior of a compressed member. The present work focussed on the structures where the compressive behavior dominates the response. The lowest natural frequency either increases or decreases with the increasing of pre-stress. It will finally approach zero where the internal axial force in one or more compressed components have reached their critical loads of buckling and the structure collapses. At this critical situation a number of eigenvalues of the tangent stiffness matrix are also below zero, which is an indicator of instability. It is, however, important to note that the different frequencies in the resonance spectrum will behave differently under variable pre-stress. It is emphasized that the compressed components in all the tensegrity structure examples considered in this study are critical for the stability of the structures, where losing one of them likely leads to collapse. This might be different for large scale 3-D tensegrity grid structures where buckling of one of its compressed components might not lead to overall collapse, [17], but it should have a significant effect on the stiffness and thereby the natural frequency. Consequently, considering one or more three-dimensional tensegrity structures is one of the future sequels of this study. Larger structures can obviously be treated with the same methods as presented here. The beam formulation is immediately transferrable to three-dimensional situations, and the size and accuracy aspects of the eigenvalue extraction can be treated with model reduction and sub-space iteration techniques. Finally and as part of the future work, the effect of the temperature variation between daytime and nighttime on the pre-stress level should be considered. This can be predicted by measuring the variation in the natural frequency if the variation in the temperature is very high. Thereby the level of pre-stress can always be tuned to the required (designed) value by means of actuators or any other type of the tuning mechanisms. 5. Concluding remarks The following conclusions can be drawn from this study:  The beam element used with the topology mentioned in this article gives a better representation of the tensegrity structure than a truss element. Consideration of a de-stiffening close to member buckling is important.  The spectrum of natural frequencies can be used as indicators for tuning the tensegrity structure towards the target design. Considering the practical application, the transversal vibrations are giving better possibilities for evaluation.  Increasing the pre-stress level for a certain geometry and material used may decrease the stiffness of the tensegrity structure, and thereby at least some of the natural frequencies. Whether this happens is dependent on the relative properties of compressed and tensioned components.  In several of the contexts mentioned, it is necessary to consider not only the lowest resonance frequency of the structure, as higher frequencies will provide further information.

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 When the tensegrity structure has a cable dominated behavior, then any increase of the level of pre-stress will raise the lowest natural frequencies, up to the limit where the structure start to show a bar dominated response, where any further increase in pre-stress will lower the lowest natural frequencies.  Because components with the relaxed prestress will most likely reveal themselves by their local transversal vibration behavior, the present formulation will give possibilities to improve the VHM evaluation technique.

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