A comparative study on the dynamics of tensegrity-membrane systems based on multiple models

Accepted Manuscript

A Comparative Study on the Dynamics of Tensegrity-Membrane Systems Based on Multiple Models Shu Yang , Cornel Sultan PII: DOI: Reference:

S0020-7683(16)30377-8 10.1016/j.ijsolstr.2016.12.009 SAS 9395

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

12 April 2016 4 November 2016 13 December 2016

Please cite this article as: Shu Yang , Cornel Sultan , A Comparative Study on the Dynamics of Tensegrity-Membrane Systems Based on Multiple Models, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.12.009

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A Comparative Study on the Dynamics of TensegrityMembrane Systems Based on Multiple Models Shu Yang1

Cornel Sultan2

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Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

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This article investigates the modeling and dynamics of tensegrity-membrane systems. As extensions of tensegrity systems, tensegrity-membrane systems are composed of membranes, bars, and tendons. Significant system complexity and coupling between system components indicate that the behavior of the whole system needs to be investigated. For this purpose a complex nonlinear finite element model is developed to describe system dynamics. Numerical analysis results given by this nonlinear finite element model are compared with the results given by two other models based on different modeling assumptions and of lower complexity. Observations related to key dynamical properties and the influence of modeling assumptions are discussed in detail.

Keywords: Tensegrity-membrane systems; Nonlinear finite element method; Controloriented modeling; Dynamics

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Introduction

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Tensegrity-membrane systems are a class of new bar-tendon-membrane structures. The key idea in creating such novel structures is to introduce membranes in tensegrity structures. It is known that

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tensegrity systems are generally lightweight and capable of significant shape changes (Sultan, 2009; Skelton and Oliveira, 2009). As extensions of classical tensegrity systems, tensegrity-membrane systems

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inherit these two major advantages, which enable these novel systems to be generally lightweight and

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to experience relatively easy folding and unfolding between packaged and erected configurations.

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Graduate Research Assistant, Aerospace and Ocean Engineering, 215 Randolph Hall, Email: [email protected]. 2

Associate Professor (Corresponding Author), Aerospace and Ocean Engineering, 215 Randolph Hall, Email: [email protected]. 1

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Clearly, these mechanical properties make tensegrity-membrane systems promising candidates for lightweight, deployable space structures that can be used in space applications such as space antennas and solar sails. Traditional space systems which incorporate membranes are generally designed as relatively rigid

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support frames composed primarily of bars with tensioned membranes installed. The mechanical

behavior of these systems is mainly determined by the support frames, and the membranes have little effect on the statics and dynamics of these structural systems. This fact simplifies system design, since

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the support frames and the membranes can be designed separately, and each design problem can be solved using well-developed design and analysis methods in structural mechanics and solid mechanics. To change the configurations of such traditional membrane systems it is necessary to change the shape of the support frames. Therefore, numerous additional components, like actuators and rigid/flexible

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connecting parts, are required for the rigid support frames to achieve system deployment. As a result, the weight of the overall system increases, diminishing the benefit of lightweightness due to the

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membranes. Moreover, the resulting systems turn out to be more complex, leading to higher risk of deployment failure (Zolesi et al., 2012). To alleviate these problems, some space membrane system

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designs employ the inflation technology, leading to systems with membranes attached to inflatable

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support frames (Fang et al., 2001; Adetona et al., 2002; Adetona et al., 2003; Berger et al., 2004). Such systems are free from mechanical connecting parts, like hinges and joints, but system configurations

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during inflation may not be accurately controlled. Therefore, additional control devices are required to achieve precise control of system components during deployment (Freeland, 1998). It should also be noted that the design separation concept used for classical space membrane systems may not give a comprehensive understanding of the behavior of the whole system. Additional analyses/experiments

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and sophisticated models are required to study the coupling between system components and to examine whether undesired coupled dynamics is introduced. In simple terms, tensegrity-membrane systems are assemblies of membranes, bars, and tendons. The attached membranes in tensegrity-membrane systems can be part of optical devices, antennas, etc., the

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bars provide sufficient system stiffness, and the tendons introduce tensile forces and some flexibility in the system to allow easy shape changes. Moreover, all of the system components can incorporate

sensing mechanisms (e.g., via embedded optic fibers) and the tendons can also serve as actuators for

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active control. In other words, each component in a tensegrity-membrane system can serve as load carrying and prestressing element, be part of the control system, and even represent the device a classical structure is usually designed to support. Since the actuators are embedded in the system, such a system is not affected by the drawbacks associated with additional mechanisms (e.g., latches, hinges,

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etc.) usually necessary for the deployment of classical membrane systems which include rigid support frames. On the other hand, a tensegrity-membrane system should be designed and analyzed as a whole

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system and not using separation ideas discussed previously. Therefore, the mathematical problems related to system design and analysis, such as the form-finding problem and system modeling, are more

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challenging. In addition, coupling between system components of tensegrity-membrane systems is more

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significant compared to classical membrane systems, so models that can capture the mechanical properties of the whole system are desired.

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The attached membranes play key roles in tensegrity-membrane systems. In the literature,

membrane design and analysis, as well as its mechanical properties, have been studied by many researchers. A design study of square solar sails was presented in Greschik and Mikulas (2002). These researchers evaluated several classical sail design concepts, and proposed their novel sail suspension 3

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design, named stripped sail architecture. They also pointed out several design issues of sail structures deserving attention. Mikulas and Adler (2003) provided a simplified approach for assessing square solar sails, which could be used in a preliminary design procedure. The vibration of tensioned membranes under air effects was studied by Kukathasan and Pellegrino (2002). The results indicated the feasibility of

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performing finite element analysis to simplify the testing of deployable membrane structures. Adler et al. (2000) focused on the static and dynamic analysis for partially wrinkled membranes. They discussed the theory and criteria for wrinkling determination and presented several analysis results for different wrinkled membrane structures. Kukathasan and Pellegrino (2003) conducted high-fidelity finite element

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simulations for wrinkled membranes. Thin-shell elements were used for the membranes and a precise simulation of the static wrinkling process was performed. Wong and Pellegrino (2006) studied wrinkled membranes based on both experiments and finite element simulations. The analytical models used in

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their work were also presented and discussed.

As mentioned previously, due to system complexity, accurately modeling tensegrity-membrane

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systems is a challenging task (Sunny et al., 2014). A detailed study on modeling tensegrity-membrane systems is the work of Yang and Sultan (2016), where a nonlinear finite element model was developed

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to investigate the dynamics of tensegrity-membrane systems. Three major modeling assumptions were

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made: a) the bars only experienced axial deformation, b) the dynamics of tendons was ignored, and c) the transverse bending stiffness and transverse shear stiffness of membranes were ignored. These

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modeling assumptions were introduced based on experience and intuition, however more in depth examination is still required to check whether these assumptions are reasonable. The nonlinear finite element method is an effective tool for the modeling and analysis of nonlinear

flexible multibody systems (Bauchau, 2011). Its effectiveness in modeling tensegrity-membrane systems 4

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has been illustrated in our previous work. It should be pointed out that the nonlinear finite element method usually requires fine meshes, so the resulting mathematical model is high dimensional. Moreover, the formulation of the nonlinear finite element method is based on an iterative numerical procedure, so the equations of motion cannot be expressed as explicit nonlinear ordinary differential

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equations. These two features of the nonlinear finite element method make the resulting model

computationally expensive, and prevent the direct use of nonlinear finite element models for control design. Yang and Sultan (2014) developed a control-oriented model for tensegrity-membrane systems. The geometrical nonlinearity of the membrane was ignored due to two considerations: the in-plane

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stiffness of a thin sheet is generally larger than its transverse stiffness and bending stiffness (Belytschko et al., 2013), and small vibrations of tensegrity-membrane systems need to be considered if appropriate controls are applied. The dimension of this control-oriented model is significantly smaller compared to that of a finite element based model. However, its reliability, accuracy, and limitations still need to be

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studied.

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A major contribution of this article is the development of a new nonlinear finite element model of great generality, referred to as the shell-beam-cable model, for tensegrity-membrane systems. We then

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use this sophisticated model to investigate key mechanical properties of tensegrity-membrane systems

made.

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and, for comparison with other simpler models, to evaluate the validity of various modeling assumptions

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For the development of the shell-beam-cable model, assumptions for system components that may

drastically restrict the applicability of the model are avoided. Therefore, shell elements are used to model membranes, bars are treated as beams, and tendons are modeled using truss elements. In other words, the transverse bending stiffness and transverse shear stiffness of membranes and bars are 5

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considered. Also, tendon dynamics is taken into account, which not only enables the possibility of studying the coupling between tendons and other system components, but also makes this shell-beamcable model an effective tool to study the influence of tendon dynamics on the whole system and to evaluate whether it is acceptable to ignore tendon dynamics. Note that, based on engineering

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experience and intuition, tendon dynamics is usually ignored in the study of tensegrity structures for simplicity, but, to the best of our knowledge, there is no published work on the justification of this modeling assumption.

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Another finite element model, named the membrane-truss-cable model, is also used in this work. This model can be treated as a simplified shell-beam-cable model, since membranes are assumed to be very thin such that the transverse bending stiffness and transverse shear stiffness can be ignored. Also, each bar is treated as a rod by only considering its deformation in the longitudinal axis. This membrane-

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truss-cable model can also be regarded as an extension of the nonlinear finite element model presented in our previous work (Yang and Sultan, 2016), since the same modeling assumptions for membranes and

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bars are used. An improvement is that the dynamics of tendons is considered, which relieves the membrane-truss-cable model of the time-consuming dynamic relaxation technique used for system

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form-finding. Mesh convergence analysis, modal analysis, and free vibration analysis are conducted to

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study the mechanical properties of tensegrity-membrane systems. The numerical results given by the two nonlinear finite element models and the control-oriented model previously developed are

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compared to examine the accuracy and reliability of the two simplified models, i.e. the membrane-trusscable model and the control-oriented model.

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Tensegrity-Membrane Systems A tensegrity-membrane system is a prestressed system composed of bars, tendons, and membranes. In this article, we study a tensegrity-membrane system with M stages, N bars in each stage, and one membrane attached to the top of the system. This system is referred to as an M-N tensegrity-membrane

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system. The bars are labeled bij, and a stage contains the bars with the same first index. For the j-th bar in the i-th stage, bij, the lower end and the upper end of the bar bij are labeled Aij and Bij, respectively. The length of bar bij is labeled lij. The system is located at a fixed base by connecting the lower ends of the bars in the first stage, i.e. A1j, to the base through frictionless rotational joints. The membrane is

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attached to the top of the system, and the membrane corners are connected to the top ends of the bars in the M-th stage.

To illustrate the shape of tensegrity-membrane systems, a system with two stages, three bars in a

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bar ends A1j are not depicted.

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stage, and a membrane (the 2-3 system) is depicted in Fig. 1. Note that the rotational joints located at

The following modeling assumptions are made:

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a) Linear elastic constitutive laws are used for the attached membranes, the bars, and the tendons.

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b) The gravitational field is ignored. No external force is applied to the system. c) A homogeneous membrane in the shape of a convex polygon in the undeformed configuration is

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used.

d) The mechanics of wrinkles generated on the membrane is not modeled. e) Tendons and the membrane cannot be compressed (i.e. they can carry only tensile loads). 7

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Fig. 1 The sketch of a two-stage three-bar tensegrity-membrane system

The inertial reference frame O-xyz is fixed at the base of the system. The origin of O-xyz is located at

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point A1N, i.e. O ≡ A1N, and the Ox axis goes through points A1N and A11. The Oy axis is perpendicular to the Ox axis in the base plane, and the Oz axis is perpendicular to the O-xy plane, respecting the right-

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hand rule.

The tendon connection pattern used in this work is: a) Tendon Sij connects the bar ends Aij and Bi, j-1 8

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b) Tendon Vij connects the bar ends Aij and Bi-1, j+1 c) Tendon Dij connects the bar ends Aij and Bi-1, j-1 d) Tendon Cij connects the bar ends Aij and Ai-1, j-1

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e) Tendon Pij connects the bar ends Bij and Bi-1, j f) Tendon Rij connects the bar ends Bij and Bi-1, j-1

The connection pattern for tendon Vij used in this work is different from that used in our previous work

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(Yang and Sultan, 2016), where tendon Vij connects the bar ends Aij and Bi-1, j. Note that tendon

connection pattern for a tensegrity-membrane system can be specified and altered by users in order to

Mathematical Models

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change system shape and mechanical properties.

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As assemblies of membranes, bars, and tendons, tensegrity-membrane systems belong to the class of flexible multibody systems. Several particularities of tensegrity-membrane systems are important in

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modeling their dynamics. First, these systems may experience large displacements and rotations. Second, the attached membranes may experience large transverse deflections. For these reasons, it is

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recommended that tensegrity-membrane systems are treated as nonlinear flexible multibody systems

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and their dynamics is studied based on the nonlinear finite element method. If only the system dynamics around equilibriums is of interest, tensegrity-membrane systems can be

treated as linear flexible multibody systems. For example a low complexity, control-oriented model can be developed by using floating reference frames and the modal expansion technique. This controloriented model has fewer generalized coordinates and requires low developmental and computational 9

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cost. Most importantly, such a model may capture accurately the essence of the system dynamics and it may be sufficient for performant control system design. In this article, three mathematical models are used to study the dynamics of tensegrity-membrane systems: two nonlinear finite element models and a control-oriented model. For the first nonlinear finite

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element model, shell elements are used to model the attached membrane, the bars are modeled as beams, and truss elements are used to model the tendons. This model is referred to as the shell-beamcable model. For the other nonlinear finite element model, membrane elements are used to model the

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attached membrane, and truss elements are used to model the bars and tendons. This model is referred to as the membrane-truss-cable model. For the control-oriented model, the bars are treated as rigid bodies, the geometrical nonlinearity of the attached membrane is ignored, and the linear finite element method is used to determine the terms related to the membrane elasticity.

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Nonlinear Finite Element Models

The nonlinear finite element analysis used in this work follows the procedures introduced in Bathe

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(1996). The total Lagrangian formulation is applied, and geometrical nonlinearity is considered. The strain and stress components are expressed in the curvilinear coordinate system O-r1r2r3, and a tilde (~)

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is placed over the terms expressed in the O-r1r2r3 frame. Note that in this section tensor notations are

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used for simplicity. All of the coordinates and displacements are measured with respect to the inertial reference frame O-xyz introduced in Section 2.

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The principle of virtual work applied to the configuration at time t+t is:



t t 0

S ij  t 0t ij 0dV 

0

V

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t t

R

(1)

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where

t t 0

S ij is the contravariant component of the second Piola-Kirchhoff stress tensor at time t+t

with respect to the configuration at time 0, and

 is the covariant component of the Green-Lagrange

t t 0 ij

strain at time t+t with respect to time 0. The term

t t

R is the virtual work due to external forces and

The stress and strain components can be decomposed as: t t 0

S ij  0t S ij  0 S ij ;

  0t ij  0  ij

t t 0 ij

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torques.

(2)

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In addition, the strain increment 0  ij can be written as the combination of a linear part 0 eij and a nonlinear part 0ij :

  0 eij  0ij

0 ij

V

C ijrl 0 erl  0 eij 0dV 



t 0

S ij  0ij 0dV 

0

t t

R



t 0

S ij  0  ij 0dV

(4)

0

V

V

t t

R includes the virtual work of inertia forces and damping forces. From Eq.

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Note that the virtual work

0

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

0

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After linearization, Eq. (1) becomes:

(3)

(4), the incremental solution at t+t can be computed. To improve the solution accuracy, an iterative

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procedure is applied, and the matrix equations for finite element analysis can be written as:





 M t tU  k   0t K L k 1  0t K NL k 1 U  k   t t R  t t F  k 1    t tU  k   t tU  k 1  U  k  ; t tU  0  tU 

(5)

where M is the mass matrix. Also KL and KNL represent the linear strain incremental stiffness matrix and the nonlinear strain incremental stiffness matrix, respectively, R represents the vector of external forces 11

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and torques, F represents the vector of nodal point forces corresponding to the element stresses, and U represents the vector of nodal displacements. More details related to the formulation of the nonlinear finite element analysis can be found in Bathe (1996) and were also presented in our previous work (Yang and Sultan, 2016).

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Shell-Beam-Cable Model The goal of developing the shell-beam-cable model is to describe the dynamics of tensegrity-

membrane systems without introducing very restrictive modeling assumptions for system components. To achieve this goal, nonlinear shell elements are used for the attached membranes, the bars are

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treated as beams, and the dynamics of tendons is modeled by truss elements. The MITC4 shell element and the four-node beam element introduced in Bathe (1996) are used in this work. Transverse bending stiffness and transverse shear stiffness are considered in these two elements.

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In the shell element, the coordinates of a generic point can be written as: 4

t

xi   hk t xik 

(6)

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k 1

r3 4  ak hk tVnik 2 k 1

where the superscript t indicates that the coordinates are measured at time t, the subscript i (i = 1, 2, 3)

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corresponds to x, y, and z directions in the O-xyz frame, and k corresponds to the k-th node of an

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element. The symbol a represents the shell thickness, and Vn represents the normal vector. The symbol r3 is the coordinate used for the interpolation of shell thickness. The symbol hk represents the

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interpolation functions for a four-node shell element: 1 1 1  r1 1  r2  ; h2  1  r1 1  r2  4 4 1 1 h3  1  r1 1  r2  ; h4  1  r1 1  r2  4 4 h1 

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(7)

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Using Eq. (6) at times 0, t, and t+t, we can express the displacements as: t

ui  t xi  0 xi ; ui 

t t

xi  t xi

(8)

After inserting Eq. (6) into Eq.(8), we obtain:

ui   hk t uik  k 1

r3 4  ak hk  tVnik  0Vnik  2 k 1

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4

t

4

r 4 ui   h u  3  ak hkVnik 2 k 1 k 1 k k i

(9)

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where Vnik  t tVnik  tVnik . The vector Vnk represents the increment of tVnk . The symbols k and k are defined as the nodal rotational degrees of freedom for node k. For small rotations, Vnk can be approximated as: Vnk   tV2kk  tV1k k . The vectors 0V1k and 0V2k are defined as:

V1k 

e2  0Vnk

V2k  0Vnk  0V1k

0

;

e2  0Vnk

(10)

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0

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If 0Vnk is parallel to e2 , we set 0V1k equal to e3 . The vectors e2 and e3 are the unit vectors in the directions of the Oy and Oz axes, respectively. The vectors

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following equation:

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t t

Vnk  tVnk 

  V   t

k

,

k 2

t t

V1k and

t t

V2k are then obtained using the

d k  tV1k d  k 

k

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Note that the symbol v represents the Gibbs vector notation, while v is the corresponding column vector.

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(11)

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For the MITC4 shell element, mixed interpolation is employed to get rid of shear locking. It has been shown that this element does not experience membrane locking (Kim and Bathe, 2008). To apply mixed interpolation, the transverse shear strain components are chosen as: 1 1 1 1 1  r2  13A  1  r2  13C ;  23  1  r1   23D  1  r1   23B 2 2 2 2

where

1

   23 r 1; D 23

1

r2 1; r3  0 r2  0; r3  0

; 13C  13 ; 

B 23

  23

r1  0; r2 1; r3 0

(12)

(13)

r1 1; r2  0; r3  0

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13A  13 r 0;

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13 

For the four-node beam element, the coordinates of a generic point can be written as: 4

t

xi   hk t xik 

(14)

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k 1

r 4 r2 4 ak hk tV1ik  3  bk hk tV2ki  2 k 1 2 k 1

where a and b represent the beam thickness, V1 and V2 represent the normal vectors, and r2 and r3 are

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the coordinates used for the interpolation of beam thickness in V1 and V2 directions, respectively. The

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symbol hk represents the interpolation functions for a four-node beam element: 1 9 1  r1 1  3r1 1  3r1  ; h2  1  r1 1  r1 1  3r1  16 16 9 1 h3  1  r1 1  r1 1  3r1  ; h4   1  r1 1  3r1 1  3r1  16 16

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h1  

(15)

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After inserting Eq. (14) into Eq.(8), the displacements can be expressed as: r 4 r2 4 ak hk  tV1ik  0V1ik   3  bk hk  tV2ki  0V2ki   2 k 1 2 k 1 k 1 4 4 4 r r ui   hk uik  2  ak hkV1ik  3  bk hkV2ki 2 2 k 1 k 1 k 1 4

t

ui   hk t uik 

14

(16)

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where V1ik  t tV1ik  tV1ik and V2k  t tV2k  tV2ki . For small angles, the vectors V1k and V2k can be expressed as: V1k  k  tV1k ; V2k  k  tV2k

(17)

where  k  k1 k 2 k 3  . Here k1, k2, andk3 are the three nodal rotational degrees of freedom in

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the Ox, Oy, and Oz axes for node k.

The reduced integration technique is used to eliminate membrane locking and shear locking. It

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should be noted that this four-node beam element still performs well when the full numerical

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integration is used (Bathe, 1996).

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Fig. 2 The MITC4 shell element and the four-node beam element

two-node truss element are: 2

t

xi   hk t xik k 1

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The two-node truss element is used to model the tendons. The coordinates of a generic point in the

(18)

h1 

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where hk represents the interpolation functions for a two-node truss element: 1 1 1  r1  ; h2  1  r1  2 2

(19)

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The displacement expressions are the same as the expressions in Eq. (8).

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Membrane-Truss-Cable Model The shell-beam-cable finite element model discussed in the previous section is a very general model for tensegrity-membrane systems. The cost of developing and solving such a model is high. To alleviate

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these deficiencies, based on the physical properties of system components for tensegrity-membrane

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systems, some modeling assumptions can be made to simplify the model. First note that membranes used in practice are very thin (the thickness of the membranes used in

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this work is 51 m, and the ratio of thickness to length and width is around 10-5). As a consequence, the transverse bending stiffness and transverse shear stiffness have little effect on the membrane dynamics and may be ignored. Thus, the attached membrane can be modeled using membrane elements. For the bars of tensegrity-membrane systems, the bar ends are either connected to the system base through frictionless rotational joints or connected to the tendon ends or the membrane corners. These boundary 16

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conditions indicate that the bars in tensegrity-membrane systems are not rigidly constrained. Thus, bending deformation and shear deformation of bars are less likely to occur, indicating that it may be possible to model the bars without considering the bending and shear stiffness and obtain a sufficiently accurate tensegrity-membrane model at a cost much lower than that for the shell-beam-cable model.

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Based on these considerations, in addition to the modeling assumptions a) - e) in Section 2, the following two modeling assumptions are made for the membrane-truss-cable model: f)

The transverse bending stiffness and transverse shear stiffness of the attached membrane are

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ignored. g) The bars are assumed to experience axial deformation only.

This membrane-truss-cable model is developed based on our previous work (Yang and Sultan, 2016).

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A significant improvement of this model is that the dynamics of tendons is included, so the coupling

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among the attached membrane, bars, and tendons can be more accurately captured and studied. The attached membrane is modeled by four-node membrane elements. The two-node truss element,

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discussed in the previous section, is used to model the bars and the tendons. The four-node membrane element can be treated as the reduced MITC4 element without nodal rotational degrees of freedom.

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are:

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Thus, according to Eq. (6), the coordinates of an arbitrary point in the four-node membrane element

4

t

xi   hk t xik k 1

where hk represents the interpolation functions expressed in Eq. (7).

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Control-Oriented Model For the control-oriented model presented in this article, the dynamics of tendons is neglected for simplicity. Also, the linear elastic model and the infinitesimal strain assumption are used for the attached membrane. Therefore, the superposition principle and the modal expansion technique can be

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applied to the membrane. For the bars in the first stage, since the lower end of bar b1j is connected to the base, only two generalized coordinates, i.e. the declination angle, 1j, and the azimuth angle, 1j, are needed to

determine the bar’s orientation. For the bars in the i-th stage (i ≥ 2), the location of the lower end of bar

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bij, i.e. xij, yij, zij, and the declination angle and the azimuth angle, i.e. ij, and ij, are used to determine the bar’s location and orientation. Thus, the vector of generalized coordinates for bars is defined as: q  11  11

xMN

yMN

zMN

MN  MN 

T

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AC

CE

PT

ED

M

Note that ij  andij  . The generalized coordinates for bar bij are depicted in Fig. 3.

Fig. 3 Generalized coordinates of bar bij 18

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In order to model the attached membrane, we first introduce a floating reference frame labeled Ofxfyfzf. The origin of Of-xfyfzf is located at the point BMN, i.e. Of ≡ BMN. The plane Of-xfyf is identical with the plane determined by BMN, BM1, and BM, N−1. The axis Ofxf goes through BMN and BM1. The axis Ofyf is perpendicular to the axis Ofxf, and the Ofzf axis can be determined based on the right-hand rule.

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According to the definition of reference frames, Of-xfyfzf is fixed on the undeformed membrane.

For each point on the membrane, three variables u, v, and w are used to describe its deflections in the xf, yf, and zf axes. Based on the method used in Soedel (2004), the deflections of the membrane, u, v,

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and w, are separated into two parts: the prestressing deflections, us, vs, and ws, and the relative

deflections, ur, vr, and wr. Modal coordinates are used to discretize the relative deflections. Thus, we can express the deflections in the following form:

u

v w  u s  u r T

vs  vr

K   u s   uii i 1 

ws  wr 

M

K

v s   vii i 1

T

K  ws   wi i  i 1 

T

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where i is the modal coordinate and ui , vi , wi are mode shapes of the relative deflections.

z  qT

1

K 

T

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PT

The independent generalized coordinates can be expressed as:

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The equations of motion for tensegrity-membrane systems can be developed using Lagrange’s equations. Holonomic constraints and Lagrange’s multipliers are used to facilitate the mathematical derivation (Greenwood, 1988). The resulting equations of motion are a set of differential-algebraic equations (DAEs). After standard mathematical manipulations (see Yang and Sultan, 2014), these DAEs can be converted into ordinary differential equations (ODEs), which can be written as: 19

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z  Fz  z, z   O

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More details about this control-oriented model can be found in Yang and Sultan (2014).

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Numerical Results and Analysis In this section, mesh convergence analysis, modal analysis, and free vibration analysis are conducted based on the three mathematical models discussed in the previous section. Two representative

tensegrity-membrane systems are studied: a 1-4 tensegrity-membrane system and a 2-3 tensegrity-

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membrane system. The materials of membranes, bars, and tendons are chosen as Kapton, steel, and Nylon, respectively. The parameters of system components are listed in Table 1. Each tendon has a circular cross section and the radius of the cross section is listed in Table 1. Note that each bar has a square cross section and the thickness listed in Table 1 represents the length/width of the cross section.

M

For the beam element used in this work, the cross section orientation is chosen by specifying vector 0V1k

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, discussed in Section 3.1.1, according to the following equation:

V1k 

e3  Vb e3  Vb

PT

0

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where e3 is the unit vector in the direction of Oz axis and Vb is the vector pointing from the upper end

AC

(i.e. Bij) to the lower end (i.e. Aij) of each bar.

Table 1. Material and dimensional parameters of the tensegrity-membrane systems

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Membrane

Bar

Tendon

Young’s modulus

165 MPa

200 GPa

2 GPa

Density

1400 kg/m3

7800 kg/m3

1150 kg/m3

Thickness

51 m

27 mm

N/A

Poisson’s ratio

0.34

0.3

N/A

Diameter of cross section

N/A

N/A

0.6 mm

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System Prestressing

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Parameters

As discussed previously, tensegrity-membrane systems are prestressed structures. A systematic method of finding equilibriums for tensegrity-membrane systems was developed in our previous work (Yang and Sultan, 2016). In this section, this method is used to find system equilibrium parameters for

M

the two systems studied in this work.

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1-4 System For the 1-4 system, the length of each bar is chosen as 2.015 m, i.e. b1j = 2.015 m (j = 1, 2, .., 4). The bar length is much larger than the bar thickness listed in Table 1. Thus, the bar longitudinal dimension

PT

has the dominant effect on its motion, while the orientation of bar cross sections has a negligible effect.

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A square membrane is used and the length/width of the membrane is 2 m. The inclination angle and the

AC

azimuth angle of each bar are specified as: 1 j  60 ; 10  2 j 4;

1 j  

10 ;

 j  1, 2, ..., 4   j  1, 2, ..., 3  j  4

Bars’ lower ends (i.e. A1j) form a square and the radius of the circumscribed circle for this square is 0.4286 m. All tendons are identical and the rest-length of each tendon is chosen as 1.503 m. 21

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These equilibrium parameters make the attached membrane symmetrically tensioned. The stress distribution of the membrane, obtained in the floating reference frame Of-xfyfzf, is plotted in Fig. 4. Note that the stress distribution is not uniform. The major principal stress (1), the minor principal stress (2), and the von Mises stress (M) are plotted in Fig. 5. The maximum value of M is 1.9062 MPa. It is less

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than the yield stress at 3% of Kapton, which is 61MPa (DuPont, 2011). This result indicates that the membrane does not yield. The major principal stress in the membrane is positive everywhere, while the minor principal stress at the regions around membrane edges is less than zero. In Fig. 5, nodes where 2 is less than zero are marked in gray. According to the stress criteria for wrinkling determination (Adler et

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al., 2000), wrinkles occur at these regions. As stated in Section 2, the wrinkling effect of the membrane is not modeled in this work. Note that Kukathasan and Pellegrino (2003) showed that an unwrinkled

AC

CE

PT

ED

M

membrane model could accurately predict the vibration of a moderately wrinkled membrane.

22

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AC

CE

PT

ED

Fig. 4 Stress distribution of the membrane for the 1-4 system

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PT

ED

Fig. 5 Principal stresses and von Mises stress of the membrane for the 1-4 system

The stress in each tendon is the same, i.e. 37.98 MPa, and it is less than the yield stress of Nylon,

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which is 94.8 MPa (Cambridge University Engineering Department, 2003). Clearly, tendons do not yield at the designed prestressing. The load exerted on each bar is 14.53 N. According to Euler’s column

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buckling formula (Shames and Dym, 1995), the critical load of bars is 21530 N. It can be seen that the bar loads are much less than the critical load, indicating that bars do not buckle. Also, the bar loads at this level do not affect the local beam vibration modes discussed by Greschik (2008) and Ashwear and Eriksson (2014). 24

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2-3 System For the 2-3 system, the length of each bar is chosen as 1.8 m, i.e. b1j = b2j = 1.8 m (j = 1, 2, 3). Similar to the 1-4 system, the bar length is much larger than the bar thickness listed in Table 1. Thus, the orientation of bar cross sections has little effect on the motion of bars. The membrane is in the shape of

azimuth angle of each bar are specified as: 1 j   2 j  55 ; 10  2 j 3;

1 j  

10 ; 24.86  2 j 3;

2j  

 j  1, 2, 3  j  1, 2   j  3  j  1, 2   j  3

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24.86 ;

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an equilateral triangle and the length of the triangle edge is 1.732 m. The inclination angle and the

The lower ends of the bars in the first stage (i.e. A1j) form an equilateral triangle and the radius of the circumscribed circle for this triangle is 0.75 m. The tendons of the same type are identical. Tendon rest-

M

lengths are listed in Table 2.

Tendon D2j

Tendon C2j

Tendon P2j

Tendon R2j

Tendon S2j

1.0661 m

0.7998 m

0.7348 m

1.1239 m

1.5042 m

0.8535 m

1.1948 m

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CE

Restlength

Tendon V2j

PT

Tendon S1j

ED

Table 2. Tendon rest-lengths of the 2-3 system

The equilibrium parameters of this 2-3 system make the attached membrane symmetrically

tensioned. The stress distribution of the membrane is plotted in Fig. 6. It can be seen that the stress distribution is not uniform. The major principal stress (1), the minor principal stress (2), and the von Mises stress (M) are plotted in Fig. 7. The maximum value of M is 1.1488 MPa. It is less than the yield 25

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stress at 3% of Kapton, indicating that the membrane does not yield. Nodes where 2 is less than zero

PT

ED

M

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are marked in gray in Fig. 7, representing the wrinkled regions.

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Fig. 6 Stress distribution of the membrane for the 2-3 system

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PT

ED

Fig. 7 Principal stresses and von Mises stress of the membrane for the 2-3 system

The maximum stress in tendons is 33.58 MPa, which is less than the yield stress of Nylon. The loads

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exerted on bar b1j and b2j are 14.47 N and 5.02 N, respectively. According to Euler’s column buckling formula, the critical load of bars is 26981 N. Clearly, tendons do not yield, bars do not buckle at the

AC

given equilibrium, and local beam vibration modes are not affected by the given prestressing.

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Mesh Convergence Analysis This section addresses the mesh convergence study for the 1-4 system and the 2-3 system. To study the influence of mesh density on the behavior of the whole system, the convergence patterns for the system natural frequencies are investigated using the shell-beam-cable model.

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The tangent stiffness matrix at the designed equilibrium is used to conduct the analysis. According to Eq. (5), the tangent stiffness matrix KT can be expressed as: KT  0t K L  0t K NL . Thus, solving the following equation yields the natural frequencies and mode shapes of a tensegrity-membrane system: 2

 KT    O

(28)

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 M

where M is the mass matrix expressed in Eq. (5). The symbol  represents a natural frequency and  is the eigenvector containing the information of the corresponding mode shape. The mode shapes are orthogonalized with respect to M. For graphical representation, each mode shape  is normalized by 

ED

to the maximum element of ||.

 1 . In other words, mode shape  is normalized with respect

M

making the infinity norm of  be 1, i.e. 

PT

The mesh convergence study is conducted for the membrane and tendons, while it is not performed for bars. There are two reasons for this decision: 1) the deformations of the membrane and tendons are

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more significant, thus requiring appropriate meshes to capture the dynamics of these flexible components. For the bars, as discussed previously, rigid body motions, which are not sensitive to mesh

AC

density, are more likely to occur; 2) the four-node beam element used in this work is of high performance and a single four-node beam element can give good results for several beam deformation problems (Bathe, 1996).

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1-4 System For the mesh convergence study of the membrane, each bar is modeled by one four-node beam element and each tendon is modeled using one two-node truss element. The natural frequencies of the first 12 modes are shown in Fig. 8. It can be seen that, when the element number of the membrane is

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greater than 400, the natural frequencies of the first 12 modes start to converge. The improvement in the values of natural frequencies is not significant as the element number further increases. Therefore,

AC

CE

PT

ED

M

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modeling the membrane using 400 elements is a reasonable choice.

Fig. 8 Mesh convergence for the membrane of the 1-4 system

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The mesh convergence results for tendons are shown in Fig. 9. The membrane is modeled using 400 elements and each bar is modeled using one beam element. The number of elements for tendons is changed to examine the influence of tendon mesh density on the whole system. It can be seen that the

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natural frequencies of the first 12 modes are identical as the element number of each tendon increases. This observation indicates that the dynamics of the whole system is not sensitive to the mesh density of

AC

CE

PT

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tendons.

Fig. 9 Mesh convergence for the tendons of the 1-4 system 30

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2-3 System To conduct the mesh convergence analysis for the membrane, each bar is modeled by one four-node beam element, and each tendon is modeled using one two-node truss element. The mesh convergence

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results for the membrane are shown in Fig. 10, where the frequencies of the first 12 system modes are presented. It can be seen that, when the number of elements is greater than 300, the natural

frequencies of most modes converge. The frequencies of Mode 11 and Mode 12 are still trying to converge when the number of elements is greater than 300, while the improvement in natural

AC

CE

PT

ED

M

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frequency values is not significant. Thus, it is reasonable to model the membrane with 300 elements.

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Fig. 10 Mesh convergence for the membrane of the 2-3 system

The mesh convergence results for tendons are shown in Fig. 11. The membrane is modeled using 300

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elements and one beam element is used for each bar. Similar to the 1-4 system, the natural frequencies of the first 12 modes are identical as the element number of each tendon increases. Thus, we can

AC

CE

PT

ED

M

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conclude that the mesh density of tendons has little effect on the dynamics of the whole system.

Fig. 11 Mesh convergence for the tendons of the 2-3 system

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Modal Analysis For the nonlinear finite element models, modal analysis is performed by solving Eq. (28) for natural frequencies and mode shapes. For the control-oriented model, the nonlinear equations of motion in Eq.

form can be expressed as: Mz + Cz  Kz  O

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(24) are linearized at the designed equilibrium. The linearized equation of motion in the second-order

(29)

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where M, C, and K are the mass matrix, the damping matrix, and the tangent stiffness matrix,

respectively. Note that the variable z in Eq. (29) represents small perturbations and should not be confused with the variables with the same symbols in the nonlinear equations of motion in Eq. (24). Natural frequencies and mode shapes can be found by solving the following equation:  K   O

M

 M

2

(30)

PT

corresponding mode shape.

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where  represents a natural frequency and  is the eigenvector containing the information of the

Modal analysis can be also carried out using the linearized equation of motion in the first-order form,

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which is more popular in the control engineering community. The linearized equation in the first-order

AC

form can be expressed as:

where xz is the state vector: xz   zT

xz  Axz

T

zT  . Here matrix A can be written as:

33

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 O A 1  M K

  M C  I

(32)

1

After setting the damping matrix C equal to zero, the eigenvalues of matrix A correspond to the natural frequencies of tensegrity-membrane systems. When a nonzero damping matrix C is taken into account,

tensegrity-membrane systems is not modeled, i.e. C = O.

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the eigenvalues of matrix A correspond to the damped natural frequencies. In this work, the damping of

1-4 System Based on the results of the mesh convergence analysis in the previous section, a 20×20 mesh is used

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for the membrane. In other words, the membrane is discretized into 400 elements. For the two

nonlinear finite element models, each bar is modeled by one four-node beam element and three truss elements are used for each tendon. For the control-oriented model, the first five modes are used to

M

model the membrane dynamics.

Natural frequencies given by the three models are shown in Table 3. It can be seen that the results

ED

given by the three models are close to each other. The relative errors with respect to the results given by the shell-beam-cable model are less than 3%. Note that the membrane transverse bending stiffness

PT

and transverse shear stiffness are ignored in the membrane-truss-cable model and the control-oriented

CE

model. Also, the tendon dynamics and the geometrical nonlinearity of the membrane are neglected in the control-oriented model. Therefore, we can conclude that, for this 1-4 system, when natural

AC

frequencies are of primary interest, these modeling assumptions for the membrane-truss-cable model and the control-oriented model do not introduce significant modeling errors.

Table 3. Natural frequencies of the 1-4 system 34

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Membrane-trusscable Model

Control-Oriented Model

Mode 1

0.1386 Hz

0.1386 Hz

0.1367 Hz

Mode 2

0.3076 Hz

0.3076 Hz

0.2997 Hz

Mode 3

0.3076 Hz

0.3076 Hz

Mode 4

1.5598 Hz

1.5599 Hz

Mode 5

1.8590 Hz

1.8679 Hz

Mode 6

2.0571 Hz

2.0569 Hz

Mode 7

2.0571 Hz

Mode 8

2.0770 Hz

Mode 9

2.1793 Hz

Mode 10

2.2130 Hz

Mode 11 Mode 12

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Shell-beam-cable Model

0.2997 Hz 1.5401 Hz 1.8552 Hz 2.0288 Hz 2.0289 Hz

2.0763 Hz

2.0471 Hz

2.1803 Hz

2.1507 Hz

2.2277 Hz

2.2062 Hz

2.2130 Hz

2.2277 Hz

2.2119 Hz

2.2523 Hz

2.2653 Hz

2.2479 Hz

ED

M

2.0569 Hz

PT

The mode shapes of the 1-4 system given by the shell-beam-cable model are shown in Fig. 12 – Fig.

CE

14. In Fig. 12, the mode shapes of Mode 1 – Mode 3 show significant coupling between the rigid body motions of the bars and the membrane. The corresponding mode shapes of the membrane in the Of-

AC

xfyfzf frame are depicted in Fig. 13. It can be seen that, in Mode 1 – Mode 3, the modes of membrane transverse vibrations are also excited. The coupling between membrane deformations and the rigid body motions of bars can be also seen in the mode shape of Mode 5. These observations clearly indicate that the system components of this 1-4 system are highly coupled. 35

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It should be noted that the tendon dynamics is included in the shell-beam-cable model, but the modes of tendon deformations do not appear in the first 12 mode shapes of the 1-4 system. This result is expected, since tendons are generally much lighter than bars and membranes, while they are usually

AC

CE

PT

ED

M

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excited compared with vibrations of bars and membranes.

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heavily tensioned. As a result, tendon vibrations are of much smaller amplitudes and less likely to be

Fig. 12 Mode shapes of the 1-4 system (Mode 1 – Mode 6)

36

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AC

CE

PT

Fig. 13 Mode shapes of the membrane for the 1-4 system in the Of-xfyfzf frame (Mode 1 – Mode 3)

37

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PT

Fig. 14 Mode shapes of the 1-4 system (Mode 7 – Mode 12)

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2-3 System For the 2-3 system, the triangular membrane is divided into 3 quadrilaterals and a 10×10 mesh is

AC

used for each quadrilateral. In other words, 300 elements are used for the membrane. For the two nonlinear finite element models, each bar is modeled by one beam element and three truss elements are used for each tendon. For the control-oriented model, the first four modes are used to model the membrane dynamics.

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Natural frequencies of the 2-3 system given by the three models are shown in Table 4. Good agreement between the natural frequencies obtained using all of the three models can be observed. The relative errors are less than 1%. Thus, similar to the 1-4 system, we can conclude that the modeling assumptions for the membrane-truss-cable model and the control-oriented model do not introduce

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significant errors in the computation of natural frequencies for this 2-3 system. Table 4. Natural frequencies of the 2-3 system

Mode 2

0.4116 Hz

Mode 3

0.4116 Hz

Mode 4

1.0942 Hz

Mode 5 Mode 6

0.2527 Hz

0.2503 Hz

0.4116 Hz

0.4076 Hz

0.4116 Hz

0.4076 Hz

1.0953 Hz

1.0843 Hz

1.3164 Hz

1.3174 Hz

1.3020 Hz

1.3164 Hz

1.3174 Hz

1.3020 Hz

1.5230 Hz

1.5264 Hz

1.5110 Hz

PT

Mode 7

Control-Oriented Model

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0.2527 Hz

M

Mode 1

Membrane-trusscable Model

ED

Shell-beam-cable Model

1.5230 Hz

1.5264 Hz

1.5121 Hz

Mode 9

1.7399 Hz

1.7404 Hz

1.7258 Hz

Mode 10

1.8186 Hz

1.8224 Hz

1.8053 Hz

Mode 11

1.9834 Hz

1.9876 Hz

1.9712 Hz

1.9834 Hz

1.9876 Hz

1.9720 Hz

AC

CE

Mode 8

Mode 12

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Mode shapes of the 2-3 system given by the shell-beam-cable model are depicted in Fig. 15 – Fig.17. The results in Fig. 15 and Fig. 16 show that the membrane rigid body motions, the membrane deformations, and the bar rigid body motions are highly coupled in the first six modes of this 2-3 system. The coupling between system components of this 2-3 system is more significant than that of the 1-4

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system. This is expected due to the increased complexity of this 2-3 system. Clearly, for this 2-3 system, it is not reasonable to design or analyze system components separately.

Similarly to the 1-4 system, the modes of tendon deformation do not appear in the first 12 mode

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shapes of the 2-3 system. This observation indicates that the dynamics of bars and membranes is

dominant for this 2-3 tensegrity-membrane system, and tendon dynamics can be ignored without introducing significant modeling errors. Note that this observation is valid for linear analysis, since modal analysis is based on linear models. The results of system free vibration analysis, which contain the

AC

CE

PT

ED

M

information of system nonlinearities, can be found in the following section.

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AC

CE

PT

Fig. 15 Mode shapes of the 2-3 system (Mode 1 – Mode 6)

41

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AC

CE

PT

Fig. 16 Mode shapes of the membrane for the 2-3 system in the Of-xfyfzf frame (Mode 1 – Mode 3)

42

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PT

Fig. 17 Mode shapes of the 2-3 system (Mode 7 – Mode 12)

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Free Vibration Analysis

In this section, free vibration analysis is carried out to study the responses of tensegrity-membrane

AC

systems in the time domain. We can also compare the free vibration results given by the shell-beamcable model with the results given by the membrane-truss-cable model and the control-oriented model to examine the accuracy and reliability of the two simplified models. For the three models, Park’s method (Park, 1975) is used to solve the corresponding equations of motion. If x is the generalized coordinate vector, the velocity and acceleration terms can be expressed as: 43

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1 10 xt t  15 xt  6 xt t  xt  2t  6t 1  10 xt t  15 xt  6 xt t  xt  2t  6t

xt t  xt t

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Since Park’s method is a 3-step method, initialization is required. The trapezoidal rule is applied for the

xt t  xt 

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initialization process: t t  xt  xt t  ; xt t  xt   xt  xt t  2 2

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For the two systems studied in this work, the time step is chosen as t = 0.01s. This time step is

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chosen in accordance with Shannon’s sampling theorem (Shannon, 1949) and the highest natural

frequency given by the control-oriented model. The highest natural frequency of the 1-4 system is 2.4322 Hz. For the 2-3 system, the highest natural frequency is 4.4607 Hz. In order to capture the complete system dynamics, the sampling frequency is chosen to be approximately 10 times higher than

M

4.4607 Hz, which gives the corresponding time step t = 0.01s. In order to perform fair comparisons, the

PT

ED

same time step is used to numerically integrate the three mathematical models.

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1-4 System Membrane vibrations are studied by monitoring the transverse deflections of three points on the membrane, which are labeled w1, w2, and w3. These transverse deflections are measured in the Of-xfyfzf

AC

frame and are characterized by: w1  w  x f , y f

x

f

0.6 (m); y f 1.6 (m)

w2  w  x f , y f

x

f

0.5 (m); y f 0.4 (m)

w3  w  x f , y f

x

f

1.5 (m); y f 0.3 (m)

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Free vibration analysis results for the 1-4 system are given here for two different cases: for Case I, the initial perturbation is 12  0.573 deg/s ; for Case II, the initial perturbation is selected as 11  0.286 deg/s . Simulation results given by the three models are plotted in Fig. 18 – Fig. 23. The

responses of the bars given by the three models are in good agreement. Note that the dynamics of

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tendons and the deformations of bars are ignored in the control-oriented model. This observation

indicates that ignoring the flexibility of bars and tendon dynamics does not cause significant errors. It can also be observed that the membrane transverse deflections obtained using the control-oriented model are close to the deflections obtained using the two nonlinear finite element models. Clearly, the

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geometrical nonlinearity of the membrane, which is ignored in the control-oriented model, is not

significant when this 1-4 system experiences the given perturbations. Note that we observe similar system behaviors for other initial perturbations of relatively the same size.

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To examine the transverse vibrations of tendons, we first introduce a local orthogonal coordinate system with basis vectors et1 , et 2 , and et 3 for each tendon. The vector et1 is defined to be the unit vector

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pointing from one tendon end to another. Then, et 2 and et 3 , which define the two transverse directions

et 2 

et1  e3 ; et 3  et1  et 2 et1  e3

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of a tendon, can be expressed as:

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where e3 is the basis vector along the Oz axis. The tendon transverse deflections of the maximum amplitude given by the shell-beam-cable model

and the membrane-truss-cable model are shown in Fig. 20 and Fig. 23. It can be seen that the results given by the two nonlinear finite element models are in good agreement. Note that the tendon vibration

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amplitude is less than 5×10-4 mm, which is much smaller than the vibration amplitude of the attached membrane and the bars. Similar results are obtained when the 1-4 system is subjected to other initial perturbations of relatively the same size. Therefore, it is reasonable to conclude that for this 1-4 system

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and for the given perturbations the tendon dynamics does not affect the dynamics of the whole system.

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Fig. 18 Responses of bar end B14 of the 1-4 system (Case I)

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Fig. 19 Membrane transverse deflections of the 1-4 system in the Of-xfyfzf frame (Case I)

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Fig. 20 Representative tendon transverse deflections of the 1-4 system (Case I)

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Fig. 21 Responses of bar end B14 of the 1-4 system (Case II)

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Fig. 22 Membrane transverse deflections of the 1-4 system in the Of-xfyfzf frame (Case II)

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Fig. 23 Representative tendon transverse deflections of the 1-4 system (Case II)

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2-3 System To study the membrane vibrations of this 2-3 system, the transverse deflections of three points on

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the membrane, w1, w2, and w3, are monitored in simulations. These transverse deflections are

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characterized as:

w1  w  x f , y f

x

w2  w  x f , y f

x

f

0.86 (m); y f 0.98 (m)

w3  w  x f , y f

x

f

0.45 (m); y f 0.26 (m)

f

0.86 (m); y f 0.5 (m)

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Simulation results are shown here for the free vibration behaviors of the 2-3 system in two different cases: for Case I, the initial perturbation is chosen as 11  0.573 deg/s ; for Case II, the initial perturbation is 11  0.286 deg/s . Simulation results given by the three models are shown in Fig. 24 – Fig. 31. Similar to the observations for the 1-4 system, it can be seen that the responses of the bars

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given by the three models are very close to each other. We also observe similar results when the 2-3 system experiences other initial perturbations of relatively the same size. Therefore, for this 2-3 system, it is reasonable to treat bars as rigid bodies when the system is subject to perturbations in this range. Moreover, membrane vibration responses are in good agreement, indicating that ignoring the

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geometrical nonlinearities of the membrane does not cause major errors.

The tendon transverse vibrations of the maximum amplitude given by the shell-beam-cable model and the membrane-truss-cable model are shown in Fig. 27 and Fig. 31. Clearly, the results given by the

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two nonlinear finite element models are close to each other. The tendon vibration amplitude is less than 1.5×10−3 mm, which is much smaller than the vibration amplitude of the attached membrane and the

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bars. Similar system response patterns are observed when other comparable initial perturbations are

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applied to this 2-3 system. Thus, we can conclude that it is reasonable to ignore the tendon dynamics for

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this 2-3 system when it experiences perturbations in this range.

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Fig. 24 Responses of bar end B21 of the 2-3 system (Case I)

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Fig. 25 Responses of bar end B11 of the 2-3 system (Case I)

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Fig. 26 Membrane transverse deflections of the 2-3 system in the Of-xfyfzf frame (Case I)

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Fig. 27 Representative tendon transverse deflections of the 2-3 system (Case I)

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Fig. 28 Responses of bar end B21 of the 2-3 system (Case II)

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Fig. 29 Responses of bar end B11 of the 2-3 system (Case II)

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Fig. 30 Membrane transverse deflections of the 2-3 system in the Of-xfyfzf frame (Case II)

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Fig. 31 Representative tendon transverse deflections of the 2-3 system (Case II)

Discussions and Conclusions

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In this article, an analysis of the dynamics of two tensegrity-membrane systems is conducted using three mathematical models, i.e. the shell-beam-cable model, the membrane-truss-cable model, and the

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control-oriented model. According to the modal analysis results, the natural frequencies given by the three models are very close to each other and the relative errors are less than 3%. The mode shape analysis of the two tensegrity-membrane systems indicates that there is strong coupling among the membrane rigid body motions, the membrane deformations, and the rigid body motions of bars. Thus, 60

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we can conclude that a tensegrity-membrane system should be designed and studied as a whole system, and it is not reasonable to design or analyze system components separately. Moreover, the modal analysis results indicate that the tendon vibration modes are not excited in the first 12 modes of the two tensegrity-membrane systems studied in this work. In the free vibration

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analysis, the vibration amplitude of tendons is much smaller than the vibration amplitude of membranes and bars. These two observations are expected, since the tendons are generally heavily tensioned and much lighter than the bars and the membrane. Therefore, for the two tensegrity-membrane systems

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studied in this work, it is reasonable to ignore the dynamics of tendons in the study of system dynamics. We remark that this is a commonly used assumption in the modeling and analysis of tensegrity systems. The free vibration results given by the three models are compared with each other. The results are in good agreement, which indicates that the control-oriented model can capture the system dynamics

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accurately when the two tensegrity-membrane systems studied in this work experience relatively small perturbations. Moreover, it can be also concluded that modeling the bars as rigid bodies does not

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introduce significant modeling errors. This conclusion coincides with another commonly used modeling

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assumption for tensegrity systems that bars can be treated as rigid bodies.

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Acknowledgments

This material is based upon work supported by the National Science Foundation under the grant

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CMMI-0952558.

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