Force Density Matrix Research Articles

Form-Finding of Tensegrity Basic Unit with Equal Cable Length

Tensegrity is a lightweight, self-stressing, and self-stabilizing structure made up of cables and bars, with each member bearing either tension or compression but not affected by shear stress. This design allows for optimal utilization of the material properties of the members. In a tensegrity basic unit, the bar members are of equal length, while the cable members come in three lengths: lower-end surface horizontal cable, upper-end surface horizontal cable, and stayed cable. The tensegrity basic unit with equal cable length simplifies this further by ensuring that all cables are the same length, resulting in a structure with only two member lengths, i.e., bar length and cable length, enhancing interchangeability. In order to find the form without the action of external forces, the force density coefficient ratio is introduced. By performing a force balance analysis on any node of the unit, the equilibrium equation of the structure is determined, incorporating the additional constraint of equal cable length. Two methods are employed to ascertain the force density coefficient ratio of each member in the unit: the theoretical derivation method based on the stable configuration condition of the tensegrity basic unit with equal cable length, and the method of solving the characteristic equations of the force density matrix. A program is developed to validate the form-finding method using basic units with three, four, five, and six bars as examples. The results show that the model accurately represents the physical structure, confirming the reliability of the form-finding methods.

Form-finding of tensegrity structures based on genetic algorithm

We develop a novel form-finding method that utilizes force density and a genetic algorithm. Firstly, the equilibrium equation is derived by using the force density method. Next, the force density matrix is decomposed through QR decomposition. Subsequently, an optimization objective function is introduced, which incorporates information about the force density values. The optimal solution for the objective function is obtained through the use of a genetic algorithm. We determine a suitable set of force density values that satisfy the requirements of the equilibrium matrix rank deficiency, force density rank deficiency, and precision, thereby establishing the equilibrium configuration of the structure. The simulation results verify the reliability of the proposed method. With its advantages of straightforward calculations, rapid convergence, and high precision, this method proves to be well-suited for the form-finding of both regular and irregular tensegrity structures.

A resource-efficient form-finding approach to tensegrity structures

PurposeThe aim of this article is to obtain a stable tensegrity structure by using the minimum knowledge of the structure.Design/methodology/approachThree methods have been formulated based on the eigen value decomposition (EVD) and singular value decomposition theorems. These two theorems are being implemented on the matrices, which are computed from the minimal data of the structure. The required minimum data for the structure is the dimension of the structure, the connectivity matrix of the structure and the initial force density matrix computed from the type of elements. The stability of the structure is analyzed based on the rank deficiency of the force density matrix and equilibrium matrix.FindingsThe main purpose of this article is to use the defined methods to find (1) the nodal coordinates of the structure, (2) the final force density values of the structure, (3) single self-stress from multiple self-stresses and (4) the stable structure.Originality/valueBy using the defined approaches, one can understand the difference of each method, which includes, (1) the selection of eigenvalues, (2) the selection of nodal coordinates from the first decomposition theorem, (3) the selection of mechanism mode and force density values further and (4) the solution of single feasible self-stress from multiple self-stresses.

A zeroing neural network model for form-finding problems: A nonlinear optimization approach

How to establish a simple and efficient nonlinear optimization problem form-finding model in noise environments is the main concern of this paper, thereby, a sequential quadratic programming form-finding (SQP) approach via a noise-tolerant zeroing neural network (NTZNN) is presented to solve the proposed problem. First, the rank of force density matrix is selected as optimization objective due to its crucial physical property. In addition, linear relations between the force density vectors and force density matrix are selected as the constraint conditions, which can ensure the tensegrity mechanism in a stable status. Furthermore, the simulations are shown that the presented SQP-NTZNN algorithm has superiority with anti-noise performance and computation efficiency, which by comparing the computational rate in the same noise environment with the novel form-finding algorithm. Eventually, the form-finding simulations under different initial conditions are formed to verify the importance of the constrained conditions, it can prove the high efficiency of the proposed nonlinear constrained optimization model which is transformed from the form-finding problem.

Deep Neural Networks for Form-Finding of Tensegrity Structures

Analytical paradigms have limited conventional form-finding methods of tensegrities; therefore, an innovative approach is urgently needed. This paper proposes a new form-finding method based on state-of-the-art deep learning techniques. One of the statical paradigms, a force density method, is substituted for trained deep neural networks to obtain necessary information of tensegrities. It is based on the differential evolution algorithm, where the eigenvalue decomposition process of the force density matrix and the process of the equilibrium matrix are not needed to find the feasible sets of nodal coordinates. Three well-known tensegrity examples including a 2D two-strut, a 3D-truncated tetrahedron and an icosahedron tensegrity are presented for numerical verifications. The cases of the ReLU and Leaky ReLU activation functions show better results than those of the ELU and SELU. Moreover, the results of the proposed method are in good agreement with the analytical super-stable lines. Three examples show that the proposed method exhibits more uniform final shapes of tensegrity, and much faster convergence history than those of the conventional one.

Robustness of rank minimization heuristics for form-finding of tensegrity structures

The form-finding process for tensegrity structures with a given topology seeks a distribution of force densities that satisfies stability and equilibrium conditions. A rank minimization problem regarding the force density matrix can yield feasible distributions of force densities. A direct computation of the matrix rank is difficult and is generally avoided in practice. In a recent contribution by Wang [Engineering Structures 2021;227:111419], a use the trace heuristic for form-finding is proposed. The trace heuristic does not guarantee recovery of the minimal rank. Insufficient rank deficiency leads to a degenerated geometry of the tensegrity. In this contribution, the approach is extended to log-det heuristic and a comparison between both methods is carried out focusing on robustness of the form-finding procedure for tensegrities. Robustness of the form-finding procedure is understood here as the ability to recover at least one feasible tensegrity design. Several examples show that the robustness is mainly influenced by the topology of the structure, constraints on force densities and the admissible set for force densities. Furthermore, the proposed extension with log-det heuristic outperforms the original method in terms of robustness, however it increases the computational cost.

Nonlinear Mechanics of a Smart Biotensegrity Human Foot Prosthesis

This work deals with the nonlinear mechanics of smart bioinspired tensegrity structures. A minimal regular tensegrity prism actuated by shape memory alloy (SMA) elements is investigated to represent a human foot. A formulation considering the force density matrix approach is used to model the equilibrium equations of the tensegrity structure based on node mapping. Lagrange multipliers are employed to represent constraints. The SMA thermomechanical behavior is described by considering a modified polynomial constitutive model. Numerical simulations are developed from an optimization procedure employing the Levenberg–Marquardt method. An investigation of the tensegrity capability to model a human foot is carried out analyzing either mechanical or physiological aspects of the tensegrity prosthesis. The mechanical performance is compared with high performance prostheses available on the market, showing that it is an interesting alternative with respect to mechanical resistance. Regarding physiology, foot movements are properly mimicked from SMA actuation.

A unifying framework for form-finding and topology-finding of tensegrity structures

This paper presents a unifying framework for the form-finding and topology-finding of tensegrity structures. The novel computational framework is based on rank-constrained linear matrix inequalities. For form-finding, given the topology (i.e., member connectivities), the determination of the member force densities is formulated into a linear matrix inequality (LMI) problem with a constraint on the rank of the force density matrix. The positive semi-definiteness and rank deficiency condition of the force density matrix are well managed by the rank-constrained LMI-based formulation. A Newton-like algorithm is employed to solve the rank-constrained LMI problem. Two methods, named direct method and indirect method, are proposed to determine the nodal coordinates once the force densities have been obtained. For topology-finding, given the geometry (i.e., nodal coordinates), the determination of the topology is also formulated into an LMI problem with a constraint on the rank of the tangent stiffness matrix. Numerical examples demonstrate that different types of form-finding problems (such as tensegrity structures with single and with multiple self-stress states, symmetric and irregular tensegrity structures) can be uniformly and efficiently solved by the proposed approach. Furthermore, three well-known tensegrity structures are reproduced to verify the effectiveness of the proposed formulation on the topology-finding of tensegrity structures.

Numerical form-finding of multi-order tensegrity structures by grouping elements

Multi-order tensegrity structures are an attractive form of compliant deployable structures. An efficient numerical form-finding method is proposed for multi-stable tensegrity structures in this paper. The current method first analyze the force density matrix for sets of more feasible force densities that satisfy the non-degeneracy conditions. Then, based on symmetrical grouping of elements, a genetic algorithm is used to minimize the eigenvalues; as a result, multiple orders of equilibrium can be found. For the investigation, two symmetric tensegrity structures are analyzed using the currently proposed method, and the method's applicability and accuracy have been examined.

Form-finding of tensegrity structures via rank minimization of force density matrix

This study proposes a general computational framework for the form-finding of tensegrity structures. The procedure is divided into two stages in which the member force densities and nodal coordinates are obtained respectively. In the first stage, the determination of force densities is transformed into a rank minimization problem regarding the force density matrix and then formulated into a semi-definite programming. The unilaterality condition of member forces and the positive semi-definiteness condition of force density matrix are incorporated as constraints. In the second stage, the determination of nodal coordinates is formulated into a constrained nonlinear programming model. The nodal positions and member lengths are assigned as constraints and auxiliary variables are introduced to homogenize the member lengths. The proposed formulation bypasses the number of self-stress states in a tensegrity structure thus applies to both tensegrity structures with single and multiple self-stress states. Different from existing studies that are based on the minimum required rank deficiency condition of force density matrix, the proposed method can handle tensegrity structures that have a force density matrix with rank deficiency greater than the required minimum number. Several examples are presented to verify the effectiveness of the proposed method on different types of tensegrity structures.

Innovative numerical form-finding of tensegrity structures

An innovative numerical platform for the form-finding of tensegrity structures is proposed. A particular objective function is introduced based on LU-decomposition of the force density matrix. The required rank deficiency of the force density matrix, which is a crucial quantity in the form-finding process, is then pursued by minimisation of the mentioned objective function. The gradient and hessian of the objective function are also derived explicitly. This critical feature enables us to efficiently employ powerful nonlinear programming approaches in the form-finding problems. Four numerical examples are provided to validate the feasibility and effectiveness of the method proposed. The results compared and contrasted from different aspects with existing analytical and some well-known numerical solutions. The comparisons verify that the method proposed is highly accurate and fast; therefore, it can be used as a compelling alternative to the available numerical form-finding methods.

Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements

An analytical form-finding method for regular tensegrity structures based on the concept of force density is presented. The self-equilibrated state can be deduced linearly in terms of force densities, and then we apply eigenvalue decomposition to the force density matrix to calculate its eigenvalues. The eigenvalues are enforced to satisfy the non-degeneracy condition to fulfill the self-equilibrium condition. So the relationship between force densities can also be obtained, which is followed by the super-stability examination. The method has been developed to deal with planar tensegrity structure, prismatic tensegrity structure (triangular prism, quadrangular prism, and pentagonal prism) and star-shaped tensegrity structure by group elements to get closed-form solutions in terms of force densities, which satisfies the super stable conditions.

Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model

A new method is proposed for simultaneous optimization of shape, topology and cross section of plane frames. Compliance against specified loads is minimized under constraint on structural volume. Difficulties caused by the melting nodes can be alleviated to some extent by introducing force density as design variables for defining the geometry, where the side constraints are assigned for force density to indirectly avoid the existence of extremely short members. Force density method is applied to an auxiliary cable-net model with different boundary and loading conditions so that the regularity of force density matrix is ensured by positive force densities. Sensitivity coefficients of the objective and constraint functions with respect to the design variables are also explicitly calculated. After the optimal geometry of the frame is obtained, the topology is further improved by removing the thin members and combining closely spaced nodes. It is demonstrated in the numerical examples of three types of frames that rational geometry and topology can be achieved using the proposed method, and the effect of bending moment on the optimal solution is also discussed.

A hybrid symmetry–PSO approach to finding the self-equilibrium configurations of prestressable pin-jointed assemblies

For pin-jointed assemblies with many members or self-stress states, the form-finding problem using conventional methods generally involves considerable computational complexities due to the large size of the solution spaces. Here, we propose an improved form-finding method for prestressable pin-jointed structures by combining symmetry-based qualitative analysis with particle swarm optimization. Expressed in the symmetry-adapted coordinate system, the nodal coordinate vectors of a structure with specific symmetry and topology are independently extracted from the key blocks of the small-sized force density matrices associated with rigid-body translations. Then, the first block of the equilibrium matrix is computed, in which the null space reveals integral self-stress states. Particle swarm optimization is introduced and adapted to find feasible prestress modes, where the uniformity and unilaterality conditions of the members are considered. Besides, the QR decomposition with column pivoting is adopted for efficient computations on the null space of these blocks. The QR decompositions of the small-sized blocks of the force density matrix and the equilibrium matrix are performed iteratively, to simultaneously find a stable self-equilibrium configuration and a feasible prestress mode. Representative examples show the presented method is computationally efficient and accurate for the form-finding of symmetric tensegrities and prestressed cable–strut structures.

Design and form-finding of compression structures with prestressing tendons

Compression-only structures such as domes and vaults can be modelled as pin-jointed networks whose equilibrium configuration is obtained using well-known form-findings methods such as the Force Density Method. As all the members of the network are in compression, its corresponding force density or force:length ratio matrix is non-singular and the final shape is obtained directly from the equilibrium equations.The introduction of prestressing tendons (members in tension) in compression structures allows for the design of more creative forms, showing an exceptional versatility in the design of new structures. However, because tension and compression members exist, the force density matrix can be singular or ill-conditioned and the solution of the equilibrium equations is not quite as simple as in the case of compression-only structures. In this work, a procedure for the design of compression structures with prestressing tendons along with several examples are presented.

Octahedron family: The double-expanded octahedron tensegrity

The octahedron family of tensegrity structures is presented in this research. The octahedron and the expanded octahedron (well-known tensegrities in the literature) are the first and second components of the family. A new tensegrity is presented: the double-expanded octahedron. This new tensegrity form was obtained following the connectivity pattern of the octahedron family presented in this work. The values of the force densities or force:length ratios that satisfy the minimum required rank deficiency of the force density matrix were computed analytically. Two types of solutions are obtained: full and folded forms. Results show that each lower member of the octahedron family is a folded form of a superior member of this family. Several examples are shown.

Self-equilibrium and super-stability of truncated regular hexahedral and octahedral tensegrity structures

This paper presents conditions for self-equilibrium as well as super-stability of the truncated regular hexahedral and octahedral tensegrity structures. Their symmetry can be described by octahedral group in group representation theory, and furthermore, their force density matrix is analytically rewritten in the symmetry-adapted form. The condition for self-equilibrium, in terms of force densities, is found by satisfying the non-degeneracy condition for a tensegrity structure. The condition for super-stability, also in terms of force densities, is further presented by guaranteeing positive semi-definiteness of the force density matrix.

Improved Form-Finding of Tensegrity Structures Using Blocks of Symmetry-Adapted Force Density Matrix

AbstractForm-finding analysis is very important for developing innovative tensegrity structures. Nevertheless, it is frequently difficult to determine simultaneously the prestress mode and the exac...

Group-theoretical form-finding of cable-strut structures based on irreducible representations for rigid-body translations

A group-theoretical method is proposed for form-finding of symmetric cable-strut structures. Only the given topology and structural symmetry are necessary for implementing this form-finding process. Irreducible representations for rigid-body translations along d directions are used to get the corresponding symmetry subspaces, which transform the original force density matrix into d small-sized block matrices. Therefore, significant simplification on the form-finding process is achieved. Numerical decomposition of these blocks is performed to find analytical sets of force densities and nodal coordinates, which satisfy the minimum required rank deficiencies of the blocks and indicate meaningful null spaces. On condition that the tangent stiffness matrix of the obtained configuration is positive definite, a non-degenerate d-dimensional stable cable-strut structure is built. Some examples specified by Cnv or Dn symmetry are presented. It shows that the proposed method is effective for searching feasible configurations of cable-strut structures with specific symmetries.

An Investigation on Optimal Initial Self-Stress Design of Tensegrity Grid Structures

To achieve the optimal feasible force density vector of a given geometry configuration tensegrity grid structure, an efficient procedure is presented for optimal initial self-stress design of tensegrity grid structures by consecutively solving two linear homogeneous systems in conjunction with a minimization problem. The nonlinear constrained optimization algorithm, known as the Interior-Point Method (I-PM), is utilized to obtain the minimal solution, leading to a set of force densities which guarantee the non-degeneracy condition of the force density matrix. The evaluation of the eigenvalues of tangent stiffness matrix is also introduced to check the geometric stability of the tensegrity grid structures. Finally, three numerical examples have been investigated comprehensively to prove the capability of the proposed method in optimal initial self-stress design of tensegrities. Furthermore, division of number of member group has been discussed in detail for the purpose of demonstrating the efficiency of the proposed method in seeking initial force densities of tensegrity grid structures.