Infinitesimal Mechanisms Research Articles

Structural Analysis of Periodic Trusses and Lattice Materials: States of Self-Stress, Mechanisms, and Mechanical Properties

Abstract In the realm of low relative density, a rigid-jointed lattice material exhibits a mechanical response analogous to that of a pin-jointed periodic truss with an identical micro-architecture. This correspondence simplifies the evaluation of the lattice's structural performance. While established matrix methods in linear algebra are capable of determining the states of self-stress, infinitesimal mechanisms, and mechanical properties of pin-jointed periodic trusses, they lack a systematic approach to deriving closed-form expressions for these properties. This paper presents a straightforward approach to address this gap. Furthermore, we introduce an inventive finite element framework that directly yields self-stress states and infinitesimal mechanisms in periodic trusses subjected to specific uniform macroscopic loading conditions. This framework allows for the determination of mechanical properties for periodic trusses by solving straightforward boundary value problems, such as uniaxial tension/compression or simple shear. The developed finite element framework offers greater simplicity compared to the matrix methods, with its benefits becoming more pronounced for complex micro-architectures. To demonstrate the effectiveness of the two newly developed methods, we conduct structural analyses on five different lattice materials, achieving successful outcomes.

Experimental investigations on mechanical propertiesof 3D-printed tensegrity-inspired metamaterialsbased on 4-strut simplex module

The present study is focused on experimental investigations of mechanical properties of 3D-printed tensegrity-inspired metamaterials. Tensegrity systems have many advantageous features, such as: light weight, high stiffness-to-mass ratio, controllability, inherent attributes of smart structures, and unique mechanical behaviour. They may be applied not only in macro-scale, but they can also be used to create cellular mechanical metamaterials and lattices in various scales. Metamaterials are understood here as human-designed artificial materials, which do not exist in nature, and whose mechanical properties result from the morphology of the inner structure rather than from chemical or phase composition. Experimental studies on tensegrity metamaterials manufactured using 3D printing techniques are hardly present in the literature. This paper presents results of uniaxial compression tests carried out on a number of 3D-printed tensegrity-based modules corresponding to the metamaterial cells, differing in the manufacturing technology, parent material, and size. The following observations were made during the tests: one of the most important parameters that has a direct impact on the results is the elongation at break of the parent material; any inaccuracies at the production stage greatly affect the mechanical behaviour of the structure; it is crucial to ensure a free deformation consistent with the infinitesimal mechanism mode of tensegrity; a post-critical behaviour of the struts was clearly visible in the performed tests.

Analytical and numerical investigation of second-order infinitesimal mechanism in rigid origami

This study investigates second-order infinitesimal mechanisms and bifurcation paths of rigid origamis with multiple-degree-of-freedom mechanisms. A truss model, consisting of pin-connected rigid bars, is employed for the infinitesimal mechanism analysis. The conditions for the existence of a second-order infinitesimal mechanism are analytically solved to investigate the combinations of the infinitesimal mechanism modes that are the potential finite mechanisms. Analytical solutions for simple crease patterns show the bifurcation of the deformation paths at the flat state, some disappeared combinations of the mechanism modes in a single path, and the relationship between the nodal displacement and axial force distribution. This comprehensive analysis, using analytical solutions, lays the foundation for understanding bifurcation and finite mechanism of rigid origami, which have not yet been fully explored. These properties are also verified in the finite deformation paths generated by large deformation analysis of a frame model consisting of frame members and hinges, which is suitable for the analysis using general-purpose finite element analysis software. The second-order infinitesimal mechanisms found in this study are confirmed to be able to be extended to the finite mechanisms.

Panel-Pin Model for Kinematic and Equilibrium Analysis of Rigid Origami

In this study, the mathematical model referred to as the panel-pin model is proposed for analyzing the infinitesimal-deformation mechanism and the large-deformation equilibrium path of a rigid origami composed of rigid faces (panels) and subjected to deformation only at its crease lines. The panel-pin model represents a rigid origami as a structure of rigid panels pin-connected at the vertices and has the following advantages: 1) consistent formulation of compatibility equations for any type of the rigid origami structure; 2) systematic computation of vertex displacements, folding angles, and their derivatives; and 3) ease of accounting for gravity acting on the panels. The infinitesimal mechanism of the panel-pin model is studied according to the standard procedure for mechanisms and linkages, and an equilibrium path is traced as the trajectory of equilibrium points obtained by minimizing the total potential energy of the model with a rotational spring along each crease line. The numerical examples with multiple degrees of freedom of the mechanism show that the multiple equilibrium paths can be obtained by assigning the initial imperfection, and this is also confirmed by the physical model.

Experimental Testing of a Tensegrity Simplex: Self-Stress Implementation and Static Loading

A physical model of the simplest three-dimensional tensegrity unit was built at human scale out of aluminum. Self-stress implementation and static loading tests were performed on this model. At each step, accurate measurements were obtained for all nodal positions and element forces. For the prestressing phase, elongations were imposed, via mechanical devices, in different combinations of elements, called prestress scenarios. Experimental results are compared to the theoretical self-stress state obtained by singular value decomposition of the equilibrium matrix and to numerical simulations using the dynamic relaxation method. It is shown that internal forces follow the same linear trend for all prestress scenarios even if the geometry is significantly impacted. Compressive tests were conducted by hanging masses from the top nodes. It is shown that there exists a unique load-displacement relation that follows the infinitesimal mechanism direction for a finite distance, which depends on the self-stress level. The paper provides a detailed overview of the simplex’s structural behavior using both experimental and numerical results while discussing the limitations of the analysis methods explored.

Dynamic Stability of Tensegrity Structures-Part I: The Time-Independent External Load.

The paper contains a parametric analysis of tensegrity structures subjected to time-independent external loads. A complete dynamic stability analysis is a three-step process. The first stage involves the identification of self-stress states and infinitesimal mechanisms. The next stage concentrates on the static and dynamic behavior of tensegrities under time-independent external loads, whereas the third is under periodic loads. In this paper, the first two stages are carried out. The structures built with the most popular tensegrity modules, Simplex and Quartex, are considered. The effect of the initial prestress on the static parameters and frequency is analyzed. To assess this behavior, a geometrically non-linear model is used.

Damage detection in tensegrity under varying temperature using interacting Particle-Ensemble Kalman Filter

Tensegrities are structural mechanisms, with dedicated compression (struts/bars) and tension members (cables). The compression members float inside the network of tension members. Tensegrities are characterized by the presence of at least one infinitesimal mechanism, which is stabilized by the pre-stress present in the members, to ensure the equilibrium of the structure. Under external load, tensegrity may change its form by altering its member pre-stress, thereby affecting its global stiffness even in the absence of damage. Moreover, tensegrities can have different stiffness properties under the same structural configuration in the absence of any damage or external load, if the pre-stress levels of the members are different. However, the change in dynamic characteristics of tensegrities is not limited to the aforementioned causes only and is also affected by ambient uncertainties. A variation in temperature may alter the dynamic characteristics of a tensegrity by influencing its material (Young’s modulus, etc.) and structural (boundary conditions, structural dimensions, etc.) properties. This can potentially lead to a false impression of tensegrity damage/health. Meanwhile, the prolonged usage of tensegrity may lead to loss of pre-stress in the cables, buckling of the bars, corrosion, and damage of the members, etc. Thus affecting the structural stiffness which leads to change in the measured dynamic properties of the tensegrity. To account for this actual damage in the tensegrity, all the mentioned major challenges that could lead to a false alarm need to be dealt with. The present study develops a vibration-based time-domain approach for tensegrity health monitoring in the presence of uncertainties due to ambient force, measurement noise, and varying temperature. An interacting filtering technique has been used, where the state variables are estimated by the Ensemble Kalman filter that resides inside the Particle filter which computes the health parameters.

Kinematic restriction method for tensegrity structures

The use of tensegrity structures in robotics has been studied in recent years thanks to their properties of light weight, efficient distribution of forces and the possibility of applying control methods for the active shape reconfiguration, with crawling and rolling as examples. The structural composition of tensegrities, particularly the use of tensile elements, results in the presence of infinitesimal mechanisms. These affine motions have been considered for the development of control strategies that follow the manifold of stable positions described by these mechanisms. However, in robotic applications, the presence of rigid body motions can cause undesired effects, such as tensegrity structures flipping over. The form finding methods available in literature consider only the statics of tensegrities and most dynamic models work assume free-standing structures, while few consider fixed structures but do not deepen into the process of removing the rigid body motions. In this paper, the formulation of the equilibrium equations of tensegrity structures is reviewed, to describe the presence of the affine motions in the vector spaces of the equilibrium matrix. A simplified method for removing rigid body motions from the equilibrium equations is detailed, and the effect on the set of affine motions is explained. To verify the validity of this process, two numerical examples are given, including a single unit and a two stage tensegrity structures.

Active Control of Stiffness of Tensegrity Plate-like Structures Built with Simplex Modules.

The aim of this study is to prove that it is possible to control the static behavior of tensegrity plate-like structures. This possibility is very important, particularly in the case of deployable structures. Here, we analyze the impact the support conditions of the structure have on the existence of specific characteristics, such as self-stress states and infinitesimal mechanisms, and, consequently, on the active control. Plates built with Simplex modules are considered. Firstly, the presence of the specific characteristics is examined, and a classification is carried out. Next, the influence of the level of self-stress state on the behavior of structures is analyzed. A geometrically non-linear model, implemented in an original program, written in the Mathematica environment, is used. The results confirm the feasibility of the active control of stiffness of tensegrity plate-like structures characterized by the presence of infinitesimal mechanisms. In the case when mechanisms do not exist, structures are insensitive to the initial prestress level. It is possible to control the occurrence of mechanisms by changing the support conditions of the structure. Based on the obtained results, tensegrity is very promising structural concept, applicable in many areas, when conventional solutions are insufficient.

Nonlinear Dynamics Analysis of Flexible Deployable Linkage Mechanisms Using the Finite Particle Method

Deployable mechanisms have notable applications in mechanical engineering, civil engineering, and space technology. Although often ignored, deployable linkage mechanism exhibits additional flexibility beyond rigid folding owing to the deformation of nonrigid components. The actual behavior of flexible deployable linkage usually involves the dynamic effect and geometric nonlinearity. Using the finite particle method (FPM), this study investigated the nonlinear responses of flexible deployable linkage mechanisms. As a particle method, the FPM is displacement based, explicit, and can avoid iterations to solve nonlinear equilibrium equations. It can be used for nonlinear analyses of structures with rigid body motion and infinitesimal mechanisms, and to determine the internal force and structural deformation. To investigate the nonrigid Bennett linkage and Bricard linkage, formulations of a three-dimensional beam element and revolute hinge element were derived for FPM analysis. Agreement between analytical solutions and numerical simulations demonstrated the efficiency of the proposed approach for nonlinear motion analysis of nonrigid mechanisms. The FPM results revealed that mechanism flexibility can cause deviation from rigid compatibility paths, and the internal force and deformation of mechanisms should be considered when designing nonrigid mechanisms.

Pop-up kirigami for stiff, dome-like structures

In architecture and engineering, surfaces with positive Gaussian curvature such as domes are used for their high stiffness-to-weight ratios and efficiency in enclosing a volume. These curved shapes are difficult to create due to time-consuming or scale-limited processes. In recent years, origami and kirigami have risen as viable routes for the rapid fabrication of complex surfaces from flat sheets; however, these methods typically lead to systems that are overly flexible due to their high number of degrees of freedom. In this paper, we present a design for a pop-up kirigami system that achieves symmetric, positive Gaussian curvature by taking advantage of an internal infinitesimal mechanism. The system is fabricated from flat sheets using a hexagonal pattern, and the sheets remain flat locally as the system deforms into a dome-like shape. We investigate the internal mechanism and deformation modes of the system, revealing the flexible mode that creates dome-like curvature. We discuss geometric variations of the system and illustrate the possible shapes that result from changing the initial pattern parameters. Finally, we demonstrate the high stiffness of the system that arises from restricting its one flexible mode in its final, dome-like shape. The proposed pop-up kirigami system offers a method for fabricating dome-like surfaces with potential applications as deployable enclosures, concave reflectors, and more.

Estimation of local failure in tensegrity using Interacting Particle-Ensemble Kalman Filter

Tensegrities form a special case of truss, wherein compression members (struts/bars) float within a network of tension members (cables). Tensegrities are characterized by the presence of at least one infinitesimal mechanism stabilized with member pre-stress to ensure equilibrium. Over prolonged usage, the cables may lose their pre-stress while the bars may buckle, get damaged, or corrode, affecting the structural stiffness leading to change in the measured dynamic properties. Upon loading, a tensegrity structure may change its form through altering its member pre-stress affecting its global stiffness, even in the absence of damage. This can potentially mask the effect of damage leading to a false impression of tensegrity health. This poses the major challenge in tensegrity health monitoring especially when the load is stochastic and unknown.Present study proposes an output-only time-domain method that makes use of tensegrity vibrational responses within a Bayesian filtering-based approach to monitor the tensegrity health in the presence of uncertainties due to ambient force, model inaccuracy, and measurement noise. For this, an interacting strategy combining Particle Filter (PF) and Ensemble Kalman Filter (EnKF) has been adopted (Interacting Particle-Ensemble Kalman Filter, IP-EnKF) in which the EnKF estimates the response states as ensembles while running within a PF envelop that estimates a set of location-based health parameters as particles. Furthermore, for a cheaper damage detection procedure, strain responses are used as measurements. The efficiency of the proposed methodology in terms of accuracy, computational cost, and robustness against noise contamination has been demonstrated using numerical experiments performed on two tensegrity modules: a simplex tensegrity and an extended-octahedron tensegrity.

Towards Recognition of Scale Effects in a Solid Model of Lattices with Tensegrity-Inspired Microstructure

This paper is dedicated to the extended solid (continuum) model of tensegrity structures or lattices. Tensegrity is defined as a pin-joined truss structure with an infinitesimal mechanism stabilized by a set of self-equilibrated normal forces. The proposed model is inspired by the continuum model that matches the first gradient theory of elasticity. The extension leads to the second- or higher-order gradient formulation. General description is supplemented with examples in 2D and 3D spaces. A detailed form of material coefficients related to the first and second deformation gradients is presented. Substitute mechanical properties of the lattice are dependent on the cable-to-strut stiffness ratio and self-stress. Scale effect as well as coupling of the first and second gradient terms are identified. The extended solid model can be used for the evaluation of unusual mechanical properties of tensegrity lattices.

Parametric Analysis of Tensegrity Plate-Like Structures: Part 2—Quantitative Analysis

The study includes a parametric analysis of a group of tensegrity plate-like structures built with modified Quartex modules. The quantitative assessment, including the calculation of the structure’s response to constant loads, was carried out. A static parametric analysis was performed, with particular emphasis on the influence of the initial prestress level on the displacements, the effort, and the stiffness of the structure. A geometrical non-linear model was used in the analysis. A reliable assessment required introducing a parameter for determining the influence of the initial prestress level on the overall stiffness of the structure at a given load. The stiffness of the structure was found to depend not only on the geometry and material properties, but also on the initial prestress level and external load. The results show that the effect of the initial prestress on the overall stiffness of the structure is greater with less load and that the effect of load is most significant with low pre-stressing forces. The analysis demonstrates that the control of static parameters is possible only when infinitesimal mechanisms occur in the structure.

Parametric Analysis of Tensegrity Plate-Like Structures: Part 1—Qualitative Analysis

The study includes parametric analysis of special spatial rod grids called tensegrity plate-like structures. Tensegrity structures consist of only compression and tension components arranged in a system, whose unique mechanical and mathematical properties distinguish them from conventional cable–strut frameworks. Complete analysis of tensegrity structures is a two-stage process. The first stage includes the identification of self-stress states and infinitesimal mechanisms (qualitative analysis). The second stage focuses on the behaviour of tensegrities under external loads (quantitative analysis). In the paper, a qualitative analysis of tensegrity plate-like structures built with modified Quartex modules was conducted. Starting from a single-module structure, more complex cases were sequentially analysed. The different ways of plate support were considered. To carry out a qualitative assessment, a spectral analysis of the truss matrices and singular value decomposition of the compatibility matrix were used. The characteristic features of tensegrity structures were identified. On this basis, the plates were classified into one of the four groups defined in the paper, i.e., ideal tensegrity, “pure” tensegrity and structures with tensegrity features of class 1 or class 2. This classification is important due to different behaviours of the structure under external actions. The qualitative analysis carried out in the paper is the basis for a quantitative analysis.

Mechanism creation in tensegrity structures by cellular morphogenesis

Tensegrity structures are self-equilibrated statically and kinematically indeterminate structures. Cellular morphogenesis represents a generative process for the composition of complex tensegrity structures based on elementary cells. This article discusses the mechanism creation by the integration of mobility conditions in the generation of tensegrity structures using cellular morphogenesis. The creation of finite and infinitesimal mechanisms in tensegrity structures is described by the adhesion of cells sharing less than d nodes (d being the dimension of the workspace) and by the fusion of cells with the removal of two edges. Parametric descriptions of the infinitesimal displacements in the case of trivial and finite mechanisms are derived by the analysis of rigid assemblies corresponding to the rigid parts of the structure. In addition, an interpretation of the degeneracy of tensegrity structures in the case of self-stressable mechanisms is also presented. Analytical solutions of the degeneracy space for configurations resulting from adhesion and the fusion of two cells with the removal of two edges are described using symbolic calculations on the rigidity matrices of tensegrity structures. Although the study focuses on selected configurations and arrangements, the generalization of the ideas and findings included in this paper can lead to the generation of tensegrity structures with predefined static as well as kinematic properties, thus further enabling the application of tensegrity systems in science and engineering.

Extended integrated force method for the analysis of prestress-stable statically and kinematically indeterminate structures

This paper presents a new force method to predict the structural response of statically and kinematically indeterminate systems that can be stabilized through prestress, i.e. prestress-stable structures. This new force method, here named Extended Integrated Force Method (IFME), extends the existing Integrated Force Method (IFM) which is only applicable to the analysis of kinematically determinate systems. The product force concept is adopted and incorporated into the IFME to model the effect of infinitesimal mechanisms. This makes the IFME capable of dealing with cases in which the external loads contain components that cannot be taken by the system in its initial configuration. As the original IFM, the IFME bypasses the well-known concept of redundant forces and basis determinant structure of the Standard Force Method (SFM) by taking the internal forces as the independent variables which are obtained simultaneously. A proof is provided to show that, when the product force is not included in the formulation, the IFME reduces to the IFM for kinematically determinate systems and it also reduces to another force method based on singular value decomposition of the equilibrium matrix which is here named SVD-FM. Compared to the better known Displacement Method (DM), the IFME is a suitable alternative and it offers a deeper insight into the structure response which is decoupled into an extensional and an inextensional part for prestress-stable kinematically indeterminate systems. Numerical examples are carried out to test accuracy and effectiveness of the IFME on kinematically indeterminate structures with multiple self-stress states and mechanism modes. Application of the IFME to active structural control of kinematically indeterminate systems is also discussed through a numerical example.

Feasible Prestress Modes for Cable-Strut Structures with Multiple Self-Stress States Using Particle Swarm Optimization

A prestressed cable-strut structure is usually regarded as a mechanism before being prestressed. Under the action of initial prestresses, the internal infinitesimal mechanisms can be rigidified, resulting in achieving the desired structural stiffness. Therefore, feasible prestress design is a key to develop and analyze novel prestressed cable-strut structures. In this study, an effective optimization method is presented to determine the optimal feasible prestress modes of a cable-strut structure with predefined geometry and multiple self-stress states. Two optimization models based on the self-stress states and the integral self-stress states are presented to compute the optimal feasible prestress modes. Thereafter, the multiobjective optimization problem is converted into a single objective optimization problem by the weight coefficient method, and the particle swarm optimization algorithm is applied to find feasible solutions. Illustrative examples verify the feasibility of the presented optimization algorithms to calculate feasible prestress modes. In comparison with the conventional optimization methods, the proposed method shows satisfactory accuracy and efficiency.

Reconfiguration method of tensegrity units using infinitesimal mechanisms

PurposeThis paper aims to present a reconfiguration strategy for actuated tensegrity structures. The main idea is to use the infinitesimal mechanisms of the structure to generate a path along which the tensegrity can change its shape while maintaining the equilibrium.Design/methodology/approachCombining the force density method with a marching procedure, the solution to the equilibrium problem is given by a set of differential equations. Beginning from an initial stable position, the algorithm calculates a small displacement until a new stable configuration is reached, and recurrently repeats the process during a given interval of time.FindingsBy means of three numerical simulations and their respective experimental example, the efficacy of this algorithm for reconfiguring the well-known three-bar tensegrity prism along different directions is shown. The proposed method shows efficiency only for small changes of string length. Further work should consider the application of this method to more complex tensegrity structures.Originality/valueThe advantage of this reconfiguration method is its simplicity for finding new stable positions for tensegrity structures, and the fact that it doesn’t need the information of the material of the structure for the computations.

Reconfiguration of multi-stage tensegrity structures using infinitesimal mechanisms

Abstract The use of tensegrity structures in soft robotics has seen an increased interest in recent years thanks to their mechanical properties, but the control of these systems remains an open problem. This paper presents a reconfiguration strategy for actuated multi-stage tensegrity structures. The algorithm works on the principle of using the infinitesimal mechanisms of the structure to generate a path of positions along which a multi-stage tensegrity structure can change its shape while maintaining the self-equilibrium. Combining the force density method with a marching procedure, the solution to the equilibrium problem is given by a set of differential equations that define the kinematic constraints of the structure. Beginning from an initial stable position, the algorithm calculates a small displacement until a new stable configuration is reached, and recurrently repeats the process during a given interval of time. By means of three numerical examples, we show the efficacy of our algorithm for reconfiguring a two-stage tensegrity mast along different directions.