Pantographic Structures Research Articles

A rate-independent internal friction to describe the hysteretic behavior of pantographic structures under cyclic loads

The cyclic behavior and dissipative properties of pantographic fabrics are analytically and experimentally investigated in this paper. To these purposes, a two-dimensional second-gradient continuum model, already developed in the literature, is taken as representative of the elastic behavior of the homogenized system. However, to take into account the dissipative phenomena, the model is enriched with a suitable Dahl-like phenomenological dissipation law, accounting for friction effects at the interconnecting pivots, that it is shown to be effective in describing the macroscopic hysteretic behavior under cyclic and quasi-static imposed displacement time-histories. Numerical results are compared against experimental ones, thus showing the effectiveness of the continuum model in reproducing the real behavior of pantographic fabrics.

A 3D extension of pantographic geometries to obtain metamaterial with semi-auxetic properties

In this work we present a three-dimensional extension of pantographic structures and describe its properties after homogenization of the unit cell. Here we rely on a description involving only the first gradient of displacement, as the semi-auxetic property is effectively described by first-order stiffness terms. For a homogenization technique, discrete asymptotic expansion is used. The material shows two positive () and one negative Poisson’s ratios (). If, on the other hand, we assume inextensible Bernoulli beams and perfect pivots, we find a vanishing stiffness matrix, suggesting a purely higher gradient material.

Novel method of estimation of inertial and dissipative parameters of a railway pantograph model

ABSTRACT An increase in electric railway vehicles service velocity requires that correct interaction between the pantograph and the catenary is ensured. This implies the need for developing mathematical models of pantographs and catenaries and determining their parameters. The article presents a method to determine parameters of mechanical joints of a railway pantograph based on analysis of pantograph subassemblies in swinging motion. The experimental tests consisted in disassembling the pantograph and creating partial subassemblies which were then analysed with respect to their damped linear or angular oscillations. The simulation analysis required developing, in CAD, 3D models of individual pantograph parts and their subassemblies. Defined were joints between particular elements, which represent of real pantograph structure. The inertia parameters of the model were determined, based on structural characteristics and physical properties of materials, used for manufacturing individual pantograph elements, and then verified experimentally. The dissipative parameters of the mechanical joints were calculated iteratively. The model of complete pantograph assembly was also subject to verification and in this case, good convergence between simulation and experiment results was achieved. The obtained parameter values can be used in simulation models of other railway pantographs having a similar structure.

Investigating the mechanical response of microscale pantographic structures fabricated by multiphoton lithography

The design of micromechanical devices that can facilitate large but recoverable deformations requires a mechanical behavior that embosoms hyperelasticity. While multiphoton lithography is the epitome of microscale fabrication, the employed materials demonstrate a linear elastic response accompanied by limited ductility. In this study, we investigate how this hindrance can be circumvented through the design of microscale pantographic structures. Pantographs possess riveting hyperelastic response inherited by their structural design, providing exorbitant reversible deformations. To prove the utility of pantographs in microscale design, finite element analysis simulations are performed to unravel the behavior of the structure as a function of its geometrical parameters. In addition, to evaluate the microscale modeling, specimens are fabricated with multiphoton lithography in a push to pull up configuration to accomplish in situ SEM microindentation tensile testing due to compression. Our findings are adduced to expound how the pantographic structures can embrace hyperelastic response even at the microscale, elucidating their feasibility for structural members in micromechanical devices that require reversible large deformations.

Research on Structural reliability and reliability sensitivity of EMU pantograph

Due to the uncertainty of the contact state between the pantograph and the power grid when the EMU is running at high speed, as well as the influence of road conditions and environment, the external load applied to the pantograph and the material properties of the pantograph have certain randomness, thus affecting the current collection quality of the pantograph and the service life of the carbon slide plate. Taking a certain type of EMU single arm pantograph as the research object, a three-dimensional model is established, ANSYS Workbench software is used to carry out the finite element analysis to determine the dangerous point and its stress response; the simulation samples of material properties and stress of dangerous points are obtained by using sampling method and test design method based on the principle of normal distribution; the reliability index and reliability of pantograph structure strength are analyzed, based on the random perturbation method and the first-order second- moment method, and using the simulation sample data extracted by Latin hypercube sampling, and the reliability accuracy is verified by Monte Carlo method. Based on the definition of reliability sensitivity, the influence degree of each input variable on the reliability change of pantograph structure is determined as follows: external load > Poisson’s ratio > material density > elastic modulus, which provides reference for improving the reliability, service life and structural optimization design of pantograph

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Based on the progress and advances of additive manufacturing technologies, design and production of complex structures became cheaper and therefore rather possible in the recent past. A promising example of such complex structure is a so-called pantographic structure, which can be described as a metamaterial consisting of repeated substructure. In this substructure, two planes, which consist of two arrays of beams being orthogonally aligned to each other, are interconnected by cylinders/pivots. Different inner geometries were taken into account and additively manufactured by means of fused deposition modeling technique using polyethylene terephthalate glycol (PETG) as filament material. To further understand the effect of different manufacturing parameters on the mechanical deformation behavior, three types of specimens have been investigated by means of displacement-controlled extension tests. Different slicing approaches were implemented to eliminate process-related problems. Small and large deformations are investigated separately. Furthermore, 2D digital image correlation was used to calculate strains on the outer surface of the metamaterial. Two finite-element simulations based on linear elastic isotropic model and linear elastic transverse isotropic model have been carried out for small deformations. Standardized extension tests have been performed on 3D-printed PETG according to ISO 527-2. Results obtained from finite-element method have been validated by experimental results of small deformations. These results are in good agreement with linear elastic transverse isotropic model (up to about [Formula: see text] of axial elongation), though the response of large deformations indicates a nonlinear inelastic material behavior. Nevertheless, all samples are able to withstand outer loading conditions after the first rupture, resulting in resilience against ultimate failure.

Stiffness optimization in nonlinear pantographic structures

Mechanical metamaterials are microstructured mechanical systems showing an overall macroscopic behaviour that depends mainly on their microgeometry and microconstitutive properties. Moreover, their exotic properties are very often extremely sensitive to small variations of mechanical and geometrical properties in their microstructure. Clearly, the methods of structural optimization, once combined with the techniques used to describe multiscale systems, are expected to determine a dramatic improvement in the quality of newly designed metamaterials. In this paper, we consider, only as a demonstrative example, planar pantographic structures which have proved to be extremely tough in extension, To describe pantographic structure behaviour in an efficient way, it has been proposed to use Piola–Hencky-type Lagrangian models, in which the understanding of the mechanics of involved microdeformation processes allows for the formulation of efficient numerical codes. In this paper, we prove that it is possible, via a suitable choice of the macroscopic shear stiffness, to increase the maximal elongation of pantographic structures, in the standard bias test, before the occurrence of rupture phenomena. The basic tool employed to this aim is a constrained optimization algorithm, which uses the numerical tool, previously developed for determining equilibrium shapes, as a subroutine. Actually, one looks for the shear stiffness distribution, which, given the imposed elongation of the pantographic structure and the force applied to it by the used hard device, minimizes the total elongation energy. The so-optimized shear stiffness distribution does prove able to extend the range of imposed elongations that the specimen can experience while remaining undamaged.

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

The central theme of this study is to investigate a remarkable capability of a second-gradient continuum model developed for pantographic structures. The model is applied to a particular type of this metamaterial, namely the wide-knit pantograph. As this type of structure has low fiber density, the applicability of such a continuum model may be questionable. To address this uncertainty, numerical simulations are conducted to analyze the behavior of a wide-knit pantographic structure, and the predicted results are compared with those measured experimentally under bias extension testing. The results presented in this study show that the numerical predictions and experimental measurements are in good agreement; therefore, in some useful circumstances, this model is applicable for the analysis of wide-knit pantographic structures.

Phenomenological aspects of quasi-perfect pivots in metallic pantographic structures

This article discusses the results of the first attempts to print metal pantographic structures with perfect pivots, i.e. with rotational hinges connecting the two families of fibers composing the network without any stiffness. On the one hand, it is observed that perfect pivots do not behave as expected theoretically. On the other hand, the force measured during a bias extension test is a few orders of magnitude lower than that measured for pantographic structures with standard pivots (where a certain stiffness is associated to the interconnecting hinges). This leads to considering the pivots as quasi-perfect (non-zero stiffness, but neither the theoretical one which can be computed by means of the geometrical features of the pivot). Numerical simulations complete the analysis by showing how, by modulating the torsional stiffness of the pivots, it is possible to reproduce the force-displacement plot both in the case of standard pivots and with quasi-perfect ones.

Shape morphing and adjustment of pantographic morphing aerofoil section structure

This study concerns with morphing structures, e.g. as applied in the aerospace industry. A morphing aerofoil structure capable of variable geometry was developed, which was shown to be able to cater for the different aerodynamic requirements at different stages of flight. In this work, the useful and relatively simple method has been applied, which provides a direct method for calculating required morphing shape displacements via finding the most effective bar through calculating bar sensitivity to displacement and calculating set of length actuations for bar assembly to control/adjust shape imperfection of prestressable structural assemblies including complex elements (macro-elements, e.g., the pantographic element), involving Matrix Condensation. The technique has been verified by experiments on the physical model of an aerofoil shaped morphing pantographic structure. Overall, experimental results agree well with theoretical prediction. Furthermore, the technique of multi-iteration adjustment was presented that effective in eliminating errors that occur in the practical adjustment process itself. It has been demonstrated by the experiments on the physical model of pantographic morphing structure. Finally, the study discusses identification of the most effective bars with the objective of minimal number of actuators or minimum actuation.

Pantographic beam: a complete second gradient 1D-continuum in plane

There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process.

Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms

We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure that we call asymmetrical pantographic structure.

Advances in pantographic structures: design, manufacturing, models, experiments and image analyses

In the last decade, the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discrete or continuous) have been introduced. In a previous publication, a large part of the already existing literature about pantographic metamaterials has been presented. In this paper, we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.

Parametric Experimentation on Pantographic Unit Cells Reveals Local Extremum Configuration

Pantographic metamaterials are known for their ability to have large deformation while remaining in the elastic regime. We have performed a set of experiments on 3D printed pantographic unit cells to parametrically investigate their response when undergoing tensile, compression, and shear loading with the aim of i) studying the role of each parameter in the resultant mechanical behavior of the sample, and ii) providing a benchmark for the mathematical models developed to describe pantographic structures. Results show the existence of local extrema in the space of the geometrical parameters, suggesting the use of optimization techniques to find optimal geometrical parameters resulting in desired functionalities. We have also performed tensile relaxation tests on the samples, with the results indicating the complexity of the dynamic behavior and the existence of multiple relaxation characteristic times. Such results can be used to for calibrating mathematical models describing pantographic structures under dynamic loadings.

Homogenization of nonlinear inextensible pantographic structures by Γ-convergence

Homogenization of nonlinear inextensible pantographic structures by Γ-convergence

Computation and experimental comparison of the deformation behavior of pantographic structures with different micro-geometry under shear and torsion

1. Abali B.E., M¨uller W.H., dell’Isola F., 2017, Theory and computation of higher gradient elasticity theories based on action principles, Archive of Applied Mechanics, 87, 9, 1495-1510. Google Scholar

Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: application to out-of-plane buckling

Mechanical properties are investigated for a class of microstructured materials with promising applications. Specifically, we consider a composite material with orthogonal, mutually interconnected fibers building a pantographic substructure. In order to predict the behavior of such a system in three-dimensional continuum, a reduced-order model is introduced by means of a bi-dimensional elastic surface accurately describing large deformations. The properties of this reduced-order model are characterized by an elastic energy density that involves second space derivatives of the displacement for capturing the resistance of twisted and bent fibers in plane as well as out of plane. For determining the coefficients in the elastic energy of the reduced-order model, we utilize a numerical inverse analysis and make use of ad hoc computational experiments performed by a direct numerical simulation on the microscale with detailed modeling of the pantographic substructure. This reduced-order model represents a homogenized material on macro-scale with its substructure on microscale. The homogenized model is capable of describing materials response at a significantly less computational cost than the direct numerical simulations.

Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions

3D printing technology has opened application perspectives which were difficult to imagine only few years ago. In this paper, we show how it is possible to design and print some microstructures in which relative displacements are allowed at micro-level. Some structural elements have been built in the aforementioned structures which can be confidently modeled as perfect pivots or as soft elastic connections. The obtained specimens can be regarded as constituted by very exotic materials, as forecast theoretically. We numerically study the behavior of pantographic structures including soft or nearly perfect pivots in large deformations, and we experimentally observe an enlarged elastic range and peculiar buckling mechanisms. The presented results are extremely promising: We consider now as proven that higher gradient metamaterials can be realized by using microstructures having micro-characteristic lengths of the order on tenth of millimeters.

A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams

We present a finite element discrete model for pantographic lattices, based on a continuous Euler–Bernoulli beam for modeling the fibers composing the pantographic sheet. This model takes into account large displacements, rotations and deformations; the Euler–Bernoulli beam is described by using nonlinear interpolation functions, a Green–Lagrange strain for elongation and a curvature depending on elongation. On the basis of the introduced discrete model of a pantographic lattice, we perform some numerical simulations. We then compare the obtained results to an experimental BIAS extension test on a pantograph printed with polyamide PA2200. The pantographic structures involved in the numerical as well as in the experimental investigations are not proper fabrics: They are composed by just a few fibers for theoretically allowing the use of the Euler–Bernoulli beam theory in the description of the fibers. We compare the experiments to numerical simulations in which we allow the fibers to elastically slide one with respect to the other in correspondence of the interconnecting pivot. We present as result a very good agreement between the numerical simulation, based on the introduced model, and the experimental measures.

Tactile-STAR: A Novel Tactile STimulator And Recorder System for Evaluating and Improving Tactile Perception.

Many neurological diseases impair the motor and somatosensory systems. While several different technologies are used in clinical practice to assess and improve motor functions, somatosensation is evaluated subjectively with qualitative clinical scales. Treatment of somatosensory deficits has received limited attention. To bridge the gap between the assessment and training of motor vs. somatosensory abilities, we designed, developed, and tested a novel, low-cost, two-component (bimanual) mechatronic system targeting tactile somatosensation: the Tactile-STAR—a tactile stimulator and recorder. The stimulator is an actuated pantograph structure driven by two servomotors, with an end-effector covered by a rubber material that can apply two different types of skin stimulation: brush and stretch. The stimulator has a modular design, and can be used to test the tactile perception in different parts of the body such as the hand, arm, leg, big toe, etc. The recorder is a passive pantograph that can measure hand motion using two potentiometers. The recorder can serve multiple purposes: participants can move its handle to match the direction and amplitude of the tactile stimulator, or they can use it as a master manipulator to control the tactile stimulator as a slave. Our ultimate goal is to assess and affect tactile acuity and somatosensory deficits. To demonstrate the feasibility of our novel system, we tested the Tactile-STAR with 16 healthy individuals and with three stroke survivors using the skin-brush stimulation. We verified that the system enables the mapping of tactile perception on the hand in both populations. We also tested the extent to which 30 min of training in healthy individuals led to an improvement of tactile perception. The results provide a first demonstration of the ability of this new system to characterize tactile perception in healthy individuals, as well as a quantification of the magnitude and pattern of tactile impairment in a small cohort of stroke survivors. The finding that short-term training with Tactile-STAR can improve the acuity of tactile perception in healthy individuals suggests that Tactile-STAR may have utility as a therapeutic intervention for somatosensory deficits.