Pantographic Structures Research Articles

Rayleigh surface waves of extremal elastic materials

Extremal elastic materials here refer to a specific class of elastic materials whose elastic matrices exhibit one or more zero eigenvalues, resulting in soft deformation modes that, in principle, cost no energy. They can be approximated through artificially designed solid microstructures. Extremal elastic materials have exotic bulk wave properties unavailable with conventional solids due to the soft modes, offering unprecedented opportunities for manipulating bulk waves, e.g., acting as phonon polarizers for elastic waves or invisibility cloaks for underwater acoustic waves. Despite their potential, Rayleigh surface waves, crucially linked to bulk wave behaviors of such extremal elastic materials, have largely remained unexplored so far. In this paper, we theoretically investigate the propagation of Rayleigh waves in extremal elastic materials based on continuum theory and verify our findings with designed microstructure metamaterials based on pantographic structures. Dispersion relations and polarizations of Rayleigh waves in extremal elastic materials are derived, and the impact of higher order gradient effects is also investigated by using strain gradient theory. This study provides a continuum model for exploring surface waves in extremal elastic materials and may stimulate applications of extremal elastic materials for controlling surface waves.

An experimental and theoretical investigation on the hyper-viscoelasticity of polyamide 12 produced by selective laser sintering.

Polyamide 12 (PA12) is vastly utilized in many additive manufacturing methods, such as Selective Laser Sintering (SLS), and a better understanding of its mechanical behaviors promotes available knowledge on the behaviors of 3D-printed parts made from this polymer. In this paper, SLS-produced standard tensile specimens are studied under monotonic and cyclic tension tests, as well as stress relaxation experiments, and the obtained force-displacement responses are shown to be consistent with a hyper-viscoelastic material model. This finding is also observed in typical pantographic structures produced by the same manufacturing parameters. To propose a constitutive model for predicting these behaviors, the convolution integral of a strain-dependent function and a time-dependent function is developed where the material parameters are determined with the use of both short-term and long-term responses of the specimens. Numerical results of the presented model for standard test specimens are shown to be in good agreements with the experimental ones under various loading conditions. To prove the capabilities of the proposed model in studying any SLS-produced part, finite element implementation of the constitutive equations is shown to provide numerical results in agreement with the empirical findings for tensile loading of the 3D-printed pantographic structure.

Leveraging systems’ non-linearity to tackle the scarcity of data in the design of intelligent fault diagnosis systems

Deep transfer learning (DTL) allows for the efficient building of intelligent fault diagnosis systems (IFDS). On the other hand, DTL methods still heavily rely on large amounts of labelled data. Obtaining such an amount of data can be challenging when dealing with machines or structures faults. This document proposes a novel approach to the design of vibration-based IFDS using DTL in condition of strong data scarcity. A periodic multi-excitation level procedure leveraging intrinsic non-linearities of real-world systems is used to produce images that can be conveniently analysed by pre-trained Convolutional Neural Networks to diagnose faults. A new data visualization method and its augmentation technique are proposed in this paper to tackle the typical lack of data encountered during the design of IFDS. Experimental validation on a railway pantograph structure provides effective support for the proposed method.

МОДЕЛЮВАННЯ ТА АНАЛІЗ РОЗГОРТАННЯ КОСМЕТИЧНОЇ СІТЧАСТОЇ АНТЕНИ

The aim of work is to create a dynamic model of a space antenna with the pantograph structure and study the processes of its deployment using open-source software. The methods of theoretical mechanics, multi-body dynamics, computational mechanics, and computer modeling were used for this research. The object for modeling is a novel mesh antenna, which is designed for mini-satellites. The most significant difference between this antenna and others is the design of the support ring in the form of a pantograph. Using the built model, antenna deployment is simulated for different cases. Values of deployment time and cable tension during the antenna deployment are analyzed.

Controlling nodal displacement of pantographic structures using matrix condensation and interior-point optimization: A numerical and experimental study

This study presents an innovative approach for the precise control of nodal displacements in pantographic structures. The method is founded on the Matrix Condensation of Force Method, seamlessly integrated with an Interior Point Optimization algorithm. This combination offers a unique advantage by allowing users to manipulate displaced nodes within a defined coordination domain. Furthermore, this approach introduces the Interior Point Optimization algorithm as an indispensable tool to eliminate inactive turnbuckles and minimize overall actuation requirements. Traditional control methods typically demand a substantial number of turnbuckles and extensive actuation efforts to attain the desired nodal coordinates. The interconnected nature of node movements, wherein changes in one node affect others, adds complexity to determining the impact of bar length alterations on each node. To address this challenge, precisely control power of the Interior Point Optimization algorithm systematically explores numerous scenarios to identify solutions that minimize both actuation and turnbuckle usage. The current technique's effectiveness is validated through rigorous comparisons with established methods, experimental modeling, and rigorous testing using SAP 2000 software. Notably, the current approach yields remarkable results, requiring a staggering 60% less actuation and reducing the reliance on turnbuckles by up to 40% compared to previous methods. This innovation promises to significantly enhance the efficiency and cost-effectiveness of controlling pantographic structures, marking a substantial advancement in this field.

სხვადასხვა სიმრუდის ზედაპირზე აგებული ტრანსფორმირებადი, პანტოგრაფული სტრუქტურები

The possibilities to design expandable-foldable structures on various curvature surfaces are considered in the given paper. The presented geometrical layouts are unfolding structures with closed and open contours. The transformable designs are made by an interconnected set of elementary pantographic structure. The elementary transformable structure consists of two levers connected by a transverse cylindrical joint. The considered designs of transformable structures can be effectively used in space structures, in particular deployable space reflectors, also in ground-based transformable structures.

A 3D pantographic metamaterial behaving as a mechanical shield: Experimental and numerical evidence

Multilayer pantographic metamaterials, in short, pantographic blocks, have shown peculiar mechanical behavior, especially when their constitutive hinges are revolving (i.e., perfect) joints. The pantographic block, which is the subject of the present paper, has been printed using a Powder Bed Fusion technology and its hinges may be modeled as perfect ones. In the reported in situ 3-point flexural test, the predictions obtained by second gradient models for its mechanical response are shown to be experimentally consistent thanks to measurements via Digital Volume Correlation. The deformation applied by the upper central support is almost entirely shielded by the pantographic block, namely, the specimen barely crosses through the reference bottom plane defined by the lower lateral supports, even when subjected to very large deformations. The mathematical model employed herein captures this observation in terms of a nonlinear ‘arching’ effect activated in the beams of the pantographic structure, provided elastic locking is introduced to prevent pantographic zero-energy modes.

A compliant leg design combining pantograph structure with leaf springs

AbstractWe proposed a compliant leg configuration that enhances the conventional pantograph design with leaf springs. The following facts characterize the proposed configuration: (1) Due to the use of the pantograph structure, the mass is centralized around the hip joint, reducing the lower leg inertia; (2) Leaf springs are chosen as elastic parts to increase energy efficiency and estimate foot-end contact forces. Compared with coil springs, leaf springs require no guide rails to deploy, and their stiffness can be easily adjusted through shape cutting. Analytical models are introduced to analyze the leg’s stiffness and estimate the contact forces only with the deflections of leaf springs. A one-leg robot based on the proposed design is built, and various experiments are conducted. Experiments regarding the stiffness calibration and the contact forces estimation showed an acceptable agreement with the analytical model. Experiments of dropping demonstrate the feasibility of the leg to perform spring-like behaviors. Experiments of periodic hopping demonstrate the feasibility of using spring deflections to detect touch-down events. For energy efficiency, it is also observed that the elastic leg has a 20% increment concerning the jumping height in the flight phase, compared with the one where leaf springs are replaced with rigid materials.

A second-gradient elasticity model and isogeometric analysis for the pantographic ortho-block

We present a continuum model for a three-dimensional pantographic structure that we call pantographic ortho-block. The model is based on the embedding of four families of elastic fibers in a three-dimensional domain. Similar to pantographic sheets, we find an energy density depending on the first and second-gradient of displacement resulting in a three-dimensional incomplete second-gradient material. For certain kinematic assumptions concerning perfect joints, we find two distinct zero-energy modes. Finally, we perform simulations based on isogeometric analysis to showcase standard and non-standard boundary conditions.

Investigation on the fluid-solid coupling effect in high-speed train’s windshield structure

The high-speed train’s windshield structure is vital to the boundary layer development and drag reduction design. Up to date, windshield structures are assumed as rigid body in most train aerodynamic numerical simulations. This research investigates the fluid-solid coupling effect of high-speed train’s windshield structure based on an 8-car formation model. The influence of speed level and windshield installation location on the flow field parameter and windshield deformation profile are systematically analysed. The results indicate that, with the development the boundary layer, the time-averaged pressure coefficient, lateral force coefficient and windshield’s displacement all gradually decrease from 1st windshield to 4th windshield along the stream direction. In the connecting region between the windshield and vestibule, a pair of negative pressure zone are formed and gradually intensify along the downstream direction, such that the time-averaged deformation, as well as the vibration amplitude of windshields experience a slight increase. According to the power spectral density analysis, the vibration frequency of all windshields’ displacement varies between 10.25 Hz and 12.5 Hz. Meanwhile, the pantograph structure will lead to the generation of vortex group, these vortices will lead to additional small low frequency peaks for windshields installed on the downstream direction of pantograph.

Fully mechanical motion-scaling instrument for microsurgery assistance: design improvement for enhancing the dynamic performance

Abstract Robot assistance can improve the outcome of microsurgery by scaling down the surgeon’s hand motions. However, the high cost of surgical robots has prevented their use in small hospitals or medical facilities in several developing countries. As a novel alternative, a fully mechanical motion-scaling instrument, which can be operated without computers and motors, was proposed based on the pantograph mechanism. However, it had several problems owing to the cumbersome and heavy structures during the prototype test. This study aims solving the problems found in the first design and proving the advantages of the improvement, based on the design and performance criteria. The pantograph structure was simplified, and the gravity compensation method was modified to reduce inertia by using a constant force spring instead of a counter-mass. The improvement was computationally predicted using a mathematical model, and the results were verified through trajectory measurements in a micropositioning task. Finally, the evaluation of dynamic performance is quantitatively presented through iterative positioning tasks.

Identification and validation of constitutive parameters of a Hencky-type discrete model via experiments on millimetric pantographic unit cells

Pantographic metamaterial design benefits from model identification procedures starting from what can be considered as the elementary unit cell of larger pantographic structures. Results from a tensile experiment and digital image correlation are utilized to identify the constitutive parameters of a discrete Hencky-type model for a millimetric pantographic cell. In the performed calibration, two different cost functions are formulated. First, the cost function is based upon measured resultant forces on the specimen boundaries. Then, the second cost function is based upon the measured pivot displacements in addition to reaction forces. The second cost functions thus exploits the pivot kinematics, which is a key feature of the deformation of pantographic structures. The identified model is further validated by predicting the reaction forces and pivot displacements of the same specimen subjected to compression. It is shown that the identification with the cost function incorporating pivot displacements is superior. It is also noted that the calibrated parameters deviate considerably from their initial guess derived from the linear Saint Venant problem, thereby indicating microscale nonlinear affects in otherwise linear reaction force-prescribed displacement responses at the macroscale.

Modelling of space antenna deployment using open source software

The goal of this article is to develop a dynamic model of a space antenna with the pantograph structure and to study the processes of its deployment using open-source software. Methods of theoretical mechanics, multibody dynamics, computational mechanics, and computer modeling were used in the research. A mesh antenna of the novel design, which is recommended for mini-satellites, is considered as the object for modeling. The most significant difference between this antenna and others is the design of the support ring in the form of a pantograph. To develop a model of the space antenna dynamics and implement it using open-source software, some simplifications were made due to the complexity of the structure. The antenna model is represented as a system of rigid and flexible bodies connected by hinges. Carbon fiber rods are modeled with the help of a flexible finite element using the method of absolute nodal coordinates, which allows one to model large deformations of the structure. Aluminum hinge assemblies are modeled as several rotation joints connected by conventional rigid elements. The main modeled properties of these hinge assemblies are the stiffness, location, and direction of the axes of rotation of the hinges. The tension forces created by the stretched mesh are modeled using springs. The cable drive of the antenna deployment mechanism is modeled as a load acting on the corresponding elements in defined local positions. An algorithm for building a model of the space antenna to simulate the reflector deployment process in the HotInt open-source software is presented. Using the built model, antenna deployment simulations are carried out for different cases, which differ in the forces used for the deployment. Values of deployment time, variations of angles between the V-folded bars, and tensions in the diagonal rods of the antenna sections during the antenna deployment are obtained. The approach proposed in the article can be implemented using free software, ensures flexibility of modeling, and reduces the model development time.

LQR Active Control of Fractional-Order Pantograph-Catenary System Based on Feedback Linearization

Fractional-order calculus has exclusive advantages in modeling the viscoelastic components with obvious fractional-order characteristics such as air springs and metal rubbers in the pantograph structure. In this paper, the air spring is tested, and fractional-order calculus is applied to the modeling of pantograph-catenary system of the high-speed train. The parameter identification method of fractional-order derivative is analytically derived. The traditional lumped mass model is improved and a coupling two-degree-of-freedom model of the fractional-order pantograph-catenary system is established. The fractional-order derivative term in the pantograph-catenary model is approximately calculated by the Oustaloup filter algorithm. Taking the time-varying nature into consideration, the catenary is treated as an extended variable to obtain an augmented model. On this basis, the system is linearized based on differential geometry theory, and an LQR controller is designed to control the pantograph-catenary system. The feedback linearized LQR control and PID control are used to control the same type of traditional pantograph, and the results are compared. Meanwhile, the control effects of feedback linearized LQR control under different pantograph parameters and at different train speeds are analyzed. The results show that the feedback linearized LQR control can present a much better control performance than PID control, and the pantograph-catenary contact force and pantograph head vibration amplitude are both reduced obviously. Even at different train speeds or under different pantograph parameters, it can also effectively reduce these control indexes and provide more robust control performance. These results help to put forward new control ideas and theoretical basis for the vibration control of the pantograph-catenary or similar dynamical system.

Experimental analysis on metamaterials boundary layers by means of a pantographic structure under large deformations

In this work, the dynamical behavior of a pantographic structure undergoing large deformations is analyzed. A harmonic force is applied on the structure, while its motion in time is recorded by means of a high-speed camera system. The resulting displacement field is obtained via the digital image correlation (DIC) method for the whole domain. A parametric study with respect to the amplitude of the imposed force exhibits the appearance of non-linear effects in the system response as well as the presence of a so-called boundary layer region. Such geometric nonlinearities are of paramount interest for studying the accuracy of a potential computational model. Boundary layers are known to be effected by the microstructure, but their roles in the system response are yet to be comprehended. The experimental results obtained in this work may be used as a benchmark case in modeling metamaterials.

A general theory for anisotropic Kirchhoff–Love shells with in-plane bending of embedded fibers

This work presents a generalized Kirchhoff–Love shell theory that can explicitly capture fiber-induced anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of classical Kirchhoff–Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing second-gradient Kirchhoff–Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor—which is symmetric and of second order—is proposed. The effective stress tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric in general. Constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations. The proposed theory is illustrated by several analytical examples.

The effect of local random defects on the response of pantographic sheets

In the last decade, pantographic structures have attracted the attention of many scientists because of their exotic mechanical behaviors. Despite several studies conducted on this field, probabilistic remarks cannot be found in the literature. This paper aims to fill this gap by providing the first stochastic study on pantographic sheets. A new simplified procedure is proposed to investigate the effects of local axial, bending, and torsional defects on their mechanical responses. The suggested approach involves a generalization of an existing theoretical result and the use of Monte Carlo simulations to derive probability density functions, mean values, standard deviations, and coefficients of variation of the random kinematic unknowns. It, surely, can be applied to any mechanical system. The achieved results allow stating that pantographic structures are not particularly affected by local random defects thanks to their redundant and complex microstructures. Moreover, the interdependencies among their substructures, beams, and pivots, make it not easy to find the positions of defects that minimize, once fixed all the other things, the randomness of the mechanical responses. A minimization problem should be solved and a work on this topic is in progress.

A general isogeometric finite element formulation for rotation‐free shells with in‐plane bending of embedded fibers

AbstractThis article presents a general, nonlinear isogeometric finite element formulation for rotation‐free shells with embedded fibers that captures anisotropy in stretching, shearing, twisting, and bending—both in‐plane and out‐of‐plane. These capabilities allow for the simulation of large sheets of heterogeneous and fibrous materials either with or without matrix, such as textiles, composites, and pantographic structures. The work is a computational extension of our earlier theoretical work that extends existing Kirchhoff‐Love shell theory to incorporate the in‐plane bending resistance of initially straight or curved fibers. The formulation requires only displacement degrees‐of‐freedom to capture all mentioned modes of deformation. To this end, isogeometric shape functions are used in order to satisfy the required ‐continuity for bending across element boundaries. The proposed formulation can admit a wide range of material models, such as surface hyperelasticity that does not require any explicit thickness integration. To deal with possible material instability due to fiber compression, a stabilization scheme is added. Several benchmark examples are used to demonstrate the robustness and accuracy of the proposed computational formulation.

Computational design and workspace analysis of a passive motion-scaling mechanism based on pantograph for microsurgery

Abstract Robot-assisted surgery has been extensively applied in various microsurgical disciplines owing to its enhanced accuracy and dexterity based on the motion-scaling function between a hand and a surgical instrument. However, the surgical robot system developed thus far incurs high manufacturing costs and restricted compatibility due to the complicated control system, including many actuators and sensors. This paper proposes a novel passive mechanism for motion scaling based on the pantograph structure to resolve these drawbacks of robotic assistance devices. As a first step, a design configuration featuring gravity compensation and the duplication of directional motion is suggested. Subsequently, the geometric dimensions required to satisfy the surgical space and structural constraints are defined. Moreover, the mass of elements required to enable gravity compensation is calculated using moment equations. After determining the principal design parameters, the motion scaling of the proposed passive model is identified using a three-dimensional computer-aided design. In addition, for an extremely precise operation in microsurgery, the dynamic reaction force is expected to be constant in any location. Therefore, to investigate dynamic dexterity, the mathematical model is formulated based on the Lagrangian dynamic equation. Then, the dynamic workspace range is analysed from the determinant of the mass matrix. In addition, the analysis results are evaluated through comparative simulations in different workspace ranges; hence, a constant reaction force can be achieved only in the dynamic workspace range.

Parameter identification of a second-gradient model for the description of pantographic structures in dynamic regime

Pantographic structures are examples of metamaterials with such a microstructure that higher-gradient terms’ role is increased in the mechanical response. In this work, we aim for validating parameters of a reduced-order model for a pantographic structure. Experimental tests are carried out by applying forced oscillation to 3D-printed specimens for a range of frequencies. A second-gradient coarse-grained nonlinear model is utilized for obtaining a homogenized 2D description of the pantographic structure. By inverse analysis and through an automatized optimization algorithm, the parameters of the model are identified for the corresponding pantographic structure. By comparing the displacement plots, the performance of the model and the identified parameters are assessed for dynamic regime. Qualitative and quantitative analyses for different frequency ranges are performed. A good agreement is present far away from the eigenfrequencies. The discrepancies near the eigenfrequencies are a possible indication of the significance of higher-order inertia in the model.