Rod Theory Research Articles

Double-eigenvalue bifurcation and multistability in serpentine strips with tunable buckling behaviors

Serpentine structures, composed of straight and circular strips, have garnered significant attention as potential designs for flexible electronics due to their remarkable stretchability. When subjected to stretching, these serpentine strips buckle out of plane, and previous studies have identified two distinct buckling modes whose order of appearance may interchange in serpentine structures with a single cell. In this study, we employ anisotropic rod theory to model serpentine strips as a multi-segment boundary value problem (BVP), with continuity conditions enforced at the interface between the straight and curved strips. We solve the BVP using methods of continuation, and our results reveal that: (1) the exchange of the two buckling modes in a single-cell serpentine strip is induced by a double-eigenvalue and associated secondary bifurcations, which also alter the stability of the two buckling modes; (2) a variety of stable states with reversible symmetry can be manually obtained in tabletop models and are found to be disconnected from the planar branch in numerical continuation. Furthermore, we demonstrate that modulating the strip thickness across different cells leads to the initiation of buckling in the thinnest section, thereby allowing for the tuning of buckling modes in serpentine strips. In structures with two cells, the sequence of the two buckling modes can also be controlled by designing serpentine strips with nonuniform height. This work could enhance the mechanical design of serpentine-interconnect-based flexible structures and could have applications in multistable actuators and mechanical memory devices.

Chained Spatial Beam Adomian Decomposition Model: A Novel Model of Flexible Slender Beams for Large Spatial Deflections

Abstract The main element of compliant mechanisms and continuum robots is flexible slender beams. However, the modeling of beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model called the spatial beam Adomian decomposition model (SBADM) for intermediate spatial deflections of a slender beam with uniform, bisymmetric sections subjected to general end-loading. Specifically, the elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko's assumptions and Cosserat rod theory. Then, the quaternion transformation and Adomian decomposition are used to solve the nonlinear governing differential equations for the beam by truncating the higher-order terms, yielding an explicit expression for spatially deflected beams within intermediate deflection ranges. Simulations demonstrate the accuracy and time-wise efficiency of the SBADM, as well as its advantages over the state-of-the-art. In addition, this paper also introduces a discretization-based scheme called the chained SBADM (CSBADM) for large spatial deflections of flexible beams. Real-world experiments with two different configurations have also been performed to validate the effectiveness of the CSBADM. The results indicate that the CSBADM can accurately calculate the load-displacement relations for large deformed beams.

An explicit nonlinear model for large spatial deflections of symmetric slender beams

Flexible slender beams are commonly used in compliant mechanisms and continuum robots. However, the modeling of these beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model for large spatial deflections of a slender beam with uniform, symmetrical sections subjected to general end-loading. The elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko’s assumptions and Cosserat rod theory. Subsequently, the nonlinear governing differential equations for the beam are derived from the quaternion representation of the rotation matrix. The explicit load–displacement relations of the beam are obtained using the improved Adomian decomposition method. This method is superior to the classical Adomian decomposition method in terms of convergence rate and domain. The convergence and superiority of the method are also rigorously demonstrated. Simulations are provided to verify the one-, two-, and three-dimensional deflections of beams. Real-world experiments have also been performed to validate our method’s effectiveness with two different beam configurations. The results indicate that the proposed method accurately estimates large spatial deflections of flexible beams.

Simulation and parameterization of nonlinear elastic behavior of cables

AbstractThis work contributes to the simulation, modeling, and characterization of nonlinear elastic bending behavior within the framework of geometrically nonlinear rod models. These models often assume a linear constitutive bending behavior, which is not sufficient for some complex flexible slender structures. In general, nonlinear elastic behavior often coexists with inelastic behavior. In this work, we incorporate the inelastic deformation into the rod model using reference curvatures. We present an algorithmic approach for simulating the nonlinear elastic bending behavior, which is based on the theory of Cosserat rods, where the static equilibrium is calculated by minimizing the linear elastic energy. For this algorithmic approach, in each iteration the static equilibrium is obtained by minimizing the potential energy with locally constant algorithmic bending stiffness values. These constants are updated according to the given nonlinear elastic constitutive law until the state of the rod converges. To determine the nonlinear elastic constitutive bending behavior of the flexible slender structures (such as cables) from the measured values, we formulate an inverse problem. By solving it we aim to determine a curvature-dependent bending stiffness characteristic and the reference curvatures using the given measured values. We first provide examples using virtual bending measurements, followed by the application of bending measurements on real cables. Solving the inverse problem yields physically plausible results.

Exact derivative propagation method to compute the generalized compliance matrix for continuum robots: Application to concentric tubes continuum robots

This paper introduces the concept of Generalized Compliance for continuum robots, specifically for those modeled with the Cosserat Rod theory. Unlike existing models based on tip compliance, the proposed approach considers interactions along the entire body of the flexible robot. The paper also presents a novel method referred to as the Low-Level Derivative Propagation Method, which is designed for the computationally efficient derivation of the Generalized Compliance matrix. The proposed method streamlines calculations and reduces integration time. The presented method, which is general and applies to various types of continuum robot models, is demonstrated on the case of a Concentric Tubes Continuum Robot. We provide detailed derivations of the equations and computation techniques leading to the derivation of the Generalized Compliance matrix, as well as a large-scale numerical validation of the method. The code used in this paper is available on the following GitHub website: https://github.com/benoitrosa/Generalized-Compliance-Computation-for-Continuum-Robots.

Analytical solution for axially loaded floating pipe piles in homogeneous elastic soil layer

An analytical solution for the analysis of open-ended floating pipe piles in a homogeneous elastic soil layer is developed in this paper, based on the Tajimi-type elastic continuum model. Solving the governing equations of the outer and inner soil results in closed-form expressions for the external and internal frictional resistances developing at the soil-pile interfaces. The one-dimensional rod theory is adopted to model the pipe pile and its underlying soil hollow column. The external and internal frictional resistances are substituted into the governing equations of the pile and soil column to derive closed-form solutions for the vertical displacement and head stiffness of the pipe pile. The accuracy of the proposed model is verified by comparisons against existing numerical studies. Finally, parametric analyses are presented to discuss the sensitivity of the behavior of axially loaded floating pipe piles to the main problem parameters.

Geometrically exact 3D arbitrarily curved rod theory for dynamic analysis: Application to predicting the motion of hard-magnetic soft robotic arm

Magnetorheological elastomers are active materials which can be actuated by the applied magnetic field. Hard magnetic soft (HMS) materials, a type of magnetorheological elastomers, show great potential in the fields of biomedical engineering and soft robotics, due to their short response time, remote operation, and shape programmability. To exploit its potential, a series of theoretical frameworks of HMS rods have been developed, but they are mainly limited to the static rod models or classical curved rod models that fail to consider the effect of the “initial curvature” on the distribution of the stress. In this work, we develop a curved rod theory to predict the 3D dynamic motion of the rod-like HMS robotics under large deformation. Based on the geometrically exact rod theory, we include the heterogeneous initial length of the longitudinal fiber caused by “initial curvature” into our model and obtain the reduced balance equations of the HMS robotics. As a result, the “tension-bending” and “shear-torsion” coupling effects of curved rods emerge in the present model. A numerical implementation of our model based on the classical Newmark algorithm is presented. To validate our model, three numerical examples, including the dynamic snap-through behavior of a bistable arch, are performed and compared with the simulation or experiment results reported in literatures, which show a good agreement. Finally, we experimentally study the 2D and 3D static and dynamic motion of a quarter arc HMS robotic arm under an applied magnetic field of 10 mT, and our model gives a satisfactory prediction, especially for static deformation.

Investigation of the influence of the elastic medium on the dynamic behavior of a three-layer elliptical cylinder under non-stationary loading

Dynamic transient processes are studied and the results of evaluating the influence of an elastic medium on the behavior of a three-layer cylindrical shell structure of elliptical cross-section under non-stationary impulse loading are given. The used model of the theory of shells and rods by S. P. Tymoshenko, taking into account independent static and kinematic hypotheses for each layer of the structure. Numerical calculations of normal deflections and normal stresses of the load-bearing layers of the structure, which determine its stress-strain state (SSS), have been performed. Variants of the structure without a polymer aggregate and with a discrete-symmetric rib-reinforced aggregate are considered. A comparative analysis of the deflections and stresses of the load-bearing layers of the structure in the absence of light aggregate and in its presence is given.Numerical results regarding the dynamics of the three-layer structure were obtained using the finite element method.

Model-based trajectory tracking of a compliant continuum robot.

Introduction: Compliant mechanisms, especially continuum robots, are becoming integral to advancements in minimally invasive surgery due to their ability to autonomously navigate natural pathways, significantly reducing collision severity. A major challenge lies in developing an effective control strategy to accurately reflect their behavior for enhanced operational precision. Methods: This study examines the trajectory tracking capabilities of a tendon-driven continuum robot at its tip. We introduce a novel feedforward control methodology that leverages a mathematical model based on Cosserat rod theory. To mitigate the computational challenges inherent in such models, we implement an implicit time discretization strategy. This approach simplifies the governing equations into space-domain ordinary differential equations, facilitating real-time computational efficiency. The control strategy is devised to enable the robot tip to follow a dynamically prescribed trajectory in two dimensions. Results: The efficacy of the proposed control method was validated through experimental tests on six different demand trajectories, with a motion capture system employed to assess positional accuracy. The findings indicate that the robot can track trajectories with an accuracy within 9.5%, showcasing consistent repeatability across different runs. Discussion: The results from this study mark a significant step towards establishing an efficient and precise control methodology for compliant continuum robots. The demonstrated accuracy and repeatability of the control approach significantly enhance the potential of these robots in minimally invasive surgical applications, paving the way for further research and development in this field.

Breaking the left-right symmetry in fluttering artificial cilia that perform nonreciprocal oscillations

Recent investigations on active materials have introduced a new paradigm for soft robotics by showing that a complex response can be obtained from simple stimuli by harnessing dynamic instabilities. In particular, polyelectrolyte hydrogel filaments actuated by a constant electric field have been shown to exhibit self-sustained oscillations as a consequence of flutter instability. Owing to the nonreciprocal nature of the emerging oscillations, these artificial cilia are able to generate flows along the stimulus. Building upon these findings, in this paper we propose a design strategy to break the left-right symmetry in the generated flows, by endowing the filament with a natural curvature at the fabrication stage. We develop a mathematical model based on morphoelastic rod theory to characterize the stability of the equilibrium configurations of the filament, proving the persistence of flutter instability. We show that the emerging oscillations are nonreciprocal and generate thrust at an angle with the stimulus. The results we find at the level of the single cilium open new perspectives on the possible applications of artificial ciliary arrays in soft robotics and microfluidics.

Angular Momentum in a Special Nonlinear Elastic Rod

In the constrained Cosserat theory of a rod with a rigid cross-section, the balance of angular momentum is satisfied by a restriction on the constitutive equations that requires a second order tensor to be symmetric. The kinematics of the rod are determined by satisfying the balances of linear and director momentum and kinematic constraints. In contrast, the Antman model for a special Cosserat theory of rods proposes constitutive equations directly for the force and mechanical moment applied to the rod and the kinematics are determined by the balances of linear and angular momentum. These two models differ by their treatment of angular momentum. This note poses and answers the question: Are the solutions of these two models identical for the same strain energy?

Out-of-Plane Instability and Vibrations of a Flexible Circular Arch Under a Moving Load

Flexible lightweight arched structures are finding increasing use as components in smart engineering applications. Such structures are prone to various types of instability under moving transverse loads. Here, we study deformation and vibration of a hinged circular arch under a uniformly moving point load using geometrically-exact rod theory to allow for large pre- and post-buckling deformations. We first consider the quasi-statics problem, without inertia. We find that for arches with relatively large opening angle ([Formula: see text]160[Formula: see text]) a sufficiently large traversing load will induce an out-of-plane flopping instability, instead of the in-plane collapse (snap-through) that dominates failure of arches with smaller opening angle. In a subsequent dynamics study, with full account of inertia, we then explore the effect of the speed of the load on this lateral buckling. We find speed to have a delaying (or even suppressing) effect on the onset of three-dimensional bending–torsional vibrations and instability. Based on numerical computations we propose a power law describing this effect. Our results highlight the role of inertia in the onset of elastic instability.

PDE-based control synthesis for a planar cable-driven continuum arm

Classical wave and beam models cannot describe the arc-like planar configuration of a continuum arm. Based on the Cosserat rod theory, a strain vector and the rotation matrix are introduced into this work. Two groups of nonlinear partial differential equations (PDEs) are obtained by analyzing kinetic and potential energy densities and virtual power done by the cable tension, which are explicit both in the spatial coordinates and in the physical parameters. Although involving algebraic–trigonometric descriptions, the obtained dynamical model is the most concise formulation but difficult to use for control design. We develop a feedforward plus proportional–derivative feedback boundary control scheme for the nonlinear PDE system to regulate the planar shape of the continuum arm and then linearize the dynamical equations about the equilibrium to assess the control performance. Some interesting results are the closed-loop stability and the well-posedness analysis of the linearized model, which have not previously been noted. Finally, numerical results validate the dynamical equations and the performance of the controller.

Investigation of deformation of the spring tooth of agricultural implements from the action of the force applied to it

The relevance of the study lies in the need to investigate the dependence of the force applied to the spring tooth on its parameters, which is an important task due to the widespread use of spring teeth in agricultural implements, such as balers, reapers, rakes, etc. The purpose of the study is to establish an analytical description of the spring tooth deformation depending on the amount of applied force. For this purpose, the theory of bending rods from the resistance of materials was applied, without simplifying it, as is common in construction, where the deflection of a beam is small compared to its length. The calculation is based on the well-known dependence of the curvature of the elastic axis of the beam (tooth) on the applied moment and the stiffness of its cross-section. The study considers a cantilevered tooth, which at the point of pinching is a spring with several turns, followed by a smooth transition to a rectilinear shape. The tooth is divided into two parts along its length: curvilinear and rectilinear. Calculation of the deformation, i.e., finding the shape of the elastic axis after the action of the applied force, is carried out for both parts separately. The need for this approach is dictated by the fact that the curvature of the elastic axis of the tooth in the free state changes abruptly from the stable value of the curvilinear part to a zero value of the straight part. The main result of the study is to find the shape of the elastic axis of individual parts of the tooth under the action of the applied force and combine them into one whole. This helps to determine the amount of movement of the free end of the tooth depending on the amount of force applied to it. The application of the obtained data can help in the development of more efficient and productive agricultural tools, and increase their durability and efficiency when interacting with the soil

Interference Morphology of Free-Growing Tendrils and Application of Self-Locking Structures.

Organisms can adapt to various complex environments by obtaining optimal morphologies. Plant tendrils evolve an extraordinary and stable spiral morphology in the free-growing stage. By combining apical and asymmetrical growth strategies, the tendrils can adjust their morphology to wrap around and grab different supports. This phenomenon of changing tendril morphology through the movement of growth inspires a thoughtful consideration of the laws of growth that underlie it. In this study, tendril growth is modeled based on the Kirchhoff rod theory to obtain the exact morphological equations. Based on this, the movement patterns of the tendrils are investigated under different growth strategies. It is shown that the self-interference phenomenon appears as the tendril grows, allowing it to hold onto its support more firmly. In addition, a finite element model is constructed using continuum media mechanics and following the finite growth theory to simulate tendril growth. The growth morphology and self-interference phenomenon of tendrils are observed visually. Furthermore, an innovative class of fluid elastic actuators is designed to verify the growth phenomena of tendrils, which can realize the wrapping and locking functions. Several experiments are conducted to measure the end output force and the smallest size that can be clamped, and the output efficiency of the elastic actuator and the optimal working pressure are verified. The results presented in this study could reveal the formation law of free tendril spiral morphology and provide an inspiring idea for the programmability and motion control of bionic soft robots, with promising applications in the fields of underwater rescue and underwater picking.

Yin-Yang spiraling transition of a confined buckled elastic sheet

DNA in viral capsids, plant leaves in buds, and geological folds are examples in nature of tightly packed low-dimensional objects. However, the general equations describing their deformations and stresses are challenging. We report experimental and theoretical results of a model configuration of compression of a confined elastic sheet, which can be conceptualized as a one-dimensional (1D) line inside a 2D rectangular box. In this configuration, the two opposite ends of a planar sheet are pushed closer, while being confined in the orthogonal direction by two rigid walls separated by a given gap. Similar compaction of sheets has been previously studied and was shown to buckle into quasiperiodic motifs. In our experiments, we observed a different phenomenon, namely the spontaneous instability of the sheet, leading to localization into a single Yin-Yang pattern. The linearized Euler Elastica theory of elastic rods, together with global energy considerations, allow us to predict the symmetry breaking of the sheet in terms of the number of motifs, compression distance, and tangential force. Surprisingly, the appearance of the Yin-Yang pattern does not require friction. Published by the American Physical Society 2024

Haptic Localization with a Soft Whisker from Moment Readings at the Base.

This article focuses on haptic localization of very lightweight and delicate objects while applying a contact force >5000 times lower than the weight of the object. A soft whisker integrated with a Force/Moment (F/M) sensor at the base, and a novel reconstruction algorithm have been proposed for this purpose. Initially, the mathematical relationships between the deformations of the whisker and the F/M sensor outputs were used to reconstruct the shape of the whisker and the position of the touched object. The Cosserat rod theory was used under the assumption that only one contact point occurs during the exploration, and friction effects are negligible. A new methodology we called moment only reading (MOR) has been tested, verified, and compared with previous methods that employed Force and Moment Readings (FMR). Experimental investigations revealed that the spatial position estimation error of the MOR method was confined within 13 mm, when the force applied ranged between 0.001 and 0.01 N. Moreover, the comparison with FMR demonstrated that MOR is capable of retrieving the position of objects even when the force readings drop below the force resolution of the sensor. Eventually, the MOR method has been applied to demonstrate the localization and grasping, with a soft gripper, of delicate crops like tomatoes and strawberries.

Determination of deflections of above-ground pipelines under loads in the longitudinal and transverse directions

The most common method of construction of field pipelines in permafrost-dominated areas is above-ground laying on supports. The size of the span between bearings is determined taking into account the condition of tuning out resonance frequencies in case of vortex excitation due to wind load. For these purposes it is necessary to determine the natural frequencies of vibrations of above-ground pipeline sections. When determining the natural frequencies of pipeline sections, it is necessary to take into account the initial position caused by additional loads in longitudinal and transverse directions (pre-stressed state of pipeline sections). A differential equation for determining the position of a pipeline section under stationary loads in longitudinal and transverse directions based on the rod theory is obtained. The numerical solution of the equation is obtained using the finite difference method. The influence of longitudinal and lateral forces is evaluated. It is shown that the influence of longitudinal and lateral loads has a significant effect on the initial position of pipeline sections. Therefore, the complex influence of loads should be taken into account for the correct determination of deflections. The initial position of the pipeline under the action of various combinations of loads can be used to evaluate the preliminary stress-strain state in the design of above-ground pipelines.

Analysis of transverse constraint force mechanics of anchor rods based on higher order theory

Abstract Research on the mechanical properties of transverse constraint force of anchor support under different loads can help to improve work safety. In this paper, based on the higher-order shear deformation theory, the higher-order whole-local theory of anchor rods is established with full consideration of the transverse normal strain. The geometric and physical equations in the intrinsic relationship are given for the wet thermodynamic behavior of anchor rods’ transverse restraining force. The equilibrium equations of the whole-local are constructed by combining them with the principle of virtual displacement. The analytical solution of the anchor core material was calculated by applying Navier’s method. The boundary displacement conditions of the anchor core material were solved. The anchor core material was modeled by combining it with ABAQUS finite element software. The mechanical analysis was carried out from the two perspectives of the buckling problem under thermal load and the parameter change of the transverse restraining force, respectively. It was found that when the thickness of the anchor bar increased from 20 mm to 50 mm, the radial displacement of the upper layer decreased by 52.95%, and there was no significant change in the positive stress at the upper and lower interfaces. The peak value of the upper interface shear stress decreases significantly from 0.55MPa to −1.41MPa when the anchor laying method is changed from a single layer to four layers. The combination of higher-order theory with ABAQUS software can effectively analyze the change of transverse constraint force of the anchor support, which can help improve its safety.

ПРОБЛЕМА РАСКРЫТИЯ ТРЕЩИН В ЖЕЛЕЗОБЕТОНЕ

The article discusses various aspects of the problem of assessing crack opening in reinforced concrete based on experimental data obtained in recent studies. The author proposed a classification of crack types, introduced a number of new hypotheses, established experimentally the effects of deformation of reinforced concrete in the crack zone. The principles formulated on their basis include schemes for the distribution of force flows between cracks, the concept of progressing main cracks and the deformation effect in reinforced concrete - a special two-console element in the local region near the crack banks, new generalized hypotheses, theorems and functionals about linear and angular deformations of the compressed and tensile zones of reinforced concrete element sections at all levels of elastic-plastic deformation. The proposed model of composite rods in the form of single strips for determining the stiffness of a section of reinforced concrete with intersecting cracks has made it possible to reduce the differential equations of the theory of composite rods by an order of magnitude when solving such problems. The connection is established and analytical dependences for displacements in the crack with opening and shear of the crack banks are given. The main vector of displacements and the angle of equidirectional forces in the reinforcement crossing the crack are determined. Calculated dependences for determining the level distances between cracks and crack opening widths are constructed. Within the framework of the general methodology of the considered problem of crack opening in reinforced concrete, using the formulated principles, a general combined numerical-analytical model of the structural mechanics of reinforced concrete is constructed, which takes into account the deformation effect in the crack modeled by a double cantilever element, the types of cracks, the spatial surface of strain distribution in the cross section with the crack and other experimentally determined features of the mechanics of reinforced concrete, cracks types, spatial surface of strain distribution in the cross-section with a crack and other experimentally determined features of reinforced concrete mechanics.