Rod Model Research Articles

Deformation Analysis of Cosserat Rods Using Piecewise Clothoid Approximation

AbstractThis paper presents an alternative to isogeometric analysis (IGA) for static analysis of planar curved slender beams. Geometrically exact Cosserat rod model is employed to establish the governing equations. In contrast to conventional IGA, which uses NURBS as the basis function, our method incorporates clothoid curves to represent the curved geometry. We use piecewise clothoid curves to approximate the initial curvature of the undeformed beam, facilitating a seamless integration into the Cosserat rod model. A straightforward solution of the governing equations is implemented using shooting method, verifying applicability across a range of problems. Interestingly, the parameters that define the clothoid segments also appear in the governing equations of the beam. This bridges the gap between the geometric design of the beams and their static simulations. In this way, we present the proposed formulation as an alternative to conventional IGA. The notable features of the method are easy implementation, good accuracy, and convergence. Moreover, the method predicts bending stress in the beam, capturing the nonlinearity of the deformation.

Forced vibrations of the shaft line based on the model of a two-stage rod with an elastic connection of sections

The paper considers a method for solving the problem of dynamics of forced vibrations of a wall line based on a model of an equivalent elastic two-stage rod with a pliable connection of sections. The model is represented by a system of partial differential equations. The solution of the equations is obtained by the Fourier method for eigenfunctions orthogonal with weight. It is established that the presence of an elastic element at the junction of the sections does not affect the orthogonality of the eigenfunctions. The influence of the elastic coupling compliance coefficient on the natural frequency spectrum of the shaft line is estimated. The obtained forms of natural vibrations of an equivalent elastic two-stage rod are consistent in appearance with the corresponding forms known in the literature for a discrete model of a similar torsional circuit. A partial solution of the equations is found for the case of the action of the harmonic load on the engine and the averaged torque on the propeller. As an example, the dynamics of forced vibrations of the shaft line of a barge of the Sosnovka type is studied. The results of calculating the natural frequencies of the considered model are compared with the classical discrete mass method. The analysis of the obtained plots of the twisting angles and torques showed that the angles are satisfactorily approximated by the first proper oscillation shape, and for moments, high shapes must be taken into account. The comparison of the simulation results with the data of the torsiogram obtained under production testing conditions indicates the need for further development of the model. Further improvement of the torsional circuit, as well as a more detailed study of the nature of the operating loads, are considered as promising areas of research.

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

Shape-programmable hard-magnetic soft actuators with high magnetic particle content via digital light processing method

In this work, we developed a novel hard-magnetic photosensitive suspension with high solid loadings of 50 wt% for digital light processing 3D printing. The suspension combining photosensitive resin and reactive diluent can retain stability for 6 h. The printed materials from the suspension achieve a low modulus of below 900 kPa and high remanence of 83.7 kA/m. The suspension’s cure behavior was studied in detail to obtain the optimal printing parameters. The printing error of the small structures with feature size of about 1000 μm is kept below 20 %. Moreover, we proposed an efficient magnetization programming strategy for hard-magnetic soft actuators with desirable shapes based on the rod model considering the non-magnetic load. The accuracy and reliability of the strategy are verified by the experiment results. Finally, we designed and fabricated a “flower” using our method, achieving its closing action in accordance with the prescribed shape.

A comprehensive numerical approach to coil placement in cerebral aneurysms: mathematical modeling and in silico occlusion classification.

Endovascular coil embolization is one of the primary treatment techniques for cerebral aneurysms. Although it is a well-established and minimally invasive method, it bears the risk of suboptimal coil placement which can lead to incomplete occlusion of the aneurysm possibly causing recurrence. One of the key features of coils is that they have an imprinted natural shape supporting the fixation within the aneurysm. For the spatial discretization, our mathematical coil model is based on the discrete elastic rod model which results in a dimension-reduced 1D system of differential equations. We include bending and twisting responses to account for the coils natural curvature and allow for the placement of several coils having different material parameters. Collisions between coil segments and the aneurysm wall are handled by an efficient contact algorithm that relies on an octree based collision detection. In time, we use a standard symplectic semi-implicit Euler time stepping method. Our model can be easily incorporated into blood flow simulations of embolized aneurysms. In order to differentiate optimal from suboptimal placements, we employ a suitable in silico Raymond-Roy-type occlusion classification and measure the local packing density in the aneurysm at its neck, wall region and core. We investigate the impact of uncertainties in the coil parameters and embolization procedure. To this end, we vary the position and the angle of insertion of the micro-catheter, and approximate the local packing density distributions by evaluating sample statistics.

Automated Method for Growing Rod Length Measurement on Ultrasound Images in Children With Early Onset Scoliosis

ObjectiveTo develop and validate machine learning algorithms to automatically extract the rod length of the magnetically controlled growing rod from ultrasound images (US) in a pilot study. MethodsTwo machine-learning (ML) models, called the "Boundary model" and "Rod model," were developed to identify specific rod segments on ultrasound images. The models were developed utilizing Mask Regional Convolutional Neural Networks (Mask RCNN). Ninety US images were acquired from 23 participants who had early onset scoliosis (EOS) surgeries; among those, 70 were used for model development, including training and validation, and 20 were used for testing by comparing the AI-based vs. manual measurements. ResultsThe average precision (AP) of the ML models was 88.5% and 60.2%, respectively. The inter-method correlation coefficient (ICC) was 0.98, and the mean absolute difference ± standard deviation (MAD ± SD) between AI and manual measurements was 0.86 ± 1.0 mm. The Bland-Altman analysis showed no bias, and 90% of the data were within the 95% confidence interval. The automated method was reliable, accurate, and fast. Measurements were displayed in 4.6 seconds after the US image was inputted. ConclusionThis was the first AI-based method to measure the MCGR rod length on US images automatically.

The stick-slip bending behavior of the multilevel helical structures: A 3D thin rod model with frictional contact

The multilevel helical structures in various engineering and natural fields offer excellent deformation flexibility and load bearing capabilities. Understanding the interplay between the local frictional contact and the geometric characteristics of the helical structure under complex external loads has attracted considerable interest. In this work, the effect of local frictional contact behaviors on the bending in multilevel helical structures is investigated by using a combination of theoretical modeling, finite element (FE) simulations, and experiments. In the case of pure bending, the kinematic parameters of the bent multi-stage helix are derived concisely by the idea of the kinematic analogy. The bending stiffness of the multi-stage helix is further obtained. In the case of the combined tension/torsion and bending, the 3D thin rod model incorporating Coulomb’s friction is established to describe the mechanical responses. It is found that the relationship between equivalent bending stiffness and the laying angle exhibits nonlinearity. A comparison with the classical Papailiou model reveals that, for helical structures at large laying angles, the influence of friction is primarily determined by the internal force in the tangential direction, which is the core assumption of the Papailiou model. However, in the case of small laying angles, the helical twisting characteristics and the contribution of the internal forces and moments in the other two directions (normal and binormal directions) to the friction cannot be ignored. Subsequently, a multilevel frictional contact transmission formulation is proposed according to the force action–reaction principle. Based on the above formulation, the non-simplified thin rod equations with Coulomb’s friction are extended to describe the multilevel stick-slip bending behaviors of the second stage cable (3*3). The dissipation capacity of helical structures is evaluated quantitatively under the hysteretic bending. Finally, the theoretical model is verified by FE simulations and experimental results. This work provides insights for unveiling the intrinsic relationship between the nonlinear bending and local frictional contact behaviors in the multilevel helical structures.

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.

Contact detection between curved fibres: high order makes a difference

Computer Graphics has a long history in the design of effective algorithms for handling contact and friction between solid objects. For the sake of simplicity and versatility, most methods rely on low-order primitives such as line segments or triangles, both for the detection and the response stages. In this paper we carefully analyse, in the case of fibre systems, the impact of such choices on the retrieved contact forces. We highlight the presence of artifacts in the force response that are tightly related to the low-order geometry used for contact detection. Our analysis draws upon thorough comparisons between the high-order super-helix model and the low-order discrete elastic rod model. These reveal that when coupled to a low-order, segment-based detection scheme, both models yield spurious jumps in the contact force profile. Moreover, these artifacts are shown to be all the more visible as the geometry of fibres at contact is curved. In order to remove such artifacts we develop an accurate high-order detection scheme between two smooth curves, which relies on an efficient adaptive pruning strategy. We use this algorithm to detect contact between super-helices at high precision, allowing us to recover, in the range of wavy to highly curly fibres, much smoother force profiles during sliding motion than with a classical segment-based strategy. Furthermore, we show that our approach offers better scaling properties in terms of efficiency vs. precision compared to segment-based approaches, making it attractive for applications where accurate and reliable forces are desired. Finally, we demonstrate the robustness and accuracy of our fully high-order approach on a challenging hair combing scenario.

Hybrid compliant control with variable-stiffness wrist for assembly and grinding application

This research presents a novel robot system that combines active and passive components to enhance compliance and dependability. The system is based on a continuous variable stiffness wrist. A wrist was created that met the requirements and a combination of active and passive control methods was suggested to insert and regulate forces effectively. The control strategy is based on the Cosserat rod model, with the fundamental concept being calculating the position and orientation of the component using data on the force exerted during contact between the parts and the stiffness of the contact between the shaft and hole components. This process converts the hard assembly into a flexible contact. Compliance is monitored via force and vision sensors, which allows for the shaft-hole assembly operation to be carried out even with attitude alignment problems, resulting in a notable decrease in the precision needed for component alignment. Initially, the camera supplies the first positional data of the shaft component for the robotic system. In addition, the performance of the wrist with variable stiffness is evaluated in terms of stiffness. Additionally, the calculation of relative deformation between components is examined using contact force information. Moreover, a robust active/passive hybrid insertion control technique, which relies on contact force, is proposed. Finally, the shaft-hole assembly task substantiates the necessity for contact force monitoring in the insertion assembly process. This control technique has demonstrated its efficacy in ensuring passive-compliant assembly performance. Furthermore, the variable stiffness wrist has been employed in robotic grinding for surfaces with curved contours to validate its effectiveness.

Pushing with Soft Robotic Arms via Deep Reinforcement Learning

Soft robots can adaptively interact with unstructured environments. However, nonlinear soft material properties challenge modeling and control. Learning‐based controllers that leverage efficient mechanical models are promising for solving complex interaction tasks. This article develops a closed‐loop pose/force controller for a dexterous soft manipulator enabling dynamic pushing tasks using deep reinforcement learning. Force tests investigate the mechanical properties of a soft robot module, resulting in orthogonal forces of N. Then, the policy is trained in simulation leveraging a dynamic Cosserat rod model of the soft robot. Domain randomization mitigate the sim‐to‐real gap while careful reward engineering induced pose and force control even without explicit force inputs. Despite the approximate simulation, the sim‐to‐real transfer achieved an average reaching distance of mm (), an average orientation error of rad () and applied pushing forces up to N. Such performance is reasonable for the intended assistive tasks of the manipulator. The experiments uncovered that the soft robot interacting with the environment exhibited torsional and counter‐balancing movements. Although not explicitly enforced, they emerged from the mechanical intelligence of the manipulator. The results demonstrate the potential of soft robotic manipulation via reinforcement learning.

MERCI: Mixed curvature-based elements for computing equilibria of thin elastic ribbons

Thin elastic ribbons represent a class of intermediary objects lying in-between thin elastic plates and thin elastic rods. Although the two latter families of thin structures have received much interest from the Computer Graphics community over the last decades, ribbons have seldom been considered and modelled numerically so far, in spite of a growing number of applications in Computer Design. In this paper, starting from the reduced developable ribbon models [Sadowsky 1929; Wunderlich 1962] recently popularised in Soft Matter Physics, we propose a both accurate and efficient algorithm for computing the statics of thin elastic ribbons. Inspired by the super-clothoid model for thin elastic rods, our method relies on compact ribbon elements whose normal curvature varies linearly with respect to arc length s , while their geodesic torsion is quadratic in s . In contrast however, for the sake of efficiency, our algorithm avoids building a fully reduced kinematic chain and instead treats each element independently, gluing them only at the final solving stage through well-chosen bilateral constraints. Thanks to this mixed variational strategy, which yields a banded Hessian, our algorithm recovers the linear complexity of low-order models while preserving the quadratic convergence of curvature-based models. As a result, our approach is scalable to a large number of elements, and suitable for various boundary conditions and unilateral contact constraints, making it possible to handle challenging scenarios such as confined buckling experiments or Möbius bands with contact. Remarkably, our mixed algorithm proves an order of magnitude faster compared to Discrete Elastic Ribbon models of the literature while achieving, in a few seconds only, high accuracy levels that remain out of reach for such low-order models. Additionally, our numerical model can incorporate various ribbon energies, including the RibExt model for quasi-developable ribbons recently introduced in Physics [Audoly and Neukirch 2021], which allows to transition smoothly between a rectangular Kirchhoff rod and a (developable) Sadowsky ribbon. Our numerical scheme is carefully validated against demanding experiments of the Physics literature, which demonstrates its accuracy, efficiency, robustness, and versatility. Our Merci code is publicly available at https://gitlab.inria.fr/elan-public-code/merci for the sake of reproducibility and future benchmarking.

Hydrodynamic behavior of inertial elongated microswimmers in a horizontal channel

In the current study, the lattice Boltzmann method was used to explore the motion of an elongated microswimmer in a horizontal channel with finite fluid inertia. By employing an extended squirmer rod model, the swimming velocity, hydrodynamic efficiency, and interaction with the channel wall of the capsule-shaped squirmer rod were simulated. It was found that the aspect ratio α and the swimming Reynolds number Res of the squirmer rod significantly affect its swimming velocity and efficiency. Specifically, as the Reynolds number increases, the pusher rod's velocity increases, whereas the puller rod's velocity decreases. Moreover, compared with the puller rod, the pusher rod has a higher efficiency with the same power consumption. With the increase of the aspect ratio α, the velocity of the squirmer rod increases gradually, the power consumption of the pusher rod and the puller rod decreases gradually, and the efficiency increases gradually, showing the characteristics of lower energy consumption and higher efficiency. During the interaction of the squirmer rod with the wall, four distinct motion modes were identified, namely, steady linear motion, motion away from the wall, damped swinging motion, and wall-attraction oscillation. The emergence of these motion modes and their transitions could be associated with the pressure distribution formed between the squirmer rod and the wall. The results provide another perspective and theoretical basis for the design of bioinspired microswimming devices and microrobots, especially in medical applications such as precision drug delivery systems.

Active shape control by plants in dynamic environments.

Plants are a paradigm for active shape control in response to stimuli. For instance, it is well known that a tilted plant will eventually straighten vertically, demonstrating the influence of both an external stimulus, gravity, and an internal stimulus, proprioception. These effects can be modulated when a potted plant is additionally rotated along the plant's axis, as in a rotating clinostat, leading to intricate shapes. We use a previously derived rod model to study the response of a growing plant and the joint effects of both stimuli at all rotation speeds. In the absence of rotation, we identify a universal planar shape towards which all shoots eventually converge. With rotation, we demonstrate the existence of a stable family of three-dimensional dynamic equilibria where the plant axis is fixed in space. Further, the effect of axial growth is to induce steady behaviors, such as solitary waves. Overall, this study offers insight into the complex out-of-equilibrium dynamics of a plant in three dimensions and further establishes that internal stimuli in active materials are key for robust shape control.

Rod models of strength of compressed concrete strips of the conjugation between beams and columns of prefabricated monolithic frames

Rod models of strength of compressed concrete strips of the conjugation between beams and columns of prefabricated monolithic frames

Multipoint kinetics model with power reactivity defect for the axial offset control in the VVER-1200 nuclear reactor during the load following mode of operation

This article proposes the multipoint kinetics model consisting with different number of point kinetics model (two points, four points, six points, eight points, ten points) in the axial direction for the VVER-1200 nuclear reactor. Each node is coupled with others through the coupling coefficients determined from the diffusion approach. For more precise description of the dynamical modes of the reactor operation, the proposed model integrates the power reactivity feedback derived from temperature reactivity coefficients and Mann’s thermal hydraulic model which assumes one fuel node adjacent to two coolant nodes. On the model with four axial points, additionally was tested the influence of the different number of delayed neutrons groups on the accuracy results and model running time during the load-following mode of operation. Moreover, the novel model of the control rods is introduced, utilizing a combination of sign functions to sequentially influence all nodes during insertion or withdrawal. The computational results show that the accuracy of the proposed model is satisfactory, and general assumptions about transients align with their physical definitions. This research contributes to the advancement of the point-like nuclear reactor modeling for improvement of the automatic power controller synthesis.

Complex stress mechanism and design method of urban rail prestressed concrete U‐beams based on finite element simulation

SummaryTo explore the complex stress mechanism of prestressed concrete U‐beams in urban rail transit, in order to improve the safety of urban rail transit construction and the economy of beam structures. The study first analyzed the complex stress mechanism of U‐beams and obtained a tension compression rod model through finite element analysis. Then, experimental research was conducted on the vertical three‐dimensional finite element stress of U‐beams, and strain cloud maps were obtained and compared with calculated values. The experimental data show that the beam can still recover to its original state after the second cycle, and the beam will not crack. This recovery mechanism means that U‐beams have high crack resistance and stability under complex stress processes. In the vertical deformation cloud map of the U‐beam, the deflection of the mid span section is the largest, with a maximum displacement of about 20.4 mm, which is very close to the measured value of 20.3 mm. In the measured data of concrete strain measuring points and the results of finite element calculation, the difference rate between measured values and calculated values of some measuring points is within 10%. The results indicate that the U‐shaped beam tension and compression rod model combined with finite element analysis has a high degree of conformity with the actual situation, and can provide technical reference for the construction of urban rail transit. The stress mechanism and design method proposed in the study have high reliability and are suitable for the design and construction of prestressed concrete U‐beams in urban rail transit construction.

On nonlinear geometric transformations of finite elements

AbstractThe paper develops a systematic procedure for formulating finite elements on manifolds. The theoretical developments give rise to a modular computational framework for composing coordinate transformations and manifold parameterizations. The procedure is demonstrated with the Cosserat rod model furnishing a novel finite element formulation that rectifies the lack of objectivity of existing finite elements without violating the director constraints or compromising the symmetry of the tangent stiffness at equilibrium. The framework is element‐independent, allowing its implementation as a wrapper to existing element libraries without modification of the element state determination procedures.

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.

Optimization and Analysis of 4-Wheeler Connecting Rod for Weight Reduction and Performance Enhancement

Abstract: Composite materials are now a day widely used in the engineering field. The general characteristics possessed by the composite materials are found to be the reason for using it in the automotive applications. The connecting rod is a major link inside of combustion engine. It connects the piston to the crankshaft and is responsible for transferring power from the piston to the crankshaft. It must work on high RPM because of which it has to bear severe stresses which make its design vital for internal combustion engine. In this project, design, & analysis of the 4wheeler connecting rod is done. The CATIA V5 R21 software has been used for designing the connecting rod 3D model and then the designed connecting rod model is imported into the ANSYS software in which the design is meshed and analyzed by using the Finite Element Method (FEM) and the result is manipulated. Static analysis of existing and composite connecting rod will be performed. Results of the analysis are also validated by using UTM testin