Elastic Rod Research Articles

Longitudinal Wave Locally Resonant Band Gaps in Metamaterial-Based Elastic Rods Comprising Multi-Degree-of-Freedom DAVI Resonators

The wave propagation and vibration transmission in metamaterial-based elastic rods with periodically attached multi-degree-of-freedom (MDOF) dynamic anti-resonant vibration isolator (DAVI) resonators are investigated. A methodology based on a combination of the transfer matrix (TM) method and the Bloch theorem is developed, yielding an explicit formulation for the complex band structure calculation. The bandgap behavior of the periodic structure arrayed with single-degree-of-freedom (SDOF) DAVI resonators and two-degree-of-freedom (2DOF) DAVI resonators are investigated, respectively. A comparative study indicates that the structure consisting of SDOF DAVI resonators periodically jointed on the metamaterial-based elastic rod can obtain an initial locally resonant band gap 500 Hz smaller than the one given in the published literature. The periodic structure containing 2DOF DAVI resonators has an advantage over the periodic structure with SDOF DAVI resonators in achieving two resonance band gaps. By analyzing the equivalent dynamic mass of a DAVI resonator, the underlying mechanism of achieving a lower initial locally resonant band gap by this periodic structure is revealed. The parameters of the 2DOF DAVI resonator are optimized to obtain the lowest band gap of the periodic structure. The numerical results show that, with the optimal 2DOF DAVI parameters, the periodic structure can generate a much lower initial locally resonant band gap compared with the case before the optimization.

Unified static equilibrium modeling and analysis of elastic rods with large deformations for complex constraints

This paper presents a unified framework to describe the static equilibrium modeling and analysis of elastic rods with large deformations. Through an innovative combination of the Hamiltonian principle in arc coordinates and nonlinear optimization based on the principle of minimum potential energy, our method enables us to support the nonlinear statics model and deal with the different complex constraints of an elastic rod, including the end restraint, self-contact, through-hole, and surface contact Based on the idea of dynamic analogy, the equilibrium shape of an elastic rod is described by the movement and rotation of the moving coordinate system along an arc relative to the static coordinate system. Euler parameters are used as generalized variables to describe the strain energy and the external force potential energy of the elastic rod. Taking the total potential energy of the elastic rod as the objective function and different contact forms as the constraint conditions, a nonlinear optimization model is introduced to describe the equilibrium shape of the elastic rod. Based on a primal–dual interior-point method, the model is solved numerically. Considering various constraints and initial conditions, the geometric shape of the elastic rod is simulated. The simulation results show that the unified model proposed in this paper can effectively deal with the static equilibrium problem of elastic rods with large deformations for multiple complex constraints.

On the number of fractured segments of spaghetti breaking dynamics

Why are pieces of spaghetti generally broken into three to ten segments instead of two as one thinks? How can one obtain the desired number of fracture segments? To answer those questions, the fracture dynamics of a strand of spaghetti is modelled by elastic rod and numerically investigated by using finite-element software ABAQUS. By data fitting, two relations are obtained: the number of fracture segments in terms of rod diameter-length ratio and fracture limit curvature with the rod diameter. Results reveal that when the length is constant, the larger the diameter and/or the smaller the diameter-length ratio D/L, the smaller the limit curvature; and the larger the diameter-length ratio D/L, the fewer the number of fractured segments. The relevant formulations can be used to obtain the desired number of broken segments of spaghetti by changing the diameter-to-length ratio.

Featured Cover

The cover image is based on the Research Article A bottom-up optimization method for inverse design of two-dimensional clamped-free elastic rods by Longhui Qin, Jianxiong Zhu, and Weicheng Huang., https://doi.org/10.1002/nme.6950.

Bending Properties of an Extensile Fluidic Artificial Muscle.

Low stiffness, large stroke, and axial force capabilities make Extensile Fluidic Artificial Muscles (EFAMs) a feasible soft actuator for continuum soft robots. EFAMs can be used to construct soft actuated structures that feature large deformation and enable soft robots to access large effective workspaces. Although FAM axial properties have been well studied, their bending behavior is not well characterized in the literature. Static and dynamic bending properties of a cantilevered EFAM specimen were investigated over a pressure range of 5–100 psi. The static properties were then estimated using an Euler-Bernoulli beam model and discrete elastic rod models. The experiments provided data for the determination of bending stiffness, damping ratio, and natural frequency of the tested specimen. The bending stiffness and the damping ratio were found to change fourfold over the pressure range. Experimentally validated bending properties of the EFAM presented insights into structural and control considerations of soft robots. Future work will utilize the data and models obtained in this study to predict the behavior of an EFAM-actuated continuum robot carrying payloads.

An elastic rod in frictional contact with a rigid cylinder

We investigate the load transmission along an elastic rod of finite cross-section in contact with a rigid cylinder, as system often referred to as the generalized capstan problem. In the presence of friction, the idealized classic capstan equation predicts that the tension along a perfectly thin and flexible filament increases exponentially along the contact region. In practical applications, however, the validity of the idealized capstan equation is compromised due to the interplay between finite rod thickness, bending stiffness, and the forces applied at the rod extremities. Here, we combine precision model experiments, finite element simulations, and theoretical modeling to investigate the contact mechanics and the force transmission along an elastic rod in frictional contact with a rigid cylinder. We study two cases when the rod is either static or sliding. First, we focus on the static case, in the absence of friction, by considering equal loads at both extremities of the rod. We show that as the loading force is increased, the nature of contact transitions from a localized region to an extended band at the surface of the cylinder. The latter is characterized by double-peaked contact force distribution. In the sliding case, friction is activated by inducing a relative motion between the rod and the cylinder. We applied a fixed loading force at one rod extremity while pulling the other extremity at a constant velocity. The driving force is monitored during sliding. For increasing loading forces, we find that the force ratio is non-monotonic and displays a local minimum, in contradiction with the constant ideal capstan prediction. This minimum force ratio coincides with the transition from a single contact point to an extended contact region. A theoretical analysis based on Euler’s elastica serves to rationalize the results from the physical and numerical experiments. In addition to predicting the nature of the contact region (single point versus extended line), our model provides quantitative predictions for the wrapping angle and the driving-to-loading force ratio. Finally, we leverage our mechanics-based framework to predictively understand the force ratio at the ends of two commercially available engineering belts (spring-steel and polyurethane) in sliding contact with a steel cylinder.

Automated Stability Testing of Elastic Rods With Helical Centerlines Using a Robotic System

Experimental analysis of the mechanics of a deformable object, and particularly its stability, requires repetitive testing and, depending on the complexity of the object’s shape, a testing setup that can manipulate many degrees of freedom at the object’s boundary. Motivated by recent advancements in robotic manipulation of deformable objects, this letter addresses these challenges by constructing a method for automated stability testing of a slender elastic rod — a canonical example of a deformable object — using a robotic system. We focus on rod configurations with helical centerlines since the stability of a helical rod can be described using only three parameters, but experimentally determining the stability requires manipulation of both the position and orientation at one end of the rod, which is not possible using traditional experimental methods that only actuate a limited number of degrees of freedom. Using a recent geometric characterization of stability for helical rods, we construct and implement a manipulation scheme to explore the space of stable helices, and we use a vision system to detect the onset of instabilities within this space. The experimental results obtained by our automated testing system show good agreement with numerical simulations of elastic rods in helical configurations. The methods described in this letter lay the groundwork for automation to grow within the field of experimental mechanics.

Rotational self-friction problem of elastic rods

The aim of this paper is to extend the modeling of a hyperelastic rod undergoing large displacements with tangential self-friction to their modeling with rotational self-friction. The discontinuity of contact force into a contact region not known in advance with taking into account the effects of friction in this problem type leads to serious difficulties in modelization, mathematical and numerical analysis. In this paper, we present an accurate modeling of rotational and tangential self-friction with Coulomb’s law and describe also an augmented Lagrangian method to present a weak variational formulation approach of this problem. We then use the minimization method of the total energy to present an existence result of solution for the nonlinear penalized formulation. Finally, we give the linearization and the finite-element discretization of the weak variational formulation that can be useful for a numerical implementation.

Dispersion relation for harmonic generation in nonlinear elastic waves

We present a theory for the dispersion of generated harmonics in a traveling nonlinear wave. The harmonics dispersion relation—derived by the theory—provides direct and exact prediction of the collective harmonics spectrum in the frequency-wavenumber domain and does so without prior knowledge of the spatial-temporal solution. The new relation is applicable to a family of initial wave functions characterized by an initial amplitude and wavenumber. We demonstrate the theory on nonlinear elastic waves in a homogeneous rod and demonstrate its extension to periodic rods. We investigate a thick elastic rod admitting longitudinal motion. In the linear limit, this rod is dispersive due to the effect of lateral inertia. The nonlinearity is introduced through either the stress–strain relation and/or the strain–displacementgradient relation. Using a theory we have developed earlier, we derive an exact general nonlinear dispersion relation for the thick rod. We then derive a special case of this relation and show that it provides an exact prediction of the generated harmonics spectrum, in frequency versus wavenumbers. Both relations are validated by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) for the general nonlinear dispersion relation and short-term, pre-breaking dispersion (by Fourier transformations) for both the general and specialized relation.

Compression-torsion coupling behaviours in double-helix nanofibers assembled from negatively curved nanographenes

The recently synthesised one-dimensional (1D) double-helix nanofibers that are self-assembled from negatively curved nanographenes, i.e., 1-H nanofibers are expected to have new applications in materials science and biology. In this paper, the mechanical behaviours of 1-H nanofibers are investigated by using molecular dynamics simulations. 1-H nanofibers are found to possess a Young’s modulus comparable to that of conventional cup-stacked carbon nanofibers. Meanwhile, a large plastic deformation is observed in the axially stretched 1-H nanofibers before their final breaking. In addition, 1-H nanofibers possess a bending stiffness significantly larger than that of many other 1D nanomaterials, making 1-H nanofibers behave more like an elastic rod or beam. More importantly, a unique compression-torsion coupling behaviour is observed in 1-H nanofibers, which is strongly dependent on the rotation direction. Specifically, when 1-H nanofibers are rotated along the helix direction, a residual tensile stress or, equivalently, an axial contraction is generated. However, no significant compression-torsion coupling behaviour is observed in 1-H nanofibers rotated along the direction reverse to helix. Furthermore, due to the compression-torsion coupling effect, a unique helical buckling behaviour can occur in 1-H nanofibers under pure axial compression, which is totally different from the conventional Euler buckling behaviour observed in many other 1D nanomaterials.

Chiral waves in structured elastic systems: dynamics of a meta-waveguide

Summary The notion of meta-surfaces is well established for waves interacting with arrays of periodic resonators. Here, an infinite elastic rod, attached to a system of gyroscopic spinners, is considered. This forms a waveguide with unusual dispersion properties and will be referred to as a chiral meta-waveguide. A class of transient and time-harmonic formulations describing waves in elastic chiral meta-waveguides is addressed. Consideration of a chiral elastic chain leads to the governing equations of the homogenised continuous system. Both analytical and numerical solutions are presented. Features observed for chiral waveforms show resemblance with formulations of quantum physics rather than linear elasticity. For an elementary example related to an elastic chiral rod, with the absence of pre-tension, it is noted that the longitudinal velocity satisfies the Klein–Gordon equation. For full vector problems, corresponding to an elastic chiral rod with pre-tension, time-harmonic Green’s matrix functions are derived for the case of large values of the gyricity parameter. This case is especially interesting because of the presence of both exponentially localised and oscillatory terms in the representation of Green’s matrix components.

Packing a flexible fiber into a cavity.

The insertion of an elastic rod or fiber into a confining cavity is studied. Such an insertion is a feature of a variety of problems, including packing and unpacking of DNA in viral capsids and the insertion of catheters during surgery. We consider a simplified geometry in which the container is a smooth (frictionless) circular cylinder of radius a. The fiber is pushed through a hole in the curved surface of the cylinder and is then assumed to stay in a cross-sectional plane perpendicular to the cylinder axis. A solution is found for the fiber shape in which most of the fiber lies against the curved interior surface of the cylinder, apart from the final end sectionof the fiber, of length 2.0888a, which crosses the interior of the cylinder before ending at the opposite side, which it meets at an angle 1.15rad to the normal. The force required to push the fiber into the cylinder is EI/2a^{2}, where E is the fiber's Young's modulus and I its cross-sectional moment of inertia. The shape of the final end sectionof the fiber is confirmed by experiment.

A magnetically driven elastic rod type bi-directional swimmer at stokes flow

A magnetically driven elastic rod type bi-directional swimmer at stokes flow

Natural frequencies of pre-buckled rods and gridshells

A discrete differential geometry (DDG)-based method is proposed to numerically study the natural frequencies of elastic rods and gridshells in their post-buckling configurations. A fully implicit numerical framework is developed based on Discrete Elastic Rods (DER) algorithm, in order to characterize the mechanical behaviors of an elastic gridshell comprised of multiple rods. When their footprints are constrained along a shrinking trajectory, the slender structures as well as their constructed network would experience a geometrically nonlinear instability and deform into an out-of-plane configuration. By checking the eigenvalues and eigenvectors of the mass and stiffness matrix, the linear vibration near the post-buckling equilibria are characterized through a numerical approach. Exploiting the efficiency and the robustness of the developed discrete method, a systematic parameter sweep is performed to quantify the vibration frequency of pre-buckled gridshells with respect to the number of rods and the pre-compressed distance. It is found that the natural frequency for both pre-deformed rods and gridshells would linearly decrease as the enlargement of compressive distance, even though the geometrically nonlinear deformations have been taken into account. Moreover, the vibration frequency almost linearly rises when the number of rods in a gridshell becomes larger. These findings could provide a fundamental insight in revealing more complex structural dynamics and facilitating the designs of buckling-induced assembly in some man-made systems, e.g., avoidance of the resonance in soft electronics.

Ballooning in spiders using multiple silk threads

In this paper, three-dimensional numerical simulations of ballooning in spiders using multiple silk threads are performed using the discrete elastic rods method. The ballooning of spiders is hypothesized to be caused by the presence of the negative electric charge of the spider silk threads and the positive electric potential field in the Earth's atmosphere. The numerical model presented here is first validated against experimental data from the open literature. After which, two cases are examined, in the first it is assumed that the electric charge is uniformly distributed along the threads while in the second, the electric charge is located at the thread tip. It is shown that the normalized terminal ballooning velocity, i.e., the velocity at which the spiders balloon after they reach steady-state, decrease linearly with the normalized lift force, especially for the tip located charge case. For the uniform electric charge case, this velocity shows a slightly weaker dependence on the normalized lift force. Moreover, it is shown in both cases that the normalized terminal ballooning velocity has no dependence on the normalized elastic bending stiffness of the threads and on the normalized viscous forces. Finally, the multithread bending process shows a three-dimensional conical sheet. Here we show that this behavior is caused by the Coulomb repelling forces owing to the threads electric charge which leads to dispersing the threads apart and thus avoid entanglement.

A bottom‐up optimization method for inverse design of two‐dimensional clamped‐free elastic rods

AbstractRod‐like structures, such as DNA, climbing plants, and cables, pervade the nature and our daily life and also belong to a frequently encountered engineering problem, which usually assume a deformed shape based on the competition between elasticity (stretching, bending, twisting) and external forces (e.g., gravity). These structures often undergo geometrically nonlinear deformation under its own weight. Computing the natural (undeformed) shape of a rod from the observed deformed configuration is a nonlinear inverse problem. Here we introduce a numerical method to solve this problem in case planar (2D) rods, that is, beams, under a clamped‐free boundary condition. A simulation model is developed based on the discrete elastic rods (DER) algorithm to address the forward deformation process from natural to deformed shapes under gravity. Based on the internal mechanical link among the nodes on the discretized elastic rod, a bottom‐up optimization method is proposed in this article for an efficient inverse solution. Associated with it, a global searching algorithm is developed to search for the optimal bending curvature along the arc length of the rod. The method is numerically validated for accuracy and sheds light on the inverse design of rod‐like structures for various applications, for example, soft robots and tails of animated characters.

An Analytical Approach to an Elastic Circular Rod Equation

The size-dependent longitudinal and torsional dynamic problems for small-scaled rods have importance in two-phase media. The special case of the elastic rod equation such as magneto-electro circular equation are seen in the literature commonly, but in this work, the generalized form of the nonlinear elastic circular equation, which was not studied in the literature, is considered. The exact solutions are obtained via Mathieu approximation method with a novel proposed ansatz. Obtained solutions are discussed and illustrated in details. We believe that the proposed results will be key part of further analytical and numerical studies for waves in the dispersive medium with reaction.

Общая устойчивость балок двутаврового постоянного сечения. Развитие расчетных положений

The paper presents the derivation of formulas for the method of calculating bent elements for general stability, given in SP 16.13330, which is based on the theory of thin-walled elastic rods and the system of differential equations of V. Z. Vlasov. Formulas are given for a single-span beam with a hinged mounting of supports. A general equation has been compiled for calculating the bending stability coefficient φ_1, which allows adapting the requirements of SP16.13330 standards to various types of load action in the beam span. The results are presented for twenty-five different types of external load action. The formulas take into account the arbitrary location of the load along the height of the beam section, from the lower edge of the lower belt to the upper edge of the upper belt. The values of the critical moment M_cr, at which a new form of equilibrium state arises in the beam, are determined, according to the conclusions of the work [6]. The critical moment M_cr is calculated in the form of a stability problem of the first kind – the loss of a stable position of an element of a rectilinear shape (the load acts along the line of the bending center, the neutral axis of the beam is rectilinear, the material is elastic). The formulas of the presented table can be used to perform verification calculations.

Weaving structure: A bending-active gridshell for freeform fabrication

Double-curved surfaces are increasingly being designed and fabricated in contemporary architecture and engineering. However, the fabrications are always time-consuming and costly. Inspired by traditional weaving handicrafts, we propose a bending-active weaving structure, which consists of continuous elastic rods and standard joints for the fabrication of large-scale freeform surfaces. Uniform rods with circular cross sections were used to avoid torsion and simplify fabrication. The advantage lies in its convenience in construction, adaptability of spatial forms, and structural efficiency. We introduced the geometric principle of weaving structures, its generation algorithm based on remeshing, structural simulation by dynamic relaxation, and lattice optimization to reduce rod curvature. Structure simulation of lattices were performed to demonstrate its potential in architectural scale construction, and the extended kagome lattice was found more efficient with less stress concentration. Fabrication methods were also discussed. Furthermore, several experimental fabrications were performed to demonstrate the advantages and potentials of the system.

A geometrically exact discrete elastic rod model based on improved discrete curvature

This paper presents a discrete, geometrically exact elastic rod model based on improved discrete curvature. The model is an extension of finite difference type discretization of Kirchhoff–Love rod from Discrete Differential Geometry, with 4 unknowns, including translational displacements of the axis and the axial cross-sectional rotation. The orthogonality of the rod axis and cross-section is guaranteed by parallel transport in pseudotime. A newly defined unified discrete curvature with five weighting function schemes is presented. Three of them are proposed by reducing the max nonlinear term of the Taylor expansion of the rotation vector. Theory and numerical tests show that these three methods can increase the accuracy of the model compared to the traditional method. Scheme 5 performs the best out of the recommended schemes. A transformation of freedoms to the translational displacements and end rotations is introduced to the Discrete Differential Geometry model. By using this technique, rigid connections between rod segments, rotational boundaries and external ending moments can be realized conveniently and naturally. Important properties such as convergence, objectivity and insensitivity to the slenderness ratio are also verified and investigated with several suitable numerical examples. Furthermore, results show the use of discrete curvature has better performance for the stiff bar system with flexible connections than the continuous beam.