Spatial chaos of an extensible conducting rod in a uniform magnetic field

The equilibrium equations for the isotropic Kirchhoff rod are known to form an integrable system. It is also known that the effects of extensibility and shearability of the rod do not break the integrable structure. Nor, as we have shown in a previous paper does the effect of a magnetic field on a conducting rod. Here we show, by means of Mel'nikov analysis, that, remarkably, the combined effects do destroy integrability; that is, the governing equations for an extensible current-carrying rod in a uniform magnetic field are nonintegrable. This result has implications for possible configurations of electrodynamic space tethers and may be relevant for electromechanical devices

Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure

On an elastic strain-limiting special Cosserat rod model

Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some attractive features of the constitutive relations (orientation preservation, transverse symmetry, and monotonicity), we exhibit several explicit equilibrium states under either an isolated end thrust or an isolated end couple. In particular, certain equilibrium states exhibit Poynting like effects, and we show that under mild assumptions on the material parameters, the model predicts an explicit tensile shearing bifurcation: a straight rod under a large enough tensile end thrust parallel to its center line can shear.Comment: 26 pages, 4 figures. Typos corrected, references and a concluding section adde

Twisted rods, helices and buckling solutions in three dimensions

The study of slender elastic structures is an archetypical problem in continuum mechanics, dynamical systems and bifurcation theory, with a rich history dating back to Euler's seminal work in the 18th century. These filamentary elastic structures have widespread applications in engineering and biology, examples of which include cables, textile industry, DNA experiments, collagen modelling etc. One is typically interested in the equilibrium configurations of these rod-like structures, their stability and dynamic evolution and all three questions have been extensively addressed in the literature. However, it is generally recognized that there are still several open non-trivial questions related to three-dimensional analysis of rod equilibria, inclusion of topological and positional constraints and different kinds of boundary conditions

On a nonlinear rod exhibiting only axial and bending deformations: mathematical modeling and numerical implementation

In this work, we present the mathematical formulation and the numerical implementation of a new model for initially straight, transversely isotropic rods. By adopting a configuration space that intrinsically avoids shear deformations and by systemically neglecting the energetic contribution due to torsion, the proposed model admits an unconstrained variational statement. Moreover, as the natural state of the rod is the trivial one and the mechanical properties are homogeneous on the cross section, the need for pull-back and push-forward operations in the formulation is totally circumvented. These features impose, however, some smoothness requirements on the stored energy function that need to be carefully considered when adopting general constitutive models. In addition to introducing the rod model, we propose a spatial discretization with the finite element method, and a time integration with a hybrid, implicit scheme. To illustrate the favorable features of the new model, we provide results corresponding to numerical simulations for plane and three-dimensional problems that are investigated in the static and dynamic settings. Finally, and to put the presented ideas in a suitable context, we compare solutions obtained with the new model against a rod model that allows for torsion and shear.publishedVersio

Buckling of chiral rods due to coupled axial and rotational growth

We present a growth model for special Cosserat rods that allows for induced rotation of cross-sections. The growth law considers two controls, one for lengthwise growth and other for rotations. This is explored in greater detail for straight rods with helical and hemitropic material symmetries by introduction of a symmetry preserving growth to account for the microstructure. The example of a guided-guided rod possessing a chiral microstructure is considered to study its deformation due to growth. We show the occurrence of growth induced out-of-plane buckling in such rods

Manipulation and mechanics of thin elastic objects

In this thesis, multiple problems concerning the equilibrium and stability properties of thin deformable objects are considered, with particular focus given to the analysis of thin elastic rods. The problems considered can be divided into two related categories: manipulation and mechanics. First, a few results concerning symmetries in geometric optimal control theory are derived, which are later used in the analysis of thin elastic objects. Then the problem of quasi-statically manipulating an elastic rod from an initial configuration into a goal configuration is considered. Based upon an analysis of symmetries, geometric and topological characterizations of the set of all stable equilibrium configurations of an elastic rod are derived. Specifically, under a few regularity assumptions, it is shown that the set of all stable equilibrium configurations without conjugate points of an extensible, shearable, anisotropic, and uniform Cosserat elastic rod subject to conservative body forces is a smooth six-dimensional manifold parameterized by a single global coordinate chart. Furthermore, in the case of an inextensible, unshearable, anisotropic, uniform, and intrinsically straight Kirchhoff elastic rod without body forces, this six-dimensional manifold is shown to be path-connected. In addition to their applications to manipulation, the geometric and topological results described above can be used to answer questions concerning the mechanics of elastic rods and other deformable objects. For an inextensible, unshearable, isotropic, and uniform Kirchhoff elastic rod, it is shown that the closure of the set of all stable equilibria with helical centerlines is star-convex, and this property is used to compute and visualize the boundary between stable and unstable helical rods. Finally, two applications of geometric optimal control theory to the analysis of constitutive equations for thin elastic objects are considered. In the first application, the Pontryagin maximum principle is used to analyze curvature discontinuities observed in inextensible surfaces. In the second application, the Pontryagin maximum principle is used to derive constitutive equations for an elastic rod subject to a local injectivity constraint, and the use of this model for analyzing highly flexible helical springs with contact between neighboring coils is considered

On the Statics, Dynamics, and Stability of Continuum Robots: Model Formulations and Efficient Computational Schemes

This dissertation presents advances in continuum-robotic mathematical-modeling techniques. Specifically, problems of statics, dynamics, and stability are studied for robots with slender elastic links. The general procedure within each topic is to develop a continuous theory describing robot behavior, develop a discretization strategy to enable simulation and control, and to validate simulation predictions against experimental results.Chapter 1 introduces the basic concept of continuum robotics and reviews progress in the field. It also introduces the mathematical modeling used to describe continuum robots and explains some notation used throughout the dissertation.The derivation of Cosserat rod statics, the coupling of rods to form a parallel continuum robot (PCR), and solution of the kinematics problem are reviewed in Chapter 2. With this foundation, soft real-time teleoperation of a PCR is demonstrated and a miniature prototype robot with a grasper is controlled.Chapter 3 reviews the derivation of Cosserat rod dynamics and presents a discretization strategy having several desirable features, such as generality, accuracy, and potential for good computational efficiency. The discretized rod model is validated experimentally using high speed camera footage of a cantilevered rod. The discretization strategy is then applied to simulate continuum robot dynamics for several classes of robot, including PCRs, tendon-driven robots, fluidic actuators, and concentric tube robots.In Chapter 4, the stability of a PCR is analyzed using optimal control theory. Conditions of stability are gradually developed starting from a single planar rod and finally arriving at a stability test for parallel continuum robots. The approach is experimentally validated using a camera tracking system.Chapter 5 provides closing discussion and proposes potential future work