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1
Kogan, Irina A. "Two Algorithms for a Moving Frame Construction." Canadian Journal of Mathematics 55, no. 2 (April 1, 2003): 266–91. http://dx.doi.org/10.4153/cjm-2003-013-2.
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AbstractThe method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.
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Yavari, Arash, and Alain Goriely. "The geometry of discombinations and its applications to semi-inverse problems in anelasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2169 (September 8, 2014): 20140403. http://dx.doi.org/10.1098/rspa.2014.0403.
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The geometrical formulation of continuum mechanics provides us with a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometrical structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space. Here, we consider the problem of discombinations (a new term that we introduce in this paper), that is, a combined distribution of fields of dislocations, disclinations and point defects. Given a discombination, we compute the geometrical characteristics of the material manifold (curvature, torsion, non-metricity), its Cartan's moving frames and structural equations. This identification provides a powerful algorithm to solve semi-inverse problems with non-elastic components. As an example, we calculate the residual stress field of a cylindrically symmetric distribution of discombinations in an infinite circular cylindrical bar made of an incompressible hyperelastic isotropic elastic solid.
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Yavari, Arash, and Alain Goriely. "Weyl geometry and the nonlinear mechanics of distributed point defects." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2148 (September 5, 2012): 3902–22. http://dx.doi.org/10.1098/rspa.2012.0342.
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The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects—where the body is stress-free—is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid.
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Polyakova, K. V. "On some extension of the second order tangent space for a smooth manifold." Differential Geometry of Manifolds of Figures, no. 53 (2022): 94–111. http://dx.doi.org/10.5922/0321-4796-2022-53-9.
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This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold. Using the perturbation of the exterior derivative and ordinary differential, mappings are introduced that enable us to construct non-symmetrical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimensional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold. A deformed external differential is widely used, which is a differential, i. e., its reapplication vanishes. We introduce a deformed external differential being a differential along the curves on the manifold, i. e., its repeated application along the curves on the manifold gives zero.
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Ushakov, Vitaly. "Developable surfaces in Euclidean space." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 66, no. 3 (June 1999): 388–402. http://dx.doi.org/10.1017/s1446788700036685.
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AbstractThe classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.
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Bracken, Paul. "Delaunay surfaces expressed in terms of a Cartan moving frame." Journal of Applied Analysis 26, no. 1 (June 1, 2020): 153–60. http://dx.doi.org/10.1515/jaa-2020-2012.
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AbstractDelaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.
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Vargas, José G., and Douglas G. Torr. "Finslerian structures: The Cartan–Clifton method of the moving frame." Journal of Mathematical Physics 34, no. 10 (October 1993): 4898–913. http://dx.doi.org/10.1063/1.530331.
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ASADI, E., and J. A. SANDERS. "INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY." Glasgow Mathematical Journal 51, A (February 2009): 5–23. http://dx.doi.org/10.1017/s0017089508004746.
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AbstractQuaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space=Sp(n+1)/Sp(1) ×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry onmodelled on$(\mk{sp}_{n+1}, \mk{sp}_{1}\,{\times}\, \mk{sp}_{n})$. The integrability structure is shown to be geometrically encoded by a Poisson–Nijenhuis structure and a symplectic operator.
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Dhas, Bensingh, Jamun Kumar N., Debasish Roy, and J. N. Reddy. "A mixed variational principle in nonlinear elasticity using Cartan’s moving frames and implementation with finite element exterior calculus." Computer Methods in Applied Mechanics and Engineering 393 (April 2022): 114756. http://dx.doi.org/10.1016/j.cma.2022.114756.
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FERNÁNDEZ, V. V., A. M. MOYA, and W. A. RODRIGUES. "GEOMETRIC AND EXTENSOR ALGEBRAS AND THE DIFFERENTIAL GEOMETRY OF ARBITRARY MANIFOLDS." International Journal of Geometric Methods in Modern Physics 04, no. 07 (November 2007): 1117–58. http://dx.doi.org/10.1142/s0219887807002478.
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We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U ⊂ M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field γ defining a parallelism structure on U ⊂ M, which represents in a well-defined way the action on U of the restriction there of some given connection ∇ defined on M. Also we give a novel and intrinsic presentation (i.e. one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection operator fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor g defined on M and also introduce the concept of a geometric structure(U, γ ,g) for U ⊂ M and study metric compatibility of covariant derivatives induced by the connection extensor γ. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives of operators in metric and geometrical structures, like ordinary and covariant Hodge co-derivatives and some duality identities are exhibited.
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Evtushik, L. E. "Cartan connections and Kawaguchi geometry of spaces obtained by the moving Frame method." Journal of Mathematical Sciences 89, no. 3 (April 1998): 1279–300. http://dx.doi.org/10.1007/bf02414872.
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Raffaelli, Matteo, Jakob Bohr, and Steen Markvorsen. "Cartan ribbonization and a topological inspection." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2220 (December 2018): 20170389. http://dx.doi.org/10.1098/rspa.2017.0389.
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We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.
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CERCHIAI, B. L., J. MADORE, and G. FIORE. "FRAME FORMALISM FOR THE N-DIMENSIONAL QUANTUM EUCLIDEAN SPACES." International Journal of Modern Physics B 14, no. 22n23 (September 20, 2000): 2305–14. http://dx.doi.org/10.1142/s0217979200001849.
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We sketch our application1of a non-commutative version of the Cartan "moving-frame" formalism to the quantum Euclidean space [Formula: see text]the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over [Formula: see text] based on the R-matrix formalism, we summarize our construction of a frame, the dual inner derivations, a metric and two torsion-free almost metric compatible covariant derivatives with a vanishing curvature. To obtain these results we have developed a technique which fully exploits the quantum group covariance of [Formula: see text]. We first find a frame in the larger algebra [Formula: see text]. Then we define homomorphisms from [Formula: see text] to [Formula: see text] which we use to project this frame in [Formula: see text].
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Impelluso, Thomas J. "The moving frame method in dynamics: Reforming a curriculum and assessment." International Journal of Mechanical Engineering Education 46, no. 2 (August 30, 2017): 158–91. http://dx.doi.org/10.1177/0306419017730633.
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Rigid body dynamics, a gateway course to the mechanical engineering major (and related majors), focuses on a view of motion that is not commensurate with the contemporary age in which mobile devices have on-board inertial firmware. The traditional approach to this topic deploys a mathematical notation, and associated algebra, that inordinately privileges the inertial frames and 2D motion. This limits the study of machines to two-dimensional problems, lends an appearance of whimsy to solutions that obfuscates the theory of motion. We propose a new mathematical approach to dynamics to reinvigorate the discipline and motivate students. The new approach uses modern mathematical tools which have been distilled to tractability: Lie Group Theory, Cartan’s Moving Frames and a new compact notation from Geometrical Physics. The reconstructed course abandons the cross product—a toxic algebraic operation due to its failure to adhere to associativity. We minimize the use of vectors and replace them with rotation matrices. Sophomores learn to solve 3D Dynamics problems with as much ease as solving 2D problems. Typical problems include the precession of tops, gyroscopes, inertial devices to prevent ship roll at sea, and 3D robot and crane kinetics. A critical aspect of this new method is the consistency: the notation is the same for 3D and 2D problems, from advanced robotics to introductory dynamics, students learn the name notational method. The first objective of this paper presents the new mathematical approach to rigid body dynamics—it amounts to an introductory, yet simplified, lecture on a new method. The second objective presents assessment over a three-year period. In the first year, we taught using the old 2D vector-based approach. In the second year, we transitioned to the new method and compared student perceptions in the first two years. In the third year, the course was refined. The goal of this effort is to retain students in mechanical engineering by offering them a new view of the discipline, rather than simple pedagogical course interventions such as e-learning or flipped classrooms. The course content is delivered using the emerging visualization technology: WebGL. WebGL represents the future of the 3D web. It requires no downloads and no plugins. Students are directed to a web site where all images for the lectures are 3D and interactive. The animations run on cell phones, laptops and other mobile devices. It is the contention of this paper that modernizing the math will do more to reduce attrition than learning interventions. This new approach reduces conceptual difficulties that accompany 2D restrictions. It opens many questions on how students perceive 3D space and invites research into how exploiting more modern mathematical math may improve learning.
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YANG, YUN, and YANHUA YU. "AFFINE MAURER–CARTAN INVARIANTS AND THEIR APPLICATIONS IN SELF-AFFINE FRACTALS." Fractals 26, no. 04 (August 2018): 1850057. http://dx.doi.org/10.1142/s0218348x18500573.
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In this paper, we define the notion of affine curvatures on a discrete planar curve. By the moving frame method, they are in fact the discrete Maurer–Cartan invariants. It shows that two curves with the same curvature sequences are affinely equivalent. Conditions for the curves with some obvious geometric properties are obtained and examples with constant curvatures are considered. On the other hand, by using the affine invariants and optimization methods, it becomes possible to collect the IFSs of some self-affine fractals with desired geometrical or topological properties inside a fixed area. In order to estimate their Hausdorff dimensions, GPUs can be used to accelerate parallel computing tasks. Furthermore, the method could be used to a much broader class.
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VACARU, SERGIU I. "FINSLER AND LAGRANGE GEOMETRIES IN EINSTEIN AND STRING GRAVITY." International Journal of Geometric Methods in Modern Physics 05, no. 04 (June 2008): 473–511. http://dx.doi.org/10.1142/s0219887808002898.
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We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.
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Belova, Е., and O. Belova. "About an analogue of Neifeld’s connection on the space of centred planes with one-index basic-fibre forms." Differential Geometry of Manifolds of Figures, no. 50 (2019): 41–47. http://dx.doi.org/10.5922/0321-4796-2019-50-6.
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This research is realized by Cartan — Laptev method (with prolongations and scopes, moving frame and exterior forms). In this paper we consider a space П of centered m-planes (a space of all centered planes of the dimension m). This space is considered in the projective space n P . For the space П we have: dim П=n + (n – m)m. Principal fiber bundle is arised above it. The Lie group is a typical fiber of the principal fiber. This group acts in the tangent space to the П. Analogue of Neifeld’s connection with multivariate glueing is given in this fibering by Laptev — Lumiste way. The case when one-index forms are basic-fibre forms is considered. We realize an analogue of the Norden strong normalization of the space П by fields of the geometrical images: (n – m – 1)-plane which is not having the common points with a centered m-plane and (m – 1)-plane which is belonging to the m-plane and not passing through its centre. It is proved that the analog of the Norden strong normalization of the space of centered planes induces this connection.
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Alexander Jacobsen Jardim, Paulo, Jan Tore Rein, Øystein Haveland, Thorstein R. Rykkje, and Thomas J. Impelluso. "Modeling Crane-Induced Ship Motion Using the Moving Frame Method." Journal of Offshore Mechanics and Arctic Engineering 141, no. 5 (February 18, 2019). http://dx.doi.org/10.1115/1.4042536.
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A decline in oil-related revenues challenges Norway to focus on new types of offshore installations. Often, ship-mounted crane systems transfer cargo or crew onto offshore installations such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation using a new theoretical approach to dynamics: the moving frame method (MFM). The MFM draws upon Lie group theory and Cartan's moving frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied aspects of these mathematical tools, the notation presented here brings these methods together; it is accessible, programmable, and simple. In the MFM, the notation for multibody dynamics and single body dynamics is the same; for two-dimensional (2D) and three-dimensional (3D), the same. Most importantly, this paper presents a restricted variation of the angular velocity to use in Hamilton's principle. This work accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass. This research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley–Hamilton theorem and Rodriguez's formula reconstruct the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.
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Coiffier, Guillaume, and Etienne Corman. "The Method of Moving Frames for Surface Global Parametrization." ACM Transactions on Graphics, June 10, 2023. http://dx.doi.org/10.1145/3604282.
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This article introduces a new representation of surface global parametrization based on Cartan’s method of moving frames . We show that a system of structure equations , characterizing the local coordinates changes with respect to a local frame system, completely characterizes the set of possible cone parametrizations. The discretization of this system provably provides necessary and sufficient conditions for the existence of a valid mapping. We are able to derive a versatile algorithm for surface parametrization, allowing feature constraints and singularities. As the first structure equation is independent of the global coordinate system, we do not require prior knowledge of cuts or cone positions. So, a single non-linear least-square problem is enough to place quantized cones while minimizing a given distortion energy. We are therefore able to take full advantage of the link between the parametrization geometry and the topology of its cone metric to solve challenging constrained parametrization problems.
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N., Jamun Kumar, Bensingh Dhas, Arun R. Srinivasa, J. N. Reddy, and Debasish Roy. "A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames." Computer Methods in Applied Mechanics and Engineering, May 2022, 115094. http://dx.doi.org/10.1016/j.cma.2022.115094.
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Arbind, A., A. R. Srinivasa, and J. N. Reddy. "A Higher-Order Theory for Open and Closed Curved Rods and Tubes Using a Novel Curvilinear Cylindrical Coordinate System." Journal of Applied Mechanics 85, no. 9 (June 14, 2018). http://dx.doi.org/10.1115/1.4040335.
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In this study, the governing equation of motion for a general arbitrary higher-order theory of rods and tubes is presented for a general material response. The impetus for the study, in contrast to the classical Cosserat rod theories, comes from the need to study bulging and other deformation of tubes (such as arterial walls). While Cosserat rods are useful for rods whose centerline motion is of primary focus, here we consider cases where the lateral boundaries also undergo significant deformation. To tackle these problems, a generalized curvilinear cylindrical coordinate (CCC) system is introduced in the reference configuration of the rod. Furthermore, we show that this results in a new generalized frame that contains the well-known orthonormal moving frames of Frenet and Bishop (a hybrid frame) as special cases. Such a coordinate system can continuously map the geometry of any general curved three-dimensional (3D) structure with a reference curve (including general closed curves) having continuous tangent, and hence, the present formulation can be used for analyzing any general rod or pipe-like 3D structures with variable cross section (e.g., artery or vein). A key feature of the approach presented herein is that we utilize a non-coordinate “Cartan moving frame” or orthonormal basis vectors, to obtain the kinematic quantities, like displacement gradient, using the tools of exterior calculus. This dramatically simplifies the calculations. By the way of this paper, we also seek to highlight the elegance of the exterior calculus as a means for obtaining the various kinematic relations in terms of orthonormal bases and to advocate for its wider use in the applied mechanics community. Finally, the displacement field of the cross section of the structure is approximated by general basis functions in the polar coordinates in the normal plane which enables this rod theory to analyze the response to any general loading condition applied to the curved structure. The governing equation is obtained using the virtual work principle for a general material response, and presented in terms of generalized displacement variables and generalized moments over the cross section of the 3D structure. This results in a system of ordinary differential equations for quantities that are integrated across the cross section (as is to be expected for any rod theory).
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Erdoğdu, Melek, and Ayşe Yavuz. "On Backlund transformation and motion of null Cartan curves." International Journal of Geometric Methods in Modern Physics 19, no. 01 (November 18, 2021). http://dx.doi.org/10.1142/s0219887822500141.
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The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]
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Flatlandsmo, Josef, Torbjørn Smith, Ørjan O. Halvorsen, and Thomas J. Impelluso. "Modeling Stabilization of Crane-Induced Ship Motion With Gyroscopic Control Using the Moving Frame Method." Journal of Computational and Nonlinear Dynamics 14, no. 3 (January 18, 2019). http://dx.doi.org/10.1115/1.4042323.
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This paper presents a new method in multibody dynamics and applies it to the challenge of stabilizing ship motion induced by onboard crane operations. Norwegian industries are constantly assessing new technologies for more efficient and safer production in the aquacultural, renewable energy, and oil and gas industries. They share a common challenge to install new equipment and transfer personnel in a safe and controllable way between ships, fish farms, and oil platforms. This paper deploys the moving frame method (MFM) to analyze the motion induced by a crane, yet controlled by a gyroscopic inertial device. We represent the crane as a simple two-link system that transfers produce and equipment to and from barges. We analyze how an inertial flywheel can stabilize the ship during the transfer. Lie group theory and the work of Elie Cartan are the foundations of the MFM. This, together with a restriction on the variation of the angular velocity used in Hamilton's principle, enables an effective way of extracting the equations of motion for an open-loop system. Furthermore, this work displays the results in three-dimensional (3D) on cell phones. The long-term results of this work lead to a robust 3D active compensation method for loading/unloading operations offshore. Finally, the simplicity of the analysis anticipates the impending time of artificial intelligence when machines, equipped with onboard central processing units and internet protocol addresses, are empowered with learning modules to conduct their operations.
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Murakami, Hidenori. "Development of an Active Curved Beam Model—Part I: Kinematics and Integrability Conditions." Journal of Applied Mechanics 84, no. 6 (April 18, 2017). http://dx.doi.org/10.1115/1.4036308.
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In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.
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Murakami, Hidenori. "Development of an Active Curved Beam Model—Part II: Kinetics and Internal Activation." Journal of Applied Mechanics 84, no. 6 (April 18, 2017). http://dx.doi.org/10.1115/1.4036317.
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Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.
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Бубякин, И. В. "MIXED MULTISCALE FINITE ELEMENT METHOD FOR PROBLEMS IN PERFORATED MEDIA WITH INHOMOGENEOUS." Журнал «Математические заметки СВФУ», no. 4 (March 23, 2018). http://dx.doi.org/10.25587/svfu.2018.4.11312.
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Статья посвящена дифференциальной геометрии подмногообразий многообразий G(m, n), m-мерных плоскостей проективного пространства $P^n$, содержащих конечное число торсов. Для исследования таких подмногообразий используется грассманово отображение многообразия G(m, n) m-мерных плоскостей проективного пространства $P^n$ на (m+1)(n−m)-мерное алгебраическое многообразие $\Omega(m, n)$ пространства $P^N$, где $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. Это отображение в сочетании с методом внешних форм Э. Картана и методом подвижного репера позволило определить зависимость строения изучаемых многообразий и конфигурации (m − 1)-мерных характеристических плоскостей и (m+ 1)-мерных касательных плоскостей торсов, принадлежащих рассматриваемым многообразиям. We consider the projective differential geometry of m-dimensional plane submanifolds of manifolds G(m, n) in projective space $P^n$ containing a finite number of developable surfaces. To study such submanifolds we use the Grassmann mapping of manifolds G(m, n) of m-dimensional planes in projective space $P^n$ to $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m, n)$ of space $P^N$, where $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. This mapping combined with the method of external Cartan’s forms and moving frame method made it possible to determine the dependence of considered manifolds structure and the configuration of the (m − 1)-dimensional characteristic planes and (m + 1)-dimensional tangential planes of developable surfaces that belong to considered manifolds.