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Academic literature on the topic 'Cartan's moving frames'

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Published: 6 September 2023

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Journal articles on the topic "Cartan's moving frames"

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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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Dissertations / Theses on the topic "Cartan's moving frames"

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Morellini, Umberto. "Il teorema di Gauss-Bonnet e il teorema di Morse." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16393/.

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Le finalità di questa trattazione sono la presentazione e la dimostrazione del Teorema di Gauss-Bonnet e, conseguentemente, del Teorema di Morse. Si tratta di risultati fondamentali di geometria differenziale che evidenziano vincoli di rigidità per campi vettoriali e funzioni differenziabili definiti su varietà, dovuti alla natura topologica di quest'ultima. L'impostazione di questo elaborato prevede preliminarmente la presentazione del metodo dei "moving frames" di Cartan, il quale, avvalendosi di strumenti quali le forme differenziali, permette lo studio della geometria locale di una varietà e, in seguito, consente di ricavare quei risultati di geometria differenziale citati in precedenza e che danno il titolo alla tesi stessa.
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Dhas, Bensingh P. "On a Few Non-Classical Perspectives in Solid Mechanics." Thesis, 2021. https://etd.iisc.ac.in/handle/2005/5076.

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Postulates in solid mechanics can be broadly classified into three groups describing the geometry of a configuration, mechanical equilibrium and thermodynamics of the deformation process. The geometry defined on a configuration provides us with tools to analyse deformation. The strain tensor in three dimensional elasticity and the curvature tensor encountered in a Kirchhoff type shell theory are consequences of the geometric hypotheses. The first part of this thesis (Chapters 2-4 to wit) uses a class of affine connections to study the deformation of elastic continua, shells and continua with point defects which are representatives of theories with flat, extrinsic and intrinsic geometries. As a first case, we study non-linear elasticity whose geometry is Euclidean. The mainstay of our approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as primal. Such a treatment invites compatibility equations so that the base space (configurations of the elastic body) can be realised as a subset of an Euclidean space; Cartan's method of moving frames and the associated structure equations establish this compatibility. The geometric understanding of stress as a co-vector valued 2-form fits squarely within this program. We also show that for a hyperelastic solid, a relationship akin to the Doyle-Eriksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the Hu-Washizu (HW) functional in terms of frames and differential forms. We also show that the compatibility of deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional. Following the same spirit, we reformulate the kinematics of Kirchhoff shells using the theory of moving frames. This reformulation permits us to treat the deformation and geometry of the shell as two equally important but distinct entities. The structure equations which represent the familiar torsion and curvature free conditions (of the ambient space) are used to combine deformation and geometry in a compatible manner. From such a point of view, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of dislocations and disclinations). Using the proposed framework, we solve a simple boundary value problem and thus demonstrate the importance of moving frames. We then study the thermo-mechanics of a solid body with point defects using Weyl geometry. Here, we assume the geometries of reference and deformed configurations to be of the Weyl type. The Weyl one-form is introduced as an additional degree of freedom that determines ratios of lengths between different tangent spaces. This one-form prevents the metric to be compatible with the connection (in the Riemannian sense). We exploit this incompatibility to characterize metrical defects in the material body. When such a defective body undergoes temperature changes, additional incompatibilities appear and interact with the defects. This interaction is modelled using the Weyl transform, which keeps the Weyl connection invariant whilst changing the non-metricity of the configuration. An immediate consequence of adopting the Weyl connection for a configuration is that the critical points of the stored energy functional are shifted. We relate this change in the equilibrium point to the residual stresses developed in the body due to point defects. In order to relate stress and strain in our non-Euclidean setting, use is made of the Doyle-Ericksen formula, which is interpreted as a relation between the intrinsic geometry of the body and the stresses developed. Thus the Cauchy stress is postulated to be conjugate to the Weyl transformed metric tensor of the deformed configuration. The evolution equation for the Weyl one-form and temperature are arrived using the laws of thermodynamics. Using this model, the self-stress generated by a point defect is calculated and compared with linear elastic solution. We also obtain conditions on the defect distribution (Weyl one-form) that render a thermo-mechanical deformation stress-free. Using this condition, specific stress-free deformation profiles for a class of prescribed temperature changes are computed. In the second part, (Chapters 6 \& 7), we discuss a phase-field approach to delamination. Pseudo-ductility encountered in laminated ceramic composites is our first focus. By pseudo-ductility, we mean the engineered ductility of ceramic laminated composites by a strategic placement of weak interfaces. The phase-field based brittle fracture model has the advantage of seamlessly modelling the growth of cracks at the interface and lamina. Using a finite element implementation of this brittle fracture model, we compute the response of laminated ceramics composites and study the influence of geometric and material properties. Important geometric properties studied in this work are lamina thickness, lamina thickness scheme and interface geometry. Material properties like Young's modulus and critical energy release rate of the interface are also studied. From these studies, the importance of weak interfaces and key interface material properties which influence the pseudo-ductility of ceramic laminated composites are established. We than apply the phase-field modelling to study delamination in orthotropic laminated composites. Here, we understand the crack phase-field as an internal variable with a thermodynamic origin. We use this modelling approach to simulate delamination in mode I, mode II and another such problem with multiple initial notches. The present approach is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.
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Books on the topic "Cartan's moving frames"

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Ivey, Thomas A. Cartan for beginners: Differential geometry via moving frames and exterior differential systems. Providence, R.I: American Mathematical Society, 2003.

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Deruelle, Nathalie, and Jean-Philippe Uzan. Differential geometry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0004.

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This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.
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From Frenet to Cartan: The Method of Moving Frames. American Mathematical Society, 2017.

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Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems. American Mathematical Society, 2016.

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Cartan for beginners : differential geometry via moving frames and exterior differential systems. American Mathematical Society, 2003.

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Book chapters on the topic "Cartan's moving frames"

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Samari, Babak, Tristan Aumentado-Armstrong, Gustav Strijkers, Martijn Froeling, and Kaleem Siddiqi. "Denoising Moving Heart Wall Fibers Using Cartan Frames." In Medical Image Computing and Computer Assisted Intervention − MICCAI 2017, 672–80. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-66182-7_77.

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Dhas, Bensingh, and Debasish Roy. "A Geometrically Inspired Model for Kirchhoff Shells with Cartan’s Moving Frames." In Lecture Notes in Civil Engineering, 163–76. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-8138-0_14.

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Faugeras, Olivier. "Cartan's moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes." In Applications of Invariance in Computer Vision, 9–46. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/3-540-58240-1_2.

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"Cartan's Method of Moving Frames: Curvilinear Coordinates in ℝ3." In Fundamental Principles of Classical Mechanics, 91–101. WORLD SCIENTIFIC, 2014. http://dx.doi.org/10.1142/9789814551496_0009.

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Conference papers on the topic "Cartan's moving frames"

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Murakami, H. "A Moving Frame Method for Multi-Body Dynamics." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-62833.

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Élie Cartan’s moving frame method, developed in differential geometry, has been applied to multi-body dynamics to derive equations of motion. The explicit representation of a body-attached orthonormal coordinate basis and its origin, referred to as a moving frame, enables the usage of the special orthogonal group, SO(3), and the special Euclidean group, SE(3), to describe kinematics and kinetics of interconnected bodies by joints and force elements. The moving frame representation using Theodore Frankel’s compact notation is adopted to alleviate theoretical complexities of the Lie group theory to which SO(3) and SE(3) belong. For the variational formulation, the restricted variation of angular velocity is derived for the moving frame method. Starting from two connected rigid bodies, it will be demonstrated that the explicit representation of moving frames renders straight-forward symbolic computations of three-dimensional kinematics and kinetics. This simplicity eliminates errors in computing analytical expressions for kinematic and kinetic variables and streamlines the coding effort for numerical solution. For controller design, if the degrees-of-freedom is small, the moving frame method allows a straight-forward derivation of equations of motion in analytical form.
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Rykkje, Thorstein R., Tord Tørressen, and Håvard Løkkebø. "Modelling Buoy Motion at Sea." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10437.

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Abstract This project creates a model to assess the motion induced on a buoy at sea, under wave conditions. We use the Moving Frame Method (MFM) to conduct the analysis. The MFM draws upon concepts and mathematics from Lie group theory — SO(3) and SE(3) — and Cartan’s notion of Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. This work accounts for the masses and geometry of all components and for buoyancy forces and added mass. The resulting movement will be displayed on 3D web pages using WebGL. Finally, the theoretical results will be compared with experimental data obtained from a previous project done in the wave tank at HVL.
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Sværen, Terje, Bård Inge Nygård, and Thomas J. Impelluso. "A Framework for Spatial 3D Collision Models: Theory and Validation." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-72981.

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Abstract This project puts forth a framework for spatial collision models for bodies. The framework leverages a new method in dynamics to calculate the relevant post-collision trajectory in 3D space. The new method exploits Cartan’s notion of moving frames and places frames of reference on all moving (and colliding) bodies. Next, it exploits Lie group theory SO(3) and its associated algebra, so(3), to relate such frames to each other. Finally, it exploits a compact notation to streamline the mathematics. This project presents a numerical validation of the results using MSC Adams and compares the rotation and translation values of the bodies with those found using the new method. ThreeD web pages display the results using WebGL.
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Rykkje, Thorstein R., Daniel Leinebø, Erlend Sande Bergaas, Andreas Skjelde, and Thomas J. Impelluso. "Inspiring Learning: Assessment of Friction in a Real-World Model Using the Moving Frame Method in Dynamics." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86189.

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This project conducts research in energy dissipation. It also demonstrates the power of the new Moving Frame Method (MFM) in dynamics to inspire undergraduate students to embark on research in engineering. The MFM is founded on Lie Group Theory to model rotations of objects, Cartan’s moving frames to model the change of a frame in terms of the frame, and a new notation from the discipline of geometrical physics. The MFM presents a consistent notation for single bodies, linked systems and robotics. This work demonstrates that this new method is accessible by undergraduate students. The MFM structures the equations of motion on the Special Euclidean Group and the Principle of Virtual work. A restriction on the virtual angular velocities to enable variational methods empowers the method. This work implements an explicit fourth order Runge-Kutta numerical integration scheme. It assesses the change in mechanical energy. In addition, this work researches the energy losses due to friction in a system of linked rigid bodies. This research also builds the physical hardware and compares the theory and experiment using 3D visualization. The authors built the structure to observe the actual motion and approximate the energy loss functions. This project demonstrates the power of WebGL to supplement analyses with visualization.
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Storaas, Torstein R., Kasper Virkesdal, Gitle S. Brekke, Thorstein Rykkje, and Thomas Impelluso. "Stabilizing Ship Motion With a Dual System Inertial Disk." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10250.

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Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. 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. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.
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Korsvik, Håkon B., Even S. Rognsvåg, Tore H. Tomren, Joakim F. Nyland, and Thomas J. Impelluso. "Dual Gyroscope Wave Energy Converter." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10266.

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Abstract This research models the energy extracted by gyroscopic wave energy converters. The goal is to assess the use of such devices to provide supplementary power to fish farms and lighting on oilrigs. This project implements the Moving Frame Method (MFM) in dynamics to model the power generated from a gyroscopic wave energy converter. The MFM leverages Lie Group Theory, Cartan’s moving frames and a new notation from the discipline of geometrical physics. This research extends previous work by incorporating two inertial disks to counter the inducement of yaw, and it improves the numerical integration scheme. Furthermore, this work makes use of a coherent data structure founded in the Special Euclidean Group, and it defines the initial disk spin as a prescribed variable. It accounts for the prescribed variables by modifying the equations of motion. Finally, it conducts an analysis of the generator moments. After obtaining the suite of descriptive equations of motion, this project integrates them using the Runge-Kutta method. Finally, a simplified 3D simulation is made using the Web Graphics Library to improve the readers’ intuitive understanding of the device.
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Rykkje, Thorstein R., Eystein Gulbrandsen, Andreas Fosså Hettervik, Morten Kvalvik, Daniel Gangstad, Torgeir Oliver Tislevoll, Stefan Aasebø, and Daniel Vatle Osberg. "Production and Analytics of a Multi-Linked Robotic System." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10434.

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Abstract This paper extends research into flexible robotics through a collaborative, interdisciplinary senior design project. This paper deploys the Moving Frame Method (MFM) to analyze the motion of a relatively high multi-link system, driven by internal servo engines. The MFM describes the dynamics of the system and enables the construction of a general algorithm for the equations of motion. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. 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. The result is a dynamic 3D analytical model for the motion of a snake-like robotic system, that can take the physical sizes of the system and return the dynamic behavior. Furthermore, this project builds a snake-like robot driven by internal servo engines. The multi-linked robot will have a servo in each joint, enabling a three-dimensional movement. Finally, a test is performed to compare if the theory and the measurable real-time results match.
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Nordvik, Andreas, Natalia Khan, Roberto Andrei Burcă, and Thomas J. Impelluso. "A Study of Roll Induced by Crane Motion on Ships: A Case Study of the Use of the Moving Frame Method." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-70111.

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This paper presents a new method in dynamics — The Moving Frame Method (MFM) — and applies it to analyze the roll, yaw and pitch of a ship at sea, as induced by an onboard moving crane. The MFM, founded on Lie Group Theory, Cartan’s Moving Frames and a compact notation from geometrical physics, enables this expedited extraction of the equations of motion. Next, the method deploys the power of the special Euclidean Group SE(3) and a restricted variation to be used in Hamilton’s Principle, to extract the equations of motion. The mathematical model is then simplified to get a clearer picture of the parameters that impact the motion of the crane. The equations of interest are numerically solved by using fourth order Runge-Kutta method to obtain the specific data for the motion induced by the crane. Then, The Cayley-Hamilton theorem is used to reconstruct the rotation matrix. To supplement the paper, a webpage is coded with a model of the crane and ship, to graphically visualize the motion in 3D. It is imperative to note that while there are many approaches to dynamics, the MFM presents a consistent method, from 2D to 3D, and across sub-disciplines. The simplification is what has enabled undergraduate students to undertake this project.
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Jardim, Paulo Alexander Jacobsen, Jan Tore Rein, Øystein Haveland, and Thomas J. Impelluso. "Modeling Crane Induced Ship Motion Using the Moving Frame Method." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86190.

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A decline in oil-related revenues challenges Norway to focus on new types of offshore installations and their maintenance. Often, ship-mounted crane systems transfer cargo or crew onto marine structures such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation. It analyzes the motion of a crane mounted on a ship using 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. This work extends a previous project that assumed many simplifications. It accounts for the masses and geometry of all components. This current approach also accounts interactive motor couples and prepares for buoyancy forces and added mass. The previous work used a symbolic manipulator, resulting in unwieldy equations. In this current phase, this research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley-Hamilton theorem and Rodriguez’s formula reconstructs the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. WebGL is a JavaScript API for rendering interactive 3D and 2D graphics within any compatible web browser without the use of plug-ins. This paper presents the results qualitatively as a 3D simulation. This research proves that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.
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Rezaei, Amir G., Shannon Lee, and Thomas J. Impelluso. "Application of Moving Frame Method to Solve 3D Dynamics Problems." In ASME 2016 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/imece2016-68000.

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Abstract:
This paper presents application of a new method to solve 3D Dynamics problems. Briefly: 1. The method uses the special orthogonal group, SO(3), and the special Euclidean group, SE(3), of the Lie Algebra. 2. The method uses Cartan’s Moving Frames 3. The method uses a new notation developed in the discipline of Geometrical Physics. The method makes 3D Dynamics easier than 2D. It offers a more efficient way to model dynamics of rigid bodies and a new approach to linked mechanisms. The new method is founded on rigorous math and avoids the historical ambiguities that accompany traditional methods (such as the vector cross product that produces pseudo-vectors). It is easily learned with the simple mathematical tools of matrix multiplication and second semester calculus. We plan to apply the method to two different 3D dynamics problems: 1. Holonomic and 2. Non-holonomic. The solutions using the new method will be compared to the traditional method in Dynamics.
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