Academic literature on the topic 'Infinitesimal generator of a Lie group'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the lists of relevant articles, books, theses, conference reports, and other scholarly sources on the topic 'Infinitesimal generator of a Lie group.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Journal articles on the topic "Infinitesimal generator of a Lie group"
1
Li, Fu-zhi, Jia-li Yu, Yang-rong Li, and Gan-shan Yang. "Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness." Abstract and Applied Analysis 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/8183079.
Full textAbstract:
Based on classical Lie Group method, we construct a class of explicit solutions of two-dimensional ideal incompressible magnetohydrodynamics (MHD) equation by its infinitesimal generator. Via these explicit solutions we study the uniqueness and stability of initial-boundary problem on MHD.
2
Tryhuk, V., V. Chrastinová, and O. Dlouhý. "The Lie Group in Infinite Dimension." Abstract and Applied Analysis 2011 (2011): 1–35. http://dx.doi.org/10.1155/2011/919538.
Full textAbstract:
A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.
3
Alfaro, Ricardo, and Jim Schaeferle. "Coefficients of prolongations for symmetries of ODEs." International Journal of Mathematics and Mathematical Sciences 2004, no. 51 (2004): 2741–53. http://dx.doi.org/10.1155/s016117120430904x.
Full textAbstract:
Sophus Lie developed a systematic way to solve ODEs. He found that transformations which form a continuous group and leave a differential equation invariant can be used to simplify the equation. Lie's method uses the infinitesimal generator of these point transformations. These are symmetries of the equation mapping solutions into solutions. Lie's methods did not find widespread use in part because the calculations for the infinitesimals were quite lengthy, needing to calculate the prolongations of the infinitesimal generator. Nowadays, prolongations are obtained using Maple or Mathematica, and Lie's theory has come back to the attention of researchers. In general, the computation of the coefficients of the (n)-prolongation is done using recursion formulas. Others have given methods that do not require recursion but use Fréchet derivatives. In this paper, we present a combinatorial approach to explicitly write the coefficients of the prolongations. Besides being novel, this approach was found to be useful by the authors for didactical and combinatorial purposes, as we show in the examples.
4
Schürmann, Michael, and Michael Skeide. "Infinitesimal Generators on the Quantum Group SUq(2)." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 04 (October 1998): 573–98. http://dx.doi.org/10.1142/s0219025798000314.
Full textAbstract:
Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.
5
Cai, J. L., and F. X. Mei. "Conformal Invariance and Conserved Quantity of the Higher-Order Holonomic Systems by Lie Point Transformation." Journal of Mechanics 28, no. 3 (August 9, 2012): 589–96. http://dx.doi.org/10.1017/jmech.2012.67.
Full textAbstract:
AbstractIn this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.
6
Gaur, Manoj, and K. Singh. "Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation." International Journal of Differential Equations 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/4270724.
Full textAbstract:
We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.
7
Ndogmo, J. C. "Some Results on Equivalence Groups." Journal of Applied Mathematics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/484805.
Full textAbstract:
The comparison of two common types of equivalence groups of differential equations is discussed, and it is shown that one type can be identified with a subgroup of the other type, and a case where the two groups are isomorphic is exhibited. A result on the determination of the finite transformations of the infinitesimal generator of the larger group, which is useful for the determination of the invariant functions of the differential equation, is also given. In addition, the Levidecomposition of the Lie algebra associated with the larger group is found; the Levi factor of which is shown to be equal, up to a constant factor, to the Lie algebra associated with the smaller group.
8
Chepngetich, Winny. "The lie symmetry analysis of third order Korteweg-de Vries equation." Journal of Physical and Applied Sciences (JPAS) 1, no. 1 (November 1, 2022): 38–43. http://dx.doi.org/10.51317/jpas.v1i1.299.
Full textAbstract:
This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.
9
Ray, S. Saha. "Painlevé analysis, group invariant analysis, similarity reduction, exact solutions, and conservation laws of Mikhailov–Novikov–Wang equation." International Journal of Geometric Methods in Modern Physics 18, no. 06 (March 26, 2021): 2150094. http://dx.doi.org/10.1142/s0219887821500948.
Full textAbstract:
In this paper, for the study of integrability, symmetry analysis, group invariant solutions and conservation laws, the Mikhailov–Novikov–Wang equation is considered. Firstly, Painlevé analysis is being employed to study the integrability properties for the considered equation so as to check the possibility that this equation passes the Painlevé test. Secondly, Lie group analysis is studied for finding the symmetries by using Lie classical group analysis method and to obtain its symmetry group, infinitesimal generator, Lie algebra commutation table, and similarity reductions. The vector fields and the symmetry reduction of this equation are calculated with the aid of Lie symmetry analysis. From the similarity reduction equation, some explicit exact solutions are derived. Finally, using the new conservation theorem proposed by Ibragimov [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007) 311–328], the conservation laws of the aforesaid equation have been constructed.
10
Tam, Honwah, Yufeng Zhang, and Xiangzhi Zhang. "New Applications of a Kind of Infinitesimal-Operator Lie Algebra." Advances in Mathematical Physics 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/7639013.
Full textAbstract:
Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including (1+1) and (2+1) dimensions.
Dissertations / Theses on the topic "Infinitesimal generator of a Lie group"
1
Ouknine, Anas. "Μοdèles affines généralisées et symétries d'équatiοns aux dérivés partielles." Electronic Thesis or Diss., Normandie, 2024. http://www.theses.fr/2024NORMR085.
Full textAbstract:
Cette thèse se consacre à étudier les symétries de Lie d'une classe particulière d'équations différentielles partielles (EDP), désignée sous le nom d'équation de Kolmogorov rétrograde. Cette équation joue un rôle essentiel dans le cadre des modèles financiers, notamment en lien avec le modèle de Longstaff-Schwartz, qui est largement utilisé pour la valorisation des options et des produits dérivés.Dans un contexte plus générale, notre étude s'oriente vers l'analyse des symétries de Lie de l'équation de Kolmogorov rétrograde, en introduisant un terme non linéaire. Cette généralisation est significative, car l'équation ainsi modifiée est liée à une équation différentielle stochastique rétrograde et progressive (EDSRP) via la formule de Feynman-Kac généralisée (non linéaire). Nous nous intéressons également à l'exploration des symétries de cette équation stochastique, ainsi qu'à la manière dont les symétries de l'EDP sont connectées à celles de l'EDSRP.Enfin, nous proposons un recalcul des symétries de l'équation différentielle stochastique rétrograde (EDSR) et de l'EDSRP, en adoptant une nouvelle approche. Cette approche se distingue par le fait que le groupe de symétries qui opère sur le temps dépend lui-même du processus Y, qui constitue la solution de l'EDSR. Cette dépendance ouvre de nouvelles perspectives sur l'interaction entre les symétries temporelles et les solutions des équationsThis thesis is dedicated to studying the Lie symmetries of a particular class of partialdifferential equations (PDEs), known as the backward Kolmogorov equation. This equa-tion plays a crucial role in financial modeling, particularly in relation to the Longstaff-Schwartz model, which is widely used for pricing options and derivatives.In a broader context, our study focuses on analyzing the Lie symmetries of thebackward Kolmogorov equation by introducing a nonlinear term. This generalization issignificant, as the modified equation is linked to a forward backward stochastic differ-ential equation (FBSDE) through the generalized (nonlinear) Feynman-Kac formula.We also examine the symmetries of this stochastic equation and how the symmetriesof the PDE are connected to those of the BSDE.Finally, we propose a recalculation of the symmetries of the BSDE and FBSDE,adopting a new approach. This approach is distinguished by the fact that the symme-try group acting on time itself depends also on the process Y , which is the solutionof the BSDE. This dependence opens up new perspectives on the interaction betweentemporal symmetries and the solutions of the equations
2
Adamo, Maria Stella. "Representable functionals and derivations on Banach quasi *-algebras." Doctoral thesis, Università di Catania, 2019. http://hdl.handle.net/10761/4117.
Full textAbstract:
Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}).
Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \cite{Bag1,Bag4,Bag5,btt_meas}).
The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \textit{fully representability} and \textit{*-semisimplicity}, appeared in the literature in \cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \textit{positivity condition}. This is shown in \cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable.
The aforementioned result about sesquilinear forms allows one to select {\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics.
Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \textit{tensor products} (see \cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple).
Several examples are investigated from the point of view of Banach quasi *-algebras.
3
Fredericks, E. "Conservation laws and their associated symmetries for stochastic differential equations." Thesis, 2009. http://hdl.handle.net/10539/6980.
Full textAbstract:
The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For
example, in mathematics of finance, its use has given insights into the relationship between call options
and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility
has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation
of experimental psychology, it has been used to build random fluctuations into deterministic
models which model the dynamics of repetitive movements in humans.
Finding the quadrature for these integrals using continuous groups or Lie groups has to take families
of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus
Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero
[1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks
and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2]
and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6].
Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner
processes after the application of infinitesimal transformations. The determining equations for first-order
SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic.
Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context
both the formal and intuitive understanding of how to construct these symmetries has led to seemingly
disparate results. The impact of Lie point-symmetries on the stock market, population growth and
weather SODE models, for example, will not be understood until these different results are reconciled as
has been attempted here.
Extending the symmetry generator to include the infinitesimal transformation of the Wiener process
for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of
this work leads to an intuitive understanding of the random time change formulae in the context of Lie
point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it
formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of
these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature.
SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety
and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on
both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised
approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus
context.
A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral
context is pursued as well. The basis of this construction relies on Lie bracket relations on both the
instantaneous drift and diffusion operators.
Book chapters on the topic "Infinitesimal generator of a Lie group"
1
Iliopoulos, J., and T. N. Tomaras. "Elements of Classical Field Theory." In Elementary Particle Physics, 24–34. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780192844200.003.0003.
Full textAbstract:
The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.
Conference papers on the topic "Infinitesimal generator of a Lie group"
1
Pokas, S., and I. Bilokobylskyi. "Lie group of the second degree infinitesimal conformal transformations in a symmetric Riemannian space of the first class." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 13th International Hybrid Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’21. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0100808.
Full text
2
Rico, J. M., J. J. Cervantes, A. Tadeo, J. Gallardo, L. D. Aguilera, and C. R. Diez. "Infinitesimal Kinematics Methods in the Mobility Determination of Kinematic Chains." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86489.
Full textAbstract:
In recent years, there has been a good deal of controversy about the application of infinitesimal kinematics to the mobility determination of kinematic chains. On the one hand, there has been several publications that promote the use of the velocity analysis, without any additional results, for the determination of the mobility of kinematic chains. On the other hand, the authors of this contribution have received several reviews of researchers who have the strong belief that no infinitesimal method can be used to correctly determine the mobility of kinematic chains. In this contributions, it is attempted to show that velocity analysis by itself can not correctly determine the mobility of kinematic chains. However, velocity and higher order analysis coupled with some recent results about the Lie algebra, se(3), of the Euclidean group, SE(3), can correctly determine the mobility of kinematic chains.
3
Lerbet, Jean. "Stability of Singularities of a Kinematical Chain." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84126.
Full textAbstract:
Since few years, the geometry of singularities of kinematical chains is better known. Using both Lie group theory and concepts of differential or analytic geometry, we already classified these singularities according to the nature of the function f associated to the chain (f is a product of exponential mappings of a Lie group). In the most general case, the tangent cone at a singularity has been explicitely given. Here, a different (and more difficult) aspect of the problem is studied. The concrete realisation of a kinematical chain is never perfect. That means that the vectors of the Lie agebra defining the function f are not exactly those of the chain: they are deformed. What happens for the singularities in this case? Are they remaining or do they disappear during the deformation? First, the mechanical problem is analysed as this of the stability of the fuction f and mathematical tools concerning stable mappings are given. Stability of a mapping means that the orbit of f under the action of diffeomorphisms in the source and in the target is an open set and its infinitesimal equivalent formulation is noted the inf-stability. Then we prove that the set of singularities of first class itself is a sbmanifold and we analyse a condition of normal crossing of the restriction of f to its manifold of singularities. Applying a result of the general theory, the stability of f is analysed.
4
Zhang, Liping, and Jian S. Dai. "Genome Reconfiguration of Metamorphic Manipulators Based on Lie Group Theory." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49906.
Full textAbstract:
This paper investigates reconfiguration which was induced by topology change as a typical character of metamorphic mechanisms in a way analogous to the concept of genome varation in biological study. Genome is the full complement of genetic information that an organism inherits from its parents, espercially the set of genes they carry. Genome variation is to study the change and variation of this complement with genetic information and genes connectivity and is analogous to mechanisms reconfiguration of metamorphic mechanisms. Metamorphic mechanisms with reconfigurable topology are usually changing their configurations and varying mobility in accordance with different sub-working phase functions. The built-in spatial biological modules are for the first time compiled and introduced in this paper based on metamorphic building blocks in the form of metamorphic cells and associated inside break-down parts as the metamorphic genes for metamorphic bio-modeling as genome. The gene sequencing labels the genetic structure composition principle of the metamorphic manipulators. The bio-inspired mechanism configuration evolution is further introduced in this paper motivated by biological concept to metamorphic characteristics as different sub-phase working mechanisms gradually change and develop into different forms in a particular situation and over a period of time, as an evolutionary process of topological change that takes place over several motion phases during which a taxonomic group of organisms showing the change of their physical characteristics. Moreover, the proposed genetic structure composition principle in metamorphic manipulators leads to the development of module evolution and genetic operations based on the displacement subgroup algebraic properties of the Lie group theory. The topology transformations can further be simulated for configuration evolution and depicted with the genetic growth and degeneration in the living nature. Genome sequential reconfiguration for metamorphic manipulators promises to be mapped from degenerating the source generator to multiple sub-phase configurations. Evolution design illustrations are given to demonstrate the concept and principles.
5
Norbach, Alexandra, Kotryna Bedrovaite Fjetland, Gina Vikum Hestetun, and Thomas J. Impelluso. "Gyroscopic Wave Energy Generator for Fish Farms and Rigs." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86188.
Full textAbstract:
Norway has an opportunity to harvest ocean wave energy through gyroscopic precession as an alternative source of renewable energy, within practical limitations. This research assesses the energy extracted by gyroscopic wave energy generators and their use to provide supplementary power to fish farms and lighting on oilrigs. This project implements the Moving Frame Method (MFM) in dynamics to model the extracted power from a gyroscopic wave energy generator. The MFM leverages Lie Group Theory, Cartan’s moving frames and a new notation from the discipline of geometrical physics. Continuing, the Principle of Virtual Work extracts the equations of motion from the structure of the Special Orthogonal Group. However, the MFM supplements its analysis with a novel application of the restriction on the variations of the angular velocities. This research extends previous work as follows: it accounts for motor torques, it opens a placeholder for buoyancy, and it solves the full 3D set of equations (without assuming negligible yaw). After showing how to obtain the suite of descriptive equations of motion, this project integrates them, however with a relatively simple integration scheme. To complete each step in the analysis, the rotation matrix is updated using the Cayley Hamilton Theorem and the Rodriguez formula. Finally, the results are displayed using the Web Graphics Library such that the actual numerical analysis and display happens on cell phones.
6
Lee, Chung-Ching, and Jacques M. Hervé. "New Schoenflies-Motion Manipulator Implementing Isosceles Triangle and Delassus Parallelogram." In ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/esda2014-20343.
Full textAbstract:
Schoenflies (X) motion is a 4D displacement Lie group including a spatial translation and any rotation whose axis is parallel to a given direction. Delassus parallelogram has four parallel screw (H) pairs with related pitches and the isosceles triangle is a special HHHP. After merging these two chains, an HHH-//-HHH generator of 2-DoF translation along a right helicoid is derived. It produces a 2-DoF motion mathematically modeled by a 2D submanifold of a 4D group of X-motion. Because of the product closure in an X-group, the 4-DoF generator with HHH-//-HHH loop serving as a subchain is revealed by adding two H pairs with axes parallel to fixed H axes. Parallel arrangement of two generators of the same X motion results in a new Schoenflies-motion manipulator with hybrid topology for 4-DoF pick-and-place operations. Four fixed H pairs (two double Hs) can actuate this manipulator and the two coaxial Hs must have distinct pitches. In addition, the possible design choices of special architectures are introduced for practical applications. Computer simulations of the new parallel manipulator with Schoenflies motion verify the effectiveness.
7
Lee, Chung-Ching, and Jacques M. Hervé. "Homokinetic Shaft-Coupling Mechanisms via Double Schoenflies-Motion Generators." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34517.
Full textAbstract:
The paper begins with introducing the 5-dimensional (5D) double Schoenflies-motion (X-X motion) set employing the group product of two 4D X-motion subgroups of displacements. Two families of primitive X-X motion generators are briefly outlined. Then, the geometric constraints for homokinetic transmission via Lie-group-algebraic properties of the displacement set are established. After that, using the described mechanical generators of X-X motion as the basic building cell, we geometrically generate two major families of homokinetic shaft-coupling mechanisms characterized by a subchain with a mechanical generator of 5D X-X motion set of displacement. The obtained constant-velocity shaft couplings (CVSC) are isoconstrained linkages with two parallel shaft axes, which will be less sensitive to manufacture errors. In addition, by means of the reordering method for displacement group compositions, more CVSC mechanisms can be further obtained. The simple or special findings stemming from the proposed general architectures are presented for the potential applications too.
8
Changizi, M. Amin, Ali Abolfathi, and Ion Stiharu. "MEMS Wind Speed Sensor: Large Deflection of Curved Micro-Cantilever Beam Under Uniform Horizontal Force." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-50560.
Full textAbstract:
Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.
9
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.
Full textAbstract:
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.
10
Taves, Jay, Alexandra Kissel, and Dan Negrut. "Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 1: Description of an Index-3 DAE Solution Approach." In ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/detc2021-72057.
Full textAbstract:
Abstract In rigid multibody dynamics simulation using absolute coordinates, a choice must be made in relation to how to keep track of the attitude of a body in 3D motion. The commonly used choices of Euler angles and Euler parameters each have drawbacks, e.g., singularities, and carrying along extra normalization constraint equations, respectively. This contribution revisits an approach that works directly with the orientation matrix and thus eschews the need for generalized coordinates used at each time step to produce the orientation matrix A. The approach is informed by the fact that rotation matrices belong to the SO(3) Lie matrix group. The numerical solution of the dynamics problem is anchored by an implicit first order integration method that discretizes, without index reduction, the index 3 Differential Algebraic Equations (DAEs) of multibody dynamics. The approach handles closed loops and arbitrary collections of joints. Our main contribution is the outlining of a systematic way for computing the first order variations of both the constraint equations and the reaction forces associated with arbitrary joints. These first order variations in turn anchor a Newton method that is used to solve both the Kinematics and Dynamics problems. The salient observation is that one can express the first order variation of kinematic quantities that enter the kinematic constraint equations, constraint forces, external forces, etc., in terms of Euler infinitesimal rotation vectors. This opens the door to a systematic approach to formulating a Newton method that provides at each iteration an orthonormal rotation matrix A. The Newton step calls for repeatedly solving linear systems of the form Gδ = e, yet evaluating the iteration matrix G and residuals e is inexpensive, to the point where in the Part 2 companion contribution the proposed formulation is shown to be two times faster for Kinematics and Dynamics analysis when compared to the Euler parameter and Euler angle approaches in conjunction with a set of four mechanisms.