scholarly journals Kinematics of a screw linkage

2021 ◽  
Vol 3 (12) ◽  
Author(s):  
Jing-Shan Zhao ◽  
Song-Tao Wei ◽  
Junjie Ji
Keyword(s):  
Angular Velocity ◽  
Linear Velocity ◽  
Kinematics Analysis ◽  
First Order ◽  
Definition Of

AbstractThis paper proposes a kinematics methodology in twist coordinates for screw linkages. Based on the definition of a twist, both the angular velocity of a link and the linear velocity of a point on it may be explicitly represented in twist coordinates. Through integration on the twist solution numerically or analytically, we may obtain the displacements. By differential or numerical differential interpolation of the twist, we can find the accelerations of the link. The most outstanding advantage of this kinematic algorithm is that only the numerical differential interpolation of the first order is required to calculate the acceleration while only the first order integration of the twist is enough to compute the displacement. This merit makes it particularly fit for developing programmes to accomplish the kinematics analysis of a spatial linkage.

Kinematics of a planar slider-crank linkage in screw form

2021 ◽  
pp. 095440622110207
Author(s):  
Jing-Shan Zhao ◽  
Songtao Wei ◽  
Junjie Ji
Keyword(s):  
Angular Velocity ◽  
Inverse Kinematics ◽  
Parallel Mechanism ◽  
Numerical Algorithms ◽  
Forward Kinematics ◽  
End Effector ◽  
First Order ◽  
Forward And Inverse Kinematics ◽  
Definition Of ◽  
Inverse Velocity

This paper investigates the forward and inverse kinematics in screw coordinates for a planar slider-crank linkage. According to the definition of a screw, both the angular velocity of a rigid body and the linear velocity of a point on it are expressed in screw components. Through numerical integration on the velocity solution, we get the displacement. Through numerical differential interpolation of velocity, we gain the acceleration of any joint. Traditionally, position and angular parameters are usually the only variables for establishing the displacement equations of a mechanism. For a series mechanism, the forward kinematics can be expressed explicitly in the variable of position parameters while the inverse kinematics will have to resort to numerical algorithms because of the multiplicity of solution. For a parallel mechanism, the inverse kinematics can be expressed explicitly in the variable of position parameters of the end effector while the forward kinematics will have to resort to numerical algorithms because of the nonlinearity of system. Therefore this will surely lead to second order numerical differential interpolation for the calculation of accelerations. The most prominent merit of this kinematic algorithm is that we only need the first order numerical differential interpolation for computing the acceleration. To calculate the displacement, we also only need the first order numerical integral of the velocity. This benefit stems from the screw the coordinates of which are velocity components. The example of planar four-bar and five-bar slider-crank linkages validate this algorithm. It is especially suited to developing numerical algorithms for forward and inverse velocity, displacement and acceleration of a linkage.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals Solving puzzles in deformed JT gravity: phase transitions and non-perturbative effects

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Clifford V. Johnson ◽  
Felipe Rosso
Keyword(s):  
Phase Structure ◽  
Scalar Potential ◽  
String Equation ◽  
Two Dimensional ◽  
Leading Order ◽  
Classical Analysis ◽  
First Order ◽  
First Order Phase Transition ◽  
Definition Of ◽  
First Order Phase

Abstract Recent work has shown that certain deformations of the scalar potential in Jackiw-Teitelboim gravity can be written as double-scaled matrix models. However, some of the deformations exhibit an apparent breakdown of unitarity in the form of a negative spectral density at disc order. We show here that the source of the problem is the presence of a multi-valued solution of the leading order matrix model string equation. While for a class of deformations we fix the problem by identifying a first order phase transition, for others we show that the theory is both perturbatively and non-perturbatively inconsistent. Aspects of the phase structure of the deformations are mapped out, using methods known to supply a non-perturbative definition of undeformed JT gravity. Some features are in qualitative agreement with a semi-classical analysis of the phase structure of two-dimensional black holes in these deformed theories.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

scholarly journals Macroscopic degeneracy and order in the 3D plaquette Ising model

2015 ◽  
Vol 29 (20) ◽  
pp. 1550109 ◽  
Author(s):  
Desmond A. Johnston ◽  
Marco Mueller ◽  
Wolfhard Janke
Keyword(s):  
Low Temperature ◽  
Magnetic Order ◽  
Finite Size ◽  
Temperature Phase ◽  
Finite Size Scaling ◽  
First Order ◽  
Magnetic Order Parameter ◽  
First Order Transition ◽  
Definition Of ◽  
Low Temperature Phase

The purely plaquette 3D Ising Hamiltonian with the spins living at the vertices of a cubic lattice displays several interesting features. The symmetries of the model lead to a macroscopic degeneracy of the low-temperature phase and prevent the definition of a standard magnetic order parameter. Consideration of the strongly anisotropic limit of the model suggests that a layered, “fuki-nuke” order still exists and we confirm this with multi-canonical simulations. The macroscopic degeneracy of the low-temperature phase also changes the finite-size scaling corrections at the first-order transition in the model and we see this must be taken into account when analyzing our measurements.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

Modelling spatial reasoning systems with shape algebras and formal logic

1997 ◽  
Vol 11 (4) ◽  
pp. 273-285 ◽  
Author(s):  
Scott C. Chase
Keyword(s):  
Spatial Reasoning ◽  
Predicate Logic ◽  
Spatial Relations ◽  
Large Set ◽  
First Order ◽  
Reasoning Systems ◽  
Geometric Objects ◽  
Small Set ◽  
Definition Of ◽  
High Level

AbstractThe combination of the paradigms of shape algebras and predicate logic representations, used in a new method for describing designs, is presented. First-order predicate logic provides a natural, intuitive way of representing shapes and spatial relations in the development of complete computer systems for reasoning about designs. Shape algebraic formalisms have advantages over more traditional representations of geometric objects. Here we illustrate the definition of a large set of high-level design relations from a small set of simple structures and spatial relations, with examples from the domains of geographic information systems and architecture.

scholarly journals Angular Vectors in the Theory of Vectors

2017 ◽  
Vol 9 (5) ◽  
pp. 71
Author(s):  
Yevhen Mykolayovych Kharchenko
Keyword(s):  
Angular Velocity ◽  
Cross Product ◽  
Vector Form ◽  
Coordinate Vector ◽  
The Cross ◽  
Definition Of ◽  
Physical Quantities

The theory of angular vectors, which allows modelling of the properties of angular physical quantities, is considered. The meaning of the cross product of vectors was radically revised and changed. Formulas for finding torque and angular velocity in a coordinate-vector form with a correct mapping of their directions were deduced. Described definition of the inverse vector and its properties. The inversed vector allows us to perform vector division operations.

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