DESIGN OF A COMPUTER-AIDED GEAR MANUFACTURING TOOL – RACK-SHAPED CUTTER

In this work, the calculation of Maag gear shaper cutter parameters is performed for spur gears with helical teeth in three variants – straight-tooth tool with machine offset, helical-tooth tool without machine offset and helical-tooth tool with machine offset. It is therefore a prerequisite that the manufactured involute gearing has helical teeth for each variant. The created CAD program is universal and can be used for construction in other combinations as well. The tool clamping angles on the tool holder, the cutting geometry of the cutter and the characteristic of the gearing are introduced into the calculation. The output of the work will be then calculated individual parameters of the Maag shaping cutter in the tooling system, necessary for its construction. The calculation are performed in program T-Flex CAD and the summary output is graphical 2D/3D representation of the rack-shaped cutter in its base, normal and side planes, always in a different design based on the change of the tool input data and gearing characteristic. The analysis of the tool´s involute profile was solved by vector calculus and matrices and by rotating the tool in the chosen coordinate systems. The specific calculation of the tool parameters will be solved using goniometric functions.

General Fractional Vector Calculus

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

Towards a unified theory of fractional and nonlocal vector calculus

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

Hybrid Computational Model to Assist in the Location of Victims Buried in the Tragedy of Brumadinho

This paper presents a hybrid computational model based on regression techniques, machine learning and physicomathematical algorithms developed for assistance in locating victims in the Brumadinho tragedy in 2019. The physicomathematical model, which provided results to help search teams, is based on integral and vector calculus, and fluid mechanics concepts. In addition, from data provided by the physicomathematical algorithm, two hybrid model were developed. One of them uses regression statistical and the other one uses support vector regression which is a type of machine learning. With good prospects of the advances in research, it is expected in future work, a more accurate model that can be used in other possible situations of dam-break. Moreover the model can be applied to situations involving computational fluid dynamics in general