Surfaces of Finite III-type in the Euclidean 3-Space

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Then, we introduce the finite Chen type surfaces of revolution with respect to the third fundamental form which Gauss curvature never vanishes.

Surfaces of Finite III-type

In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces with respect to the third fundamental form of the surface. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.

About grid generation in constructions bounded by the surfaces of revolution

For constructions bounded by the surfaces of revolution, structured grid generation technique is presented. Its technology has been elaborated within the variational approach for constructing optimal grids satisfying optimality criteria: closeness of grids to uniform ones, closeness of grids to orthogonal ones and adaptation to a given function. Grid generation has been designed for numerical solution of the differential equations modeling the vortex processes of multi-component hydrodynamics. In the technology, the three-dimensional construction in which it is required to construct a grid is represented in the form of the curvilinear hexahedron defining its configuration. The specific feature of the required configurations is that some faces of a curvilinear hexahedron lie in one plane and along edges of adjoining faces grid cells degenerate into prisms. Grid generation in the considered constructions has started to be developed by the elaboration of algorithms for the volume of revolution which has become the basic construction. The basic construction is obtained by the rotation through 180? around the axis of a generatrix consisting of straight line segments, arcs of circles and ellipses. Then the deformed volumes of revolutions are considered along with the generalizations of the volume of revolution which represent constructions obtained by the surfaces of revolution with parallel axis of rotation. The aim of the further development of the technology is to consider more and more complicated constructions and elaborate the technology for them. In the presentation, the current state of the development of the technology is given. Examples of generated grids are supplied.

Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by . First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.

PARAMETRIC BENDING VIBRATIONS OF ELASTIC STRUCTURES WITH VARIABLE SECTIONS ON VIBRATION MOUNTINGS

The article is devoted to the study of the oscillatory motion of an elastic structure with variable sections on rolling bearings with straightened surfaces, which simulates a number of technical solutions, which has received its practical embodiment in the problem of ensuring the seismic resistance of building structures and vibration isolation of massive bodies. Equations of motion of an elastic structure with variable sections on rolling bearings bounded by high-order surfaces of revolution are obtained. The resonant modes of parametric vibrational motion of an elastic structure with variable sections are investigated using the Ritz variational method. In the resonant zone of the oscillatory movements of the base, the movement of the elastic structure will be small, and when the natural frequencies of the elastic structure coincide with the frequency of the disturbance, the phenomenon of resonance will appear. The amplitude of the parametric disturbance of the resonant zones of bending vibrations of elastic structures expands with increasing The amplitude-frequency characteristics of parametric oscillations depend on the parameters of the structure. An increase in the parameters of the base of wedge-shaped elastic structures leads to a shift of the resonance curves towards an increase in the disturbance frequencies