Curvilinear Axes Research Articles

Response Characteristics of Tunnels with Curvilinear Axes under Obliquely Incident Lateral SV Seismic Waves

Response Characteristics of Tunnels with Curvilinear Axes under Obliquely Incident Lateral SV Seismic Waves

Математическое описание частного случая квазивращения фокуса эллипса вокруг эллиптической оси

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Математическое описание частного случая квазивращения фокуса эллипса вокруг эллиптической оси

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Математическое описание метода вращения точки вокруг криволинейной оси второго порядка

Previously, the method of rotating of flat geometric objects around curvilinear axes was described by us. The next step in the path of our research should be the development of methods for the automated creation of surfaces digital models obtained by the described rotation method. We have created models of surfaces, the axis and the forming curve of which are circles lying in the same plane. Several cases of mutual disposition for such circles were analyzed. Modeling was carried out using constructive techniques. Surfaces were created using the “surface by section” operation. The centers of such circular sections belong to the axis of rotation, if it is a circle. Using the special tools incorporated in the KOMPAS-3D program, we have cut the surfaces modeled in this way by planes, and obtained a number of flat sections. Taking into account the difficulties occurring during the study of such complex geometric objects by means of flat graphic constructions, as well as graphic computer modeling, we have realized the need to create a mathematical apparatus describing these objects’ shape. The required mechanism should be applicable to any pair of second-order curves interconnected as “axis — generatix”. We have considered an elementary example – the rotation of a point around a curve elliptical axis. In this paper a solution for the problem of finding a system of equations describing a set of point positions, which it will successively take when rotating around the elliptic axis, is presented. It is possible to apply a similar mathematical apparatus to axes having the form of other quadrics, for example, hyperbolas or parabolas, as well as to generatices consisting of more than one point, that is, to forming curves.

Laboratory research of the stress-strain state of the drill string in the local bend of the well

The object of research is the operation of the drill string in the local bend of the well. One of the most problematic aspects in this case is the establishment of its stress-strain state. This is very important information, which, in particular, is used to make decisions on the duration and possibility of operating the drill string in a given geological and technical conditions. Based on a critical literature review, physical modeling of the object of study was chosen to solve this problem. In particular, a special laboratory stand was designed, manufactured and tested, which provides: load model of the drill string axial tensile force and torque; modeling of curvilinear axes of wells with possible local excesses, based on the results of industrial inclinometry and profilometry; measurement of stresses and deformations of the model of the column and their interpretation to the values of the natural object. As a model of a drill string, a copper tube with a weighting agent was used to provide criterial similarity with a full-scale object. To measure normal stresses on a drill string model, strain gauging was used. At the same time, analog voltage values on strain gauges are digitized and transferred to a personal computer for further processing and interpretation. With the help of the proposed laboratory stand, experimental studies of the stress-strain state of a 127-mm drill string model in a local bend of well No. 10 of the Odesa deposit (Ukraine) were carried out. It has been established that the normal bending stresses that arise in this case can more than three times exceed the same values obtained without taking into account the presence of the specified curvature of the borehole axis. Thanks to the proposed experimental method, it is possible to study the deformations and stresses that occur in the drill strings during their work in wells with local axial kinks. There is no need to apply complex analytical transformations and algorithms. In addition, the model has the same physical nature as the object of study. This is an extremely weighty argument, especially in the study of complex cases of interaction between the drill string and the borehole walls.

Heat transfer investigations on methane-air premixed flame jet exiting from a circular nozzle and impinging over semi-cylindrical surfaces

The flame jet impingement on curved surfaces finds applications in various direct flame heating situations. The design of heating equipment demands the estimation of the surface temperature and of the convective heat flux to the target surface at steady state for various impingement parameters. The characterisation of flame jets in terms of Nusselt number and effectiveness aids the designer to numerically estimate the steady state surface temperature and heat flux by specifying the convective boundary condition with spatial distribution of reference temperature. The present study reports the heat transfer characteristics of single flame jet impinging on convex and concave surfaces of a semi-cylindrical target plate. The effects of mixture Reynolds number, equivalence ratio, curvature ratio and burner to plate spacing on the heat transfer behavior are investigated. The uniformity of the heat flux distribution and the overall thermal efficiency are determined from the local heat flux distributions. The flow fields along the curvilinear axes influence the overall thermal performance. The concave surface produces higher thermal efficiency and better uniformity of heat flux over the impingement surface as compared with the convex surface.

Nature and onset age of neotectonic regime in the northern core of Isparta Angle, SW Turkey

The Isparta Angle (IA) is a reverse Λ-shaped morphotectonic structure located to the north of Antalya Gulf in the Eastern Mediterranean Sea. It resulted from the northward curvature of the originally E–W-trending Tauride orogenic belt owing to the nappe emplacements and related clockwise and anti-clockwise rotations in a time period of Early Paleocene to Early Pliocene. The IA is included in the southwest Anatolian tensional neotectonic domain and characterized by a series of grabens and horsts bounded by active normal faults of dissimilar length and trend. The evolutionary history of the graben-horst system is episodic. It is evidenced by two graben fills. These are older and modern (younger) graben fills separated by an intervening angular unconformity. The modern graben fill is nearly flat-lying (non-deformed) whereas older graben fill was deformed into a series of anticlines and synclines with ENE-trending curvi-linear axes by a short-term compressive tectonic regime operated in NNW–SSE direction during Late Pliocene. The diagnostic structures taking a part in the development of grabens and shaping the northern section of the IA are the margin-boundary normal faults. They occur in numerous single and several fault zones displaying a basin ward facing step-like land shape. Most of fault segments, particularly the master faults, are active and have a capacity of creating destructive earthquakes with a magnitude (up to Mw = 7.0). This is evidenced by both the historical and instrumental period earthquakes. Both the focal mechanism solution of earthquakes and the stereographic plots of slip-plane data, measured on the active margin-boundary faults of various grabens comprising the IA, on the Schmidt lower hemisphere net obviously reveal that the IA is under the influence of the tensional neotectonic regime, not a compressive tectonic regime, i.e. the sinistral strike-slip shearing along the Pliny arc has not propagated yet onshore, and its commencement age is Early Quaternary.

A unified aberration theory for magnetic deflection systems with rectilinear or curvilinear axes

A unified deflection aberration theory has been further developed for magnetic deflection systems with curvilinear or rectilinear axes. By using variational method, primary-order deflection aberrations with respect to curvilinear or rectilinear axis can be universally calculated by means of gradient operations on eikonal (the function of optical length). Conventional pure-order rectilinear-axis theory can be derived from a curvilinear axis eikonal of mixed-order in the rectilinear limit. By introducing general aberration integration functions in terms of mathematical operators, all primary-order deflection aberrations have been explicitly expressed in compact formulae suitable for computer computations. It is to be expected that the unified deflection aberration theory may be useful in designing high-quality magnetic deflection systems for high-definition television color picture tubes.

Fluid flow in a rotating channel of square section

Laminar flow in a channel rotating about a transverse axis has been studied numerically [1–3] and analytically [4–7] at small Reynolds numbers. The drag coefficient of rotating channels with straight and curvilinear axes has been measured [4, 8, 9]. The present paper gives the results of an experimental investigation into the kinematics of water flow in a channel rotating with different intensities. The flow was visualized by means of hydrogen bubbles and a dye. A study was made of the process of flow separation in a rapidly rotating channel into a core with homogeneous velocity distribution in the direction parallel to the rotation axis and thin shear layers on the walls normal to this axis. The values of the dimensionless numbers were found that correspond to flow rearrangement accompanied by formation of longitudinally oriented vortex structures in the region of higher pressure, and also the values of the rotation parameter needed for the almost complete suppression of turbulence in the region of lower pressure. A general analysis is made of the forms of instability in the different regions of the flow and of the possible flow regimes in a rotating channel.

A natural numerical differential geometry scheme for a doubly curved shell element

The natural strain concept of Argyris is used to calculate are lengths and curvatures on each side of a doubly curved triangular shell element. These are able to be used directly to describe natural extensional, flexural and shear strains in the element. These are then transformed to orthogonal curvilinear strains on the shell surface, for which purpose direction cosines of the sides relative to curvilinear axes are estimated using simple but apparently accurate approximations based on the projection of the curved element on the plane defined by the vertex nodes.

A review of the structure and stratigraphy of the Moinian of Ardgour, Moidart and Sunart—Argyll and Inverness-shire

Synopsis The structural geometry is the result of four episodes of deformation (F 1 –F 4 ), with most major folds belonging to the second or third episodes. Major F 1 folds are probably absent (on stratigraphical evidence) but slides of the F 1 episode are not ruled out. Complex F 2 geometries—with curvilinear axes, and antiforms which turn into synforms when traced along traces—are found where F 2 folds are ‘folded’ by F 3 folds, and also remote from F 3 fold hinges. Intense inhomogeneous flattening explains these features, and it is considered that this flattening may be part of the second, rather than the third deformation. In the prevailing high-temperature conditions, the amount of flattening seems to be a function of rock-type and it is possible to distinguish between the simpler structures in thick psammites and the more complicated, highly flattened, structures in pelitic/striped rocks. The major rock units probably represent a stratigraphic sequence, from bottom to top, 1. Morar Division, 2. Glenfinnan Division and 3. Loch Eil Division.