Analytical solution of the Grad Shafranov equation in an elliptical prolate geometry

2019 ◽  
Vol 85 (2) ◽  
Author(s):  
F. Crisanti
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Laplace Equation ◽  
Vector Potential ◽  
Fusion Development ◽  
Geometrical Properties ◽  
Shafranov Equation ◽  
Toroidal Coordinates

The analytical solution in toroidal coordinates of the Grad Shafranov equation has been at the origin of the tokamak breakthrough in the fusion development. Unfortunately, the standard toroidal coordinates have a circular poloidal section, which does not fit the elongated cross-section of the present tokamak experiments. In axisymmetry, the vacuum Grad Shafranov equation coincides with the Laplace equation for the toroidal component of the vector potential. In the present paper the solutions for the Laplace equation and that for the vacuum Grad Shafranov equation are tackled in the elliptical prolate toroidal cap-cyclide coordinates framework. The following report of the geometrical properties and of the metric of these coordinates allows us to work out the analytical solution of both equations in terms of the Wangerin functions.

scholarly journals Analytical solution for torsion of cylindrical orthotropic annular wedge-shaped bars reinforced by thin shells

2021 ◽  
Author(s):  
István Ecsedi ◽  
Attila Baksa
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Analytical Method ◽  
Torsional Rigidity ◽  
Circular Ring ◽  
Thin Shells ◽  
Elastic Wedge ◽  
Curved Boundary ◽  
Cylindrically Orthotropic ◽  
Boundary Surfaces

AbstractThis paper deals with the Saint-Venant torsion of elastic, cylindrically orthotropic bar whose cross section is a sector of a circular ring shaped bar. The cylindrically orthotropic homogeneous elastic wedge-shaped bar strengthened by on its curved boundary surfaces by thin isotropic elastic shells. An analytical method is presented to obtain the Prandtl’s stress function, torsion function, torsional rigidity and shearing stresses. A numerical example illustrates the application of the developed analytical method.

MATHEMATICAL MODEL OF TWO-DIMENSIONAL LAYERED PRESSURE FLOW IN THE AUGER CHANNEL

2018 ◽  
Author(s):  
А.В. ГУКАСЯН ◽  
В.С. КОСАЧЕВ ◽  
Е.П. КОШЕВОЙ
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Cross Section ◽  
Dynamic Pressure ◽  
Two Dimensional ◽  
Flow Dynamic ◽  
Pressure Flow ◽  
Rectangular Channels ◽  
Wide Range ◽  
Pressure Characteristics

Получено аналитическое решение двумерного слоистого напорного течения в канале шнека, позволяющее моделировать расходно-напорные характеристики прямоугольных каналов шнековых прессов с учетом гидравлического сопротивления формующих устройств и рассчитывать расходно-напорные характеристики экструдеров в широком диапазоне геометрии витков как в поперечном сечении, так и по длине канала. Obtained the analytical solution of two-dimensional layered pressure flow in the screw channel, allow to simulate the flow-dynamic pressure characteristics of rectangular channels screw presses taking into account the hydraulic resistance of the forming device and calculate the mass flow-dynamic pressure characteristics of the extruders in a wide range of the geometry of the coils, as in its cross section and along the length of the channel.

Hydrodynamic Pressure of Thrust Step Bearings

2009 ◽  
Author(s):  
Shuangbiao Liu ◽  
W. Wayne Chen ◽  
Diann Y. Hua
Keyword(s):  
Analytical Solution ◽  
Coordinate System ◽  
Laplace Equation ◽  
Analytical Solutions ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Hydrodynamic Pressure ◽  
Finite Width ◽  
Cylindrical Coordinate ◽  
Lubrication Models

Step bearings are frequently used in industry for better load capacity. Analytical solutions to the Rayleigh step bearing and a rectangular slider with a finite width are available in literature, but none for a fan-shaped thrust step bearing. This study starts with a known solution to the Laplace equation in a cylindrical coordinate system, which is in the form of infinite summation. An analytical solution to pressure is derived in this paper for hydrodynamic lubrication problems encountered in the fan-shaped step bearing. The presented solutions can be useful for designers to maximize bearing performance as well as for researchers to benchmark numerical lubrication models.

scholarly journals Improved Stresses Analysis of a Functionally Graded Beam under Prestressed CFRP Plate

2018 ◽  
Vol 27 (1) ◽  
pp. 096369351802700 ◽  
Author(s):  
Samir Brairi ◽  
Bachir Kerboua ◽  
Ismail Bensaid
Keyword(s):  
Analytical Solution ◽  
Mechanical Load ◽  
Fe Analysis ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Interfacial Stresses ◽  
Geometrical Properties ◽  
Cfrp Plate ◽  
Practical Reality ◽  
New Research

In this paper, a new analytical solution is presented to predict the interfacial stresses of a functionally graded beam reinforced by a prestressed CFRP plate under thermo-mechanical load. A finite element (FE) analysis is also employed to validate the results of the analytical solution, the results from both models agreed very closely. Also, a parametric study is carried out in order to identify the effects of various material and geometrical properties on the magnitude of interfacial stresses. The presented results show that the interfacial stresses are highly concentrated at the end of the laminate, which can lead to a debonding at this location. Also, the material and geometrical properties have a significant impact on the magnitude of interfacial stresses. This new research approaches the practical reality of the structures in their environment by taking into consideration a combination of neglected terms by the other studies. Therefore, the results presented in this paper can serve as a benchmark for future analyses of functionally graded beams strengthened by prestressed Carbon fibre-rein-forced polymer (CFRP) plates and improve the rehabilitation, mechanical and corrosion resistance.

scholarly journals Improved Stresses Analysis of a Functionally Graded Beam under Prestressed CFRP Plate

2018 ◽  
Vol 27 (6) ◽  
pp. 096369351802700
Author(s):  
Samir Brairi ◽  
Bachir Kerboua ◽  
Ismail Bensaid
Keyword(s):  
Analytical Solution ◽  
Mechanical Load ◽  
Fe Analysis ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Interfacial Stresses ◽  
Geometrical Properties ◽  
Cfrp Plate ◽  
Practical Reality ◽  
New Research

In this paper, a new analytical solution is presented to predict the interfacial stresses of a functionally graded beam reinforced by a prestressed CFRP plate under thermo-mechanical load. A finite element (FE) analysis is also employed to validate the results of the analytical solution, the results from both models agreed very closely. Also, a parametric study is carried out in order to identify the effects of various material and geometrical properties on the magnitude of interfacial stresses. The presented results show that the interfacial stresses are highly concentrated at the end of the laminate, which can lead to a debonding at this location. Also, the material and geometrical properties have a significant impact on the magnitude of interfacial stresses. This new research approaches the practical reality of the structures in their environment by taking into consideration a combination of neglected terms by the other studies. Therefore, the results presented in this paper can serve as a benchmark for future analyses of functionally graded beams strengthened by prestressed Carbon fibre-rein-forced polymer (CFRP) plates and improve the rehabilitation, mechanical and corrosion resistance.

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