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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[biomedical, computational] Symmetric graphical representation of a variety of mathematical expressions, but the most well known is the quadratic equation, for instance, mathematically expressed as y = ax2 + bx + c, or x = y2, respectively, in general notation: Ax2 + Bxy + cy2 + Dx + Ey + F = 0, where x and y are Cartesian coordinates and the remaining letters are constants. The parabola has an axis of symmetry that intersects with the graphical representation of the parabola in the vertex. The parabola is in the family of conic intersects: circle, ellipse, parabola, and hyperbola. In mechanics, the parabolic trajectory is well known and can be defined by the vector sum of the gravitational attraction and the propulsion force, where propulsion will be in vertical and horizontal directions. A horizontally released projectile (e.g., bullet from a gun) will have a vertical dependence on gravity (gravitational accelerationg) as a function of time (t), describing only one side of the parabolic track defined as y = −(1/2)gt2 (see Figure P.5).
Centroids and Moments of Inertia of Areas
Published in Robert L. Mott, Joseph A. Untener, Applied Strength of Materials, 2016
Robert L. Mott, Joseph A. Untener
Where two axes of symmetry do not occur, the method of composite areas can be used to locate the centroid. For example, consider the shape shown in Figure 6–4. It has a vertical axis of symmetry but not a horizontal axis of symmetry. Such areas can be considered to be a composite of two or more simple areas for which the centroid can be found by applying the following principle:
Centroids and Moments of Inertia of Areas
Published in Robert L. Mott, Joseph A. Untener, Applied Strength of Materials, Sixth Edition SI Units Version, 2017
Robert L. Mott, Joseph A. Untener
Where two axes of symmetry do not occur, the method of composite areas can be used to locate the centroid. For example, consider the shape shown in Figure 6–4. It has a vertical axis of symmetry but not a horizontal axis of symmetry. Such areas can be considered to be a composite of two or more simple areas for which the centroid can be found by applying the following principle:This principle uses the concept of the moment of an area, that is, the product of the area times the distance from a reference axis to the centroid of the area. The principle statesThis can be stated mathematically as ATY¯=∑Aiyi whereAT is the total area of the composite shapeY¯ is the distance to the centroid of the composite shape measured from a reference axisAi is the area of one component part of the shapeyi is the distance to the centroid of the component part from the reference axisThe subscript i indicates that there may be several component parts, and the product Aiyi for each must be formed and then summed together, as called for in Equation (6–1). Since our objective is to compute Y¯, Equation (6–1) can be solved: Y¯=∑AiyiAT A tabular form of writing the data helps keep track of the parts of the calculations called for in Equation (6–2). An example will illustrate the method.
Could a virtual bowtie filter improve image quality? A breast simulation study
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2021
Déte van Eeden, Frederik Carl du Plessis
where is the vertex of the parabola, and is the axis of symmetry. The represents the horizontal shift from and represents the vertical shift. This function was fitted to the intensity profile of the bowtie filter and parameters and was altered to obtain a new attenuation correction factor (ACF). The bowtie intensity with the quadratic fit and the iterative process used to determine the optimal ACF is seen below in Figure 6.
Scale model to measure stress under circular footings resting on sand
Published in International Journal of Geotechnical Engineering, 2021
Danny Useche-Infante, Gonzalo Aiassa-Martínez, Pedro Arrúa, Marcelo Eberhardt
The horizontal displacements in the soil-foundation interface were restricted to zero, assuming a perfect roughness of the interface and given the symmetry of the foundation. To achieve a smooth response curve, the vertical displacements at the nodes were applied in 100 increments. The initial condition of the soil (geostatic stresses) was established by applying the force of gravity to the soil in the first part of the analysis. Edge conditions of the model are specified to simulate the conditions of the laboratory tests that normal displacements are not allowed in the right edge of the model; the normal and tangential displacements at the bottom edge are restricted; while free movement is allowed at the top edge of the model. The left edge corresponds to the axis of symmetry.
Ground effects on the hypervelocity jet flow and the stability of projectile
Published in Engineering Applications of Computational Fluid Mechanics, 2018
Zijie Li, Hao Wang, Jianwei Chen
In the computational domain, the cannon tube was specified as a solid wall boundary condition, and the pressure-outlet boundary condition was applied at the domain boundary around the muzzle flow field. As mentioned previously, in order to perform the necessary calculations, a number of C-programming language based user-defined subroutines were created. The muzzle was applied to the pressure-inlet boundary condition, which was obtained through the aftereffect period User-defined Function (UDF) program. The six-degrees-of-freedom (6DOF) UDF program is used to control the projectile while moving in the tube and exiting the chamber. The symmetry boundary condition was specified along the axis of symmetry. Along the domain boundaries, impermeable wall and temperature near wall conditions were applied. The precursor pressure and temperature around the muzzle flow field were set at 101,325 Pa and 300 K, respectively.