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Global Coordinate System
Published in Sumit Sharma, Composite Materials, 2021
Consider, as shown in Figure 8.1a, the familiar view of an isolated element in the principal material coordinate system. Figure 8.1b shows a similar element but one that is isolated in an x-y-z global coordinate system. The fibers are oriented at an angle θ with respect to the +x axis of the global system. The fibers are parallel to the x-y plane, and the 3 and z axes coincide. The fibers assumed their orientation by a simple rotation of the principal material system about the 3 axis. The orientation angle θ will be considered positive when the fibers rotate counterclockwise from the +x axis toward the +y axis. Often the fibers not being aligned with the edges of the element are referred to as an off-axis condition, generally meaning the fibers are not aligned with the analysis coordinate system (i.e., off the +x axis). Though we will use the notation for a rectangular Cartesian coordinate system as the global system (i.e., x-y-z), the global coordinate system can be considered to be any orthogonal coordinate system. The use of a Cartesian system is for convenience only, and the development is actually valid for any orthogonal coordinate system [1].
Computer Modelling of Microwave Sources
Published in R A Cairns, A D R Phelps, P Osborne, Generation and Application of High Power Microwaves, 2020
External circuits are handled by coupling elements at the surface of the active computational domain to external circuit elements. The external circuit elements have no physical dimension, but provide a relationship between surface current and tangential electric fields. Figure 11 illustrates the coupling for the 2-D TM field case. Assembly of the finite element node equation for the ‘N’ node with the external circuit gives ∂tdN2=−H−I−Iσ where dN and H are the tensor field components, I is the particle current contribution to the node and Iσ is the surface current. For simplicity we here consider only orthogonal coordinates.
Heat Conduction
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
The method of conformal mapping can also be applied to the transformation of a standard coordinate system to other orthogonal curvilinear coordinates, for example, elliptic cylindrical coordinates or bipolar coordinates (see Figure 2.16). An orthogonal coordinate system is a system of curvilinear coordinates wherein each family of surfaces intersects the others at right angles. Elliptic cylindrical coordinates are an orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. As a result, the coordinate surfaces become prisms of confocal ellipses and hyperbolae. Orthogonal curvilinear coordinates are useful in solving the heat conduction equation when boundaries are more closely aligned with these coordinates than Cartesian, cylindrical, or spherical coordinates.
Continuum theory of a nematic liquid crystal with a nonideal physical surface and the surfacelike elasticity
Published in Liquid Crystals, 2023
To consider an arbitrary geometry of a liquid crystal we introduce the curvilinear orthogonal coordinate system with metric tensor . Let the boundary coincide with the coordinate surface Then and are the orthogonal coordinates on and on other surfaces and the outer normal to each of these ‘parallel’ surfaces is directed along the coordinate line i.e., , Figure 1 This is the so-called semigeodesic coordinate system with and the Cartesian spherical and cylindrical coordinates (cyclic permutation!) are its most known examples. In this section we keep for the sake of the presentation symmetry.
Assessment of passenger long-term vibration discomfort: a field study in high-speed train environments
Published in Ergonomics, 2022
Yong Peng, Zhifa Wu, Chaojie Fan, Jiahao Zhou, Shengen Yi, Yuexiang Peng, Ke Gao
Previous studies have confirmed that vibration can cause local discomfort in human body parts, such as the back and buttock (Basri and Griffin 2012; Beard and Griffin 2016). However, the main focus of the local discomfort analysis was to highlight areas of discomfort and the local discomfort cannot simply be equated with overall comfort. In addition, in the studies of Mansfield et al. (2014), the exposure time was proved to be an important factor affecting passenger vibration comfort. Moreover, a component of interaction between the vibration amplitude and exposure time was used to represent the acceleration of development of discomfort during vibration exposure. However, the RMS values are measured from vibration platform input excitation rather than the passenger-seat surface. Thus, a novel index combined exposure duration and vibration (CEtv) was proposed to assess the long-term high-speed train passenger vibration discomfort. The CEtv was calculated as follows: where Te is exposure time and the awpvtv-sum is defined as the sum frequency-weighted RMS acceleration in orthogonal coordinates on the cushion surface and seatback surface and calculated with Equation (8):
Reduction of the classical electromagnetism to a two-dimensional curved surface
Published in Journal of Modern Optics, 2019
Now we would like to reduce the whole set of equations to a certain two-dimensional, and in general curved surface. This reduction will be performed in the following way. We first choose in the 3D space a system of orthogonal coordinates well-fitted to the considered limiting surface. They will be called u, v, w, referred to also by indexes as 1, 2, 3. The w coordinate plays a special role: the corresponding tangent vector (to say it precisely: the tangent vector to the 3D space connected with fixing u and v and varying w, which is, however, normal to the considered sub-surface) defines the direction along which the projection is going to be done. The additional condition we impose while choosing the coordinates is that (we use the notation ). This means that this tangent vector may rotate, while changing u and v, but it cannot change its length. Such a choice gives: