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Orthogonal Coordinate Systems
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
Cylindrical coordinates are an alternate way of describing points in a three-dimensional space. In this coordinate system, one of the rectangular coordinate planes, namely, the x–y plane, as described by the Cartesian coordinate system, is replaced by a polar plane. In the cylindrical coordinate system, everything is measured with respect to a fixed point called the pole and an axis called the polar axis. The pole is the equivalent to the origin in the Cartesian coordinate system and the polar axis corresponds to the positive direction of the x-axis. The cylindrical coordinates of a point are then the ordered triplet (r,θ,z), as defined in Figure 3.5.
Dynamic analysis of a thick hollow cylinder made of two-dimensional functionally graded material using time-domain spectral element method
Published in Mechanics of Advanced Materials and Structures, 2019
For a cylinder, the cylindrical coordinate system (r, θ, and z) is usually used to describe its geometry and deformation. As shown in Figure 1, a cylinder made of 2D-FGM has the radius r and height L. It is noted that the geometry of the cylinder and the loading considered here are not functions of the circumferential coordinate θ. Thus, the deformation becomes symmetrical with respect to the z-axis. In this paper, the cylinder is considered as a three-dimensional solid, the theory of linear elasticity is adopted. Thus, no prior assumption on displacements is imposed like structural theories except the assumption of axisymmetric deformation which is introduced to reduce the number of unknown variables in the governing differential equations. In the following analysis, when the classic FEM is used for comparison purposes, the same assumption and solid elements are used. Therefore, by neglecting the body force, the governing equations of motions can be written in a two-dimensional cylindrical coordinate (r, z) system, defined as and where t and ρ are time and material density, respectively; u = u(r, z) and w = w(r, z) denote the displacement components along the radial and axial directions; σrr, σzz, σθθ, and τrz are the stress components.