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Robotics
Published in Jian Chen, Bingxi Jia, Kaixiang Zhang, Multi-View Geometry Based Visual Perception and Control of Robotic Systems, 2018
Jian Chen, Bingxi Jia, Kaixiang Zhang
In 3D Euclidean space, the position of the origin of coordinate frame F′ relative to coordinate frame F can be denoted by the following: 3×1 vector [xyz]T. The components of this vector are the Cartesian coordinates of F′ in the F frame, which are the projections of the vector xf onto the corresponding axes. Besides the Cartesian coordinate system, the position of a rigid body can also be expressed in spherical or cylindrical coordinates. Such representations are generally used for the analysis of specific mechanisms such as the spherical and cylindrical robot joints as well as the omnidirectional cameras. As shown in Figure 1.2, the spherical coordinate system can be viewed as the three-dimensional version of the polar coordinate system, and the position of a point is specified by three numbers: the radial distance r from the point to the coordinate origin, its polar angle θ measured from a fixed zenith direction, and the azimuth angle ϕ of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith. As shown in Figure 1.3, the position of a point in the cylindrical coordinate system is specified by three numbers: the distance ρ from the point to the chosen reference axis (generally called cylindrical or longitudinal axis), the direction angle ϕ from the axis relative to a chosen reference direction, and the distance z from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.
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.
Two- and three-dimensional simulation of natural convection flow of CuO-water in a horizontal concentric annulus considering nanoparticles’ Brownian motion
Published in Numerical Heat Transfer, Part A: Applications, 2019
Wei Wang, Ben-Wen Li, Zheng-Hua Rao, Gang Liu, Sheng-Ming Liao
A closed horizontal annular container, as shown in Figure 1, is considered with radius ratio and aspect radio where is the height of cylinders, and are the radii of inner and outer cylinders, respectively. The inner and outer cylinder surfaces are kept isothermal with the inner wall temperature higher than that of the outer while the two end walls are considered to be adiabatic for 3-D configuration. As for 2-D configuration, the enclosure is assumed to be infinite long in the axial direction, i.e., so that the computational domain is simplified into a 2-D annular cross-section. A cylindrical coordinate system () is adopted in such a way that the radial and axial coordinates are measured from the center and bottom of both cylinders, respectively, whilst the angular coordinate is measured anti-clockwise from the upward vertical line. The water-based nanofluid containing CuO nanoparticles is Newtonian, incompressible and treated as single-phase fluid, which means that the nanoparticles and base liquid are in local thermal equilibrium, and no slip motion occurs between the solid and liquid phases. The buoyancy-induced flow in the annular space is assumed to be laminar, with negligible viscous dissipation. Constant thermophysical properties are considered for the nanofluid, and the Boussinesq approximation is used for the density variation in the buoyancy force.