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Spherical harmonic expansion of hydrodynamic hull forces
Published in C. Guedes Soares, T.A. Santos, Trends in Maritime Technology and Engineering Volume 1, 2022
V. Ferrari, S. Sutulo, C. Guedes Soares
The inverse transformations are: u=Ucosβsinγ,v=Usinβsinγ,rL=Ucosγ. The main advantage of the spherical coordinate system is that, first of all it, is robust since it is well defined for any type of manoeuvre and, secondly, it enables the application of the spherical harmonic expansion. This technique, as will be seen later, is well studied in literature and applied in many fields. There exists also a cylindrical harmonic expansion (Byerly 1893), which in theory could be applied to the cylindrical coordinate system mentioned previously. However this cylindrical harmonics are mathematically more complex, being based on Bessel functions, and seem to be much less studied in literature.
Coordinate Systems and Vector Algebra
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
Figure 3.8b shows the shape of elemental volume (dv) along with its three orthogonal sides (dr,rsinθ⋅dφ,andrdθ). Its value in the spherical coordinate system is given as:dv=(dr)⋅(r⋅dθ)⋅(r⋅sinθ⋅dφ)=r2sinθ⋅dr⋅dθ⋅dφ
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.
Thermoelastic behavior of an isotropic solid sphere under a non-uniform heat flow according to the MGT thermoelastic model
Published in Journal of Thermal Stresses, 2022
Ahmed E. Abouelregal, Anouar Saidi, Hamid Mohammad-Sedighi, Ali H. Shirazi, Abdullah H. Sofiyev
In this section, the problem of an isotropic solid sphere with radius is considered as a case study of the proposed model. The solid sphere medium is exposed to a thermal shock in the presence of a time-dependent heat flow. The spherical coordinate system is taken into account where indicates the radial direction, is the co-latitude and the longitude of a spherical coordinate system. The state functions can be expressed in distance and time due to the spherical symmetry (Figure 3).
Design and characteristic analysis of CNT thin film thermoacoustic transducer spherical array panel for low intensity focused ultrasound
Published in Journal of Thermal Stresses, 2021
Zhenhuan Zhou, Jinxin Wang, Dalun Rong, Zhenzhen Tong, Xinsheng Xu, C. W. Lim
Consider an acoustic transducer array consisting of multiple rectangular CNT thin films arranged in a spherical array panel as illustrated in Figure 2. A spherical coordinate system is adopted, of which the origin O is located at the center of the sphere as shown in Figure 2a. The corresponding Cartesian coordinate system is also indicated. The radius of sphere, length, and width of each rectangular CNT thin film element are denoted as and respectively. To derive the analytical solution of the induced acoustic field, each element of the array is regarded as an area source consisting of an infinite number of infinitesimal film sources. Therefore, the sound pressure of the overall thermoacoustic array at the observation point can be obtained by the Huygens principle.
Stability optimization of grid shells based on regional sensitivity differences
Published in Mechanics of Advanced Materials and Structures, 2020
Shiying Chen, Weiguo Li, Jin Quan, Qing Li
The imperfection values at each node of an imperfection sample in the X, Y, and Z directions were generated by normrnd (0, ω/2, 1), a normal distribution function in MATLAB; here, ω represents the imperfection amplitude, and denotes the number of joints. To facilitate parametric modeling in the spherical coordinate system, we created a transformation program to convert the imperfection values of the rectangular coordinate system to those of the spherical coordinate system. Figure 2 shows the imperfection values at each node of the two imperfection samples. Imperfection values with maximum imperfection amplitudes were selected from set A and those with minimum imperfection amplitudes from set I. The Y-axis values and in Figure 2 were taken along the direction of and respectively (Figure 3).