scholarly journals Two‐line geometric derivation of the time derivatives of unit vectors in orthogonal curvilinear coordinate systems

1991 ◽  
Vol 59 (12) ◽  
pp. 1146-1147 ◽  
Author(s):  
R. A. Gordon
Keyword(s):  
Coordinate Systems ◽  
Curvilinear Coordinate ◽  
Derivatives Of ◽  
Unit Vectors

Tutorial / Article didactique Derivatives of curvilinear coordinate unit vectors: A concise matrix-vector approach

2000 ◽  
Vol 77 (12) ◽  
pp. 969-983
Author(s):  
P Mohazzabi
Keyword(s):  
Angular Velocity ◽  
Coordinate System ◽  
Time Evolution ◽  
Curvilinear Coordinate System ◽  
Curvilinear Coordinate ◽  
Orthogonal Curvilinear Coordinate System ◽  
Matrix Vector ◽  
Derivatives Of ◽  
Unit Vectors ◽  
Vector Approach

The time evolution of a curvilinear coordinate system can be identifiedwith an angular velocity matrix or vector, which contains the complete differentiationrule for all unit coordinate vectors of the curvilinear system, providinga concise method for their computations. When generalized the technique provides a powerful tool fordifferentiation, with respect to any parameter, of not only unit coordinate vectors but any vector that remains fixed in a general orthogonal curvilinear coordinate system. Several curvilinear systems, as well as the normal and tangentialcoordinate system are examined.PACS No.: 46.00

Identification of principal screws of two- and three-screw systems

2007 ◽  
Vol 221 (12) ◽  
pp. 1701-1715 ◽  
Author(s):  
J-S Zhao ◽  
F Chu ◽  
Z-J Feng
Keyword(s):  
Screw Theory ◽  
Coordinate Systems ◽  
Third Order ◽  
Analytical Methodology ◽  
Partial Derivatives ◽  
Spatial Mechanisms ◽  
Analysis Theory ◽  
Definition Of ◽  
Derivatives Of ◽  
Homogeneous Linear

The current paper proposes a unified analytical methodology to identify the principal screws of two- and three-screw systems. Based on the definition of the pitch of a screw, it first obtains an identical homogeneous quadric equation. According to functional analysis theory, it is known that the partial derivatives of an identical quadric equation with respect to its variables must be zero. Therefore, the paper deduces a set of linear homogeneous equations that are made up of the partial derivatives of the quadric equation. With the existing criteria of non-zero solutions for homogeneous linear algebra equations, it ultimately obtains the formulas of the principal pitches and the associated principal screws of the system. The most outstanding contribution of this methodology is that it proposes a unified analytical approach to identify the principal pitches and the principal coordinate systems of the second-order and the third-order screw systems. This should be a new contribution to the screw theory and will boost its applications to the kinematics analysis of robots and spatial mechanisms.

scholarly journals Implementation of an Approximate Conformal UPML in 2-D DGTD

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Linqian Li ◽  
Bing Wei ◽  
Qian Yang ◽  
Debiao Ge
Keyword(s):  
Good Accuracy ◽  
Computation Time ◽  
Elliptic Cylinder ◽  
Auxiliary Equation ◽  
Coordinate Systems ◽  
Absorbing Boundary ◽  
Computational Region ◽  
Curvilinear Coordinate ◽  
Orthogonal Curvilinear Coordinate System ◽  
Absorbing Layer

Using the numerical discrete technique with unstructured grids, conformal perfectly matched layer (PML) absorbing boundary in the discontinuous Galerkin time-domain (DGTD) can be set flexibly so as to save lots of computing resources. Based on the DGTD equations in an orthogonal curvilinear coordinate system, the processes of parameter transformation for 2-D UPML between the coordinate systems of elliptical and Cartesian are given; and the expressions of transition matrix are derived. The calculation scheme of conductivity distribution in elliptic cylinder absorbing layer is given, and the calculation coefficient of DGTD in elliptic UPML is calculated. Furthermore, the 2-D iterative formulas of DGTD and that of auxiliary equation in the elliptical cylinder UPML are derived; the conformal UPML calculation in DGTD is realized. Numerical results show that very good accuracy and computational efficiency are achieved by using the method in this paper. Compared to the rectangular computational region, both the memory and computation time of conformal UPML absorbing boundary are reduced by more than 20%.

Appendix A: General Curvilinear Coordinate Systems and Higher-Order Tensors

2021 ◽  
pp. 525-572
Author(s):  
Pierre J. Carreau ◽  
Daniel C.R. De Kee ◽  
Raj P. Chhabra
Keyword(s):  
Higher Order ◽  
Coordinate Systems ◽  
Curvilinear Coordinate ◽  
Higher Order Tensors
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