Open Access
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The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.
15Spatial arbitrarily curved microbeams with the modified couple stress theory: Formulation of equations of motion
Spatial arbitrarily curved microbeams with the modified couple stress theory: Formulation of equations of motion
Development a model of robot movement with five degrees of freedom for a warehouse
The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.
Read more14Coherent Propagation of Polarized Millimeter Waves Through Falling Hydrometeors
Coherent propagation of polarized millimeter waves through falling nonspherical hydrometeors is considered on the basis of Foldy's approximation. It is assumed that the distribution of hydrometeors over orientations is locally axially symmetric, the axis of symmetry being given by the local direction of air flow around the hydrometeors. An efficient rigorous method is described to compute the orientationally averaged forward-scattering amplitude matrix. This method is based on Waterman's T-matrix approach and fully exploits rotational properties of the T-matrix. First, the T-matrix is analytically averaged over hydrometeor orientations in the local coordinate system with the Z -axis along the direction of local air flow. Then, the elements of the forward-scattering amplitude matrix with respect to the local and laboratory coordinate systems are calculated via simple analytical expressions. Analytical solutions of the propagation equation for the coherent electric field are discussed. Cross polarization of...
Read more19Fast Solution of Scattering From Moving Target by Dynamic-Octree-Based SBR Algorithm
This letter proposes a shooting and bouncing ray (SBR) method based on a dynamic octree structure to accelerate the solution of scattering from a moving target. In this algorithm, three coordinate systems are established, i.e., the world coordinate system (WCS), the local coordinate system (LCS), and the perspective projection coordinate system (PPCS). First, an octree structure is created to store the target facets in the LCS that moves with the target in the WCS. The octree remains relatively stationary with the target, which avoids rebuilding the structure at every moment. Second, to accelerate the shadow elimination, the scanline algorithm is used to convert the 3-D processing into a 2-D graphic display in the PPCS. Third, high-order ray tracing is completed in the LCS. A forward–backward ray-tracing technology is adopted to guarantee the precision of the illuminated facet identification. Finally, the information of illuminated facets is transferred from the LCS to the WCS, and the scattering of target is calculated in the WCS. Meanwhile, some numerical simulations of our method are compared with those of the FEKO-MLFMM and the traditional octree-SBR method. The results show that the proposed method greatly reduces the computation time while maintaining satisfactory accuracy.
Read more3Improving Maps of Physical and Virtual Radio Transmitters
In multipath assisted positioning, each multipath component arriving at a receiver is regarded as being sent by a virtual transmitter in a line-of-sight condition. These virtual transmitters can be used for localizing a user. In the general case, their locations are unknown. With simultaneous localization and mapping (SLAM), the locations of physical and virtual transmitters can be estimated jointly with the location of a user. However, the convergence time can be high. In scenarios such as malls or museums, multiple users travel through the same scenario on different trajectories. Different users can cooperate by exchanging maps of physical and virtual transmitters in order to decrease the convergence time and increase the positioning performance. Though, each user is in its own local coordinate system. When a map of transmitters is handed from one user to another, the relative rotation and translation of the coordinate systems of the two users need to be estimated based on the transmitters in the two users’ maps. Within this paper, we propose an algorithm to merge and update an obtained transmitter map with the transmitters estimated by the current user. We evaluate our proposed method by simulations in an indoor mall. In a crowdsourcing based approach, maps are handed from one user to another, updated, and handed to the next user. Our evaluations show that exchanging transmitter maps improves the positioning performance, although each user is in its own local coordinate system not knowing the parameters relating its own coordinate system to the ones of obtained transmitter maps.
Read more1Using Local Coordinate Systems for Dimensional Analysis in the Machining
The article considers the implementation of the attached coordinate systems for machining at CNC machines. The main objective of this study is to suggest the improvements of processing accuracy. The problem arises when the machinable surface does not belong to a technological base. We offer the method based on preliminary measurement of workpiece dimensions. Further, the coordinate value enters into a CNC unit. The NC program uses the local coordinate systems, which are linked with measured dimensions of selected surfaces. In this case, the machining accuracy does not depend on errors connected with previous processing stages. Local coordinate systems can be effectively used for machining “rotary body”-type components at turning CNC machines or for machining conditions of box-like parts at milling CNC machines. We also offer the special device for adjustment “0” point outside a machine tool. As a result, the proposed method allows for a dimension chain shortening and accuracy improving.
Read more14Anatomically-based skeletal coordinate systems for use with impact biomechanics data intended for anthropomorphic test device development
Anatomically-based skeletal coordinate systems for use with impact biomechanics data intended for anthropomorphic test device development
Read more8Plane formation by synchronous mobile robots without chirality
We consider a distributed system consisting of autonomous mobile computing entities called robots moving in the three-dimensional space (3D-space). The robots are anonymous, oblivious, fully-synchronous and have neither any access to the global coordinate system nor any explicit communication medium. Each robot cooperates with other robots by observing the positions of other robots in its local coordinate system. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. This problem is not always solvable because of the impossibility of symmetry breaking. While existing results assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. The robots without chirality can never break the symmetry consisting of rotation symmetry and reflection symmetry. Such symmetry in 3D-space is fully described by 17 symmetry types each of which forms a group. We extend the notion of symmetricity [Suzuki and Yamashita, SIAM J. Compt. 1999] [Yamauchi et al., PODC 2016] to cover these 17 symmetry groups. Then we give a characterization of initial configurations from which the fully-synchronous robots without chirality can form a plane in terms of symmetricity.
Read more27An edge‐based smoothed finite element method (ES‐FEM) using 3‐node triangular elements for 3D non‐linear analysis of spatial membrane structures
An edge‐based smoothed finite element method using 3‐node triangular membrane elements (ES‐FEM‐T3) is proposed to analyze three‐dimensional (3D) spatial membrane structures under large deflection, rotation, and strain. In our 3D formulation, co‐rotational local coordinate systems associated with the edge‐based smoothing domains are constructed. Edge‐based gradient smoothing for the spatial membrane structure is performed in global Cartesian coordinate system and transformed into the co‐rotational local coordinate system. The smoothed strain and stress can then be properly evaluated in the co‐rotational local coordinate system after the elimination of the rigid body motions simply by coordinates transformation. Explicit time integration scheme is used to compute the transient response of the 3D spatial membrane structure to time‐domain excitations, and the dynamic relaxation method is employed to obtain steady‐state solutions. The numerical results demonstrate that the proposed method produces much better solution accuracy and computational efficiency than the standard FEM using T3 membrane element. Copyright © 2010 John Wiley & Sons, Ltd.
Read more4The Effect of Using Multiple Coordinate Systems and Datum Transformations on the Calculated Coordinates in Palestine
The recent developments in spatial data collection, management and software require the availability of proper geodetic infrastructures for integrating different types and sources of coordinates without causing effective changes in positions. Nowadays, positions are mostly collected by GNSS data collectors based on WGS84/ITRF reference systems. The data are then subjected to transformations and projections to a locally used system. Another possibility is direct data collection based on the local coordinate system by classical surveys using land surveying, photogrammetry, laser scanning, etc. The spatial data management is commonly operated using Geographic Information Systems (GIS) software for mapping, analysis, planning, and other services. The conversions between different coordinate systems should be well defined to guarantee the consistency of the coordinates on all systems and tools. In Palestine, the classical and local surveys are all based on the local coordinate system Pal1923Grid for engineering, cadastral and planning applications. The different GNSS RTK-service providers use different definitions and transformation methods between WGS84 or the International Terrestrial Reference Frames (ITRF) and the local Palestine1923Grid, whereas the Land authority has adopted a group of parameters to be implemented on the Global Navigation Satellite Systems (GNSS) data collectors, which do not fit with Palestine1923Grid properties. Additionally, different transformation methods are used in GIS applications for converting the coordinates between the different systems using WGS84 as an intermediate system. Here, the coordinates of a group of the geodetic network in the West Bank of Palestine are used to assess the accuracy of the different transformations and systems by comparing the transformed coordinates using the GNSS system and the originally registered coordinates. Furthermore, a grid of points covering the coordinate system extents is used to describe the differences between the transformations and systems. It was found that the parameters provided by GNSS service providers have results that are consistent with each other and the geodetic network in the West Bank of Palestine compared to GIS-software parameters. By contrast, all systems have extremely deteriorated coordinates in the Gaza strip and the further parts of the Pal1923Grid extents.
Read more1Orientation mechanics of liquid crystals. Transition to a local coordinate system
Equations of orientation mechanics of liquid crystals, written in a local rotational coordinate system related to the director orientation, are considered. A condition is obtained, which ensures qualitative similarity between solutions predicted by an approximate (single-constant) model and solutions obtained with the original formulation of the problem. Equations that provide a more exact approximation to the general model are proposed. The use of matrices in recording of energy relations allows a fairly easy transition to other coordinate systems and studying more sophisticated models for the description of the orientation state of liquid crystals. A transition to a local coordinate system makes it possible to compute three-dimensional liquid-crystal structures on moderate-performance personal computers.
Read more4Spherical Gravity Forwarding of Global Discrete Grid Cells by Isoparametric Transformation
For regional or even global geophysical problems, the curvature of the geophysical model cannot be approximated as a plane, and its curvature must be considered. Tesseroids can fit the curvature, but their shapes vary from almost rectangular at the equator to almost triangular at the poles, i.e., degradation phenomena. Unlike other spherical discrete grids (e.g., square, triangular, and rhombic grids) that can fit the curvature, the Discrete Global Grid System (DGGS) grid can not only fit the curvature but also effectively avoid degradation phenomena at the poles. In addition, since it has only edge-adjacent grids, DGGS grids have consistent adjacency and excellent angular resolution. Hence, DGGS grids are the best choice for discretizing the sphere into cells with an approximate shape and continuous scale. Compared with the tesseroid, which has no analytical solution but has a well-defined integral limit, the DGGS cell (prisms obtained from DGGS grids) has neither an analytical solution nor a fixed integral limit. Therefore, based on the isoparametric transformation, the non-regular DGGS cell in the system coordinate system is transformed into the regular hexagonal prism in the local coordinate system, and the DGGS-based forwarding algorithm of the gravitational field is realized in the spherical coordinate system. Different coordinate systems have differences in the integral kernels of gravity fields. In the current literature, the forward modeling research of polyhedrons (the DGGS cell, which is a polyhedral cell) is mostly concentrated in the Cartesian coordinate system. Therefore, the reliability of the DGGS-based forwarding algorithm is verified using the tetrahedron-based forwarding algorithm and the tesseroid-based forwarding algorithm with tiny tesseroids. From the numerical results, it can be concluded that if the distance from observations to sources is too small, the corresponding gravity field forwarding results may also have ambiguous values. Therefore, the minimum distance is not recommended for practical applications.
Read more2ERROR ANALYSIS AND DIFFERENCE EQUATIONS ON CURVILINEAR COORDINATE SYSTEMS
ERROR ANALYSIS AND DIFFERENCE EQUATIONS ON CURVILINEAR COORDINATE SYSTEMS
1The diffraction of the field of dipole radiation by two spheres
The diffraction of the field of dipole radiation by two spheres
2Displacements, Strains, and Stresses
We consider composite materials consisting of continuous or discontinuous fibers embedded in a matrix. Such a composite is heterogeneous, and the properties vary from point to point. On a scale that is large with respect to the fiber diameter, the fiber and matrix properties may be averaged, and the material may be treated as homogeneous. This assumption, commonly employed in macromechanical analyses of composites, is adopted here. Hence, the material is considered to be quasi-homogeneous, which implies that the properties are taken to be the same at every point. These properties are not the same as the properties of either the fiber or the matrix but are a combination of the properties of the constituents. In this chapter, equations are presented for calculating the displacements, stresses, and strains when the structure undergoes only small deformations and the material behaves in a linearly elastic manner. Continuous fiber-reinforced composite materials (and structures made of such materials) often have easily identifiable preferred directions associated with fiber orientations or symmetry planes. It is therefore convenient to employ two coordinate systems: a local coordinate system aligned, at a point, either with the fibers or with axes of symmetry, and a global coordinate system attached to a fixed reference point (Fig. 2.1). In this book the local and global Cartesian coordinate systems are designated respectively by x 1 , x 2 , x 3 and the x, y, z axes.
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