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Tectonics
Published in Aurèle Parriaux, Geology, 2018
The orientation of a straight line is determined by two angles: its azimuth (or strike) α (horizontal angle with respect to north, measured in a clockwise manner, 0° ⩽ α ⩽ 360°) and its dip β (angle with respect to the horizontal of a vertical plane containing the straight line, positive in the downward direction, 0° ⩽ β ⩽ 90°). It is conventionally noted by a symbol identifying the type of line (fold axis, lineation, direction of a landslide) followed by α/β (three numbers for α and two numbers for β to avoid confusion). Straight lines can thus represent several structural elements and also topographic elements such as rivers and parts of engineering projects such as the axis of a borehole or a tunnel. In a stereographic projection (§ 12.5.3), a straight line is represented by a point.
Tectonics
Published in Aurèle Parriaux, Geology, 2018
The orientation of a straight line is determined by two angles: its azimuth (or strike) α (horizontal angle with respect to north, measured in a clockwise manner, 0° ⩽ α ⩽ 360°) and its dip β (angle with respect to the horizontal of a vertical plane containing the straight line, positive in the downward direction, 0° ⩽ β ⩽ 90°). It is conventionally noted by a symbol identifying the type of line (fold axis, lineation, direction of a landslide) followed by α/β (three numbers for α and two numbers for β to avoid confusion). Straight lines can thus represent several structural elements and also topographic elements such as rivers and parts of engineering works such as the axis of a borehole or a tunnel. In a stereographic projection (§ 12.5.3), a straight line is represented by a point.
Large-scale molecular dynamics simulations of nanomachining
Published in Angelos P. Markopoulos, J. Paulo Davim, Advanced Machining Processes, 2017
Stefan J. Eder, Ulrike Cihak-Bayr, Davide Bianchi
Orientation can be calculated using several methods: as sequence of rotations of the object around an axis of a reference frame, for example, Euler angles, or as rotation of an object around an axis, for example, Rodrigues vectors and quaternions. Euler angles have two main advantages: they require minimum storage information, as only the minimum amounts of data, that is, three values, have to be saved, and direct coloring is possible, as each triad of angles can be associated with an RGB color. The major drawback is that the axes of rotation are usually codependent, which can generate the so-called Gimbal lock, where two axes of rotation degenerate into one, thus becoming indistinguishable and creating coloring artifacts. Both Rodrigues vectors and quaternions express the same type of rotation mechanism. As Rodrigues vectors store the minimum information of three values, some of the symmetry operations can cause infinities. By contrast, quaternions are four-dimensional (4D) vectors that, despite providing redundant information, prevent the disadvantages of both other methods. The PTM employed here uses quaternions for the calculation of the orientations of the crystal lattice.
Torsion and vertical curvature of motion-trajectory curves
Published in Mechanics Based Design of Structures and Machines, 2023
The plane which contains planar curve has fixed orientation and is motion-independent. Orientation of this plane can be described using three constant angles or one constant angle and a unit vector along which this constant rotation is performed. Because bi-normal vector is also normal to curve plane, curve plane can be defined by two unit vectors orthogonal to bi-normal vector These two vectors can be selected as where and The orthogonal matrix can be used to define orientation of curve plane in terms of three constant Euler angles, or in terms of constant angle of rotation about an axis of rotation defined by constant unit vector using Rodrigues’ formula where is the skew symmetric matrix associated with the vector In both cases, because elements of matrix are known from recorded motion trajectories, set of algebraic equations can be obtained and used to determine constant orientation parameters.
Part decomposition efficiency expectation evaluation in additive manufacturing process planning
Published in International Journal of Production Research, 2021
Yaroslav Garashchenko, Miroslaw Rucki
To perform the calculations, the system Magics with the EOSPACE module was chosen. Materialise Magics is a versatile, industry-leading data preparation software for Additive Manufacturing that allows to convert files to STL, repair errors, edit the design and prepare the build platform (Materialise Magics 2020). The EOSPACE module is able to distribute automatically a group of 3D models inside one or several platforms (multi-platform placement) in order to achieve compaction according to different criteria. Default criterion of optimisation is maximal nesting density. Determination of the 3D model position in the working space is supported with ability to set its orientation or rotation. The dimensions of the working platform of the AM machine were assumed LX = 380 mm, LY = 330 mm, which corresponded with the SLS Vanguard Si2 produced by company 3D Systems. From 5 to 40 similar parts were loaded, dependent on the utilisation of the workspace volume. To simplify the task, spatial orientation was limited to an angular resolution Δϕ = 90°. A minimum distance between the parts, as well as between the part and platform edges was set to lmin = 5 mm.
Decentralized force and motion control of multiple cooperative manipulators
Published in Automatika, 2021
Hamid AzizZadeh, Mohammad Bagher Menhaj, Heidar Ali Talebi
Representing every EE in object frame means , there is due to constrains on relative displacement of EE about the object. Kinematic constraints could be shown as wherein denotes rotation matrix from frame to frame . For representation of orientation, we use unit quaternion , where is the real part and is the vector part of it. The represents cross product. As shown above, angular velocity of the object and end-effectors are equal, lead to and also angular acceleration of them show another constraint .