The explicit representation to the principal rotation angle and the principal rotation axis

1999 ◽  
Vol 20 (6) ◽  
pp. 653-659 ◽  
Author(s):  
Dui Guansuo ◽  
Zhuo Xiaobao
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Explicit Representation

On the factorization of rotations with examples in diffractometry

1990 ◽  
Vol 428 (1875) ◽  
pp. 451-472 ◽  
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Axial Direction ◽  
The Other ◽  
Rotation Vector ◽  
Coordinate Transformations ◽  
Rotation Axes ◽  
Analytic Expressions ◽  
Direction Cosines

An analysis of compound rotations, such as occur in eulerian cradles, is presented in terms of a calculus of rotation axes, without reference to the associated coordinate transformations. The general case of three rotation shafts mounted on one another, with any relation between them at datum zero, is presented. The problem and its solution may be represented entirely in terms of a plane octagon in which four sides have directions that are instrumental constants and the other four sides have lengths that are instrumental constants. When the first four sides are given lengths that express both the rotation angle and the axial direction of the required rotation, then the remaining four sides have directions that directly express the rotations in the drive shafts, that will generate the required rotation. Analytic expressions are given for the shaft setting angles in the general case. If the first and third axes are parallel and the intermediate one perpendicular to these at datum zero (as in the four-circle diffractometer) then these reduce to θ 1 = arctan ( μ, σ ) + [arctan ( λ , v ) - ψ -½8π], θ 2 = 2 s arcsin ( λ 2 + v 2 )½, θ 3 = ( μ, σ ) - [arctan ( λ , v ) - ψ - ½8π], s = ± 1, 0 ≤ arcsin ( λ 2 + v 2)½ ≤ ½π, in which λ, μ, v and σ are the four components of a rotation vector constructed such that λ, μ and v are the direction cosines of the rotation axis multiplied by sin½ θ for a rotation angle θ and σ is cos½ θ . ψ is a constant determined by the choice of directions to which λ and v are measured. The results for the general case are also expressed in terms of more conventional variables.

The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space

2016 ◽  
Vol 13 (10) ◽  
pp. 1650116 ◽  
Author(s):  
Derya Kahvecí ◽  
Yusuf Yayli ◽  
Ísmaíl Gök
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Symmetric Matrix ◽  
Unit Length ◽  
Exponential Map ◽  
Spatial Displacement ◽  
Split Quaternions ◽  
Rodrigues Formula ◽  
Lightlike Axis ◽  
The Given

The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in [Formula: see text] and angle, the spatial displacement is represented by a [Formula: see text] semi-orthogonal rotation matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-skew-symmetric matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

scholarly journals Solar Tracking Error Analysis of Fresnel Reflector

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jiantao Zheng ◽  
Junjie Yan ◽  
Jie Pei ◽  
Guanjie Liu
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Tracking Error ◽  
The Sun ◽  
Angle Error ◽  
Tracking Errors ◽  
Solar Altitude ◽  
Solar Tracking ◽  
Main Factors ◽  
The Difference

Depending on the rotational structure of Fresnel reflector, the rotation angle of the mirror was deduced under the eccentric condition. By analyzing the influence of the sun tracking rotation angle error caused by main factors, the change rule and extent of the influence were revealed. It is concluded that the tracking errors caused by the difference between the rotation axis and true north meridian, at noon, were maximum under certain conditions and reduced at morning and afternoon gradually. The tracking error caused by other deviations such as rotating eccentric, latitude, and solar altitude was positive at morning, negative at afternoon, and zero at a certain moment of noon.

Development and Research of Intermittent Motion Planetary Mechanisms with Elliptical Gearwheels

2019 ◽  
pp. 77-88
Author(s):  
A.A. Prikhodko ◽  
A.I. Smelyagin
Keyword(s):  
Angular Velocity ◽  
Rotational Motion ◽  
Machine Tools ◽  
Rotation Angle ◽  
Rotation Axis ◽  
Kinematic Chain ◽  
Planetary Gears ◽  
Input Shaft ◽  
Urgent Task ◽  
Intermittent Motion

Actuators converting rotational motion of the input shaft into intermittent motion of the output one are widely used in mechanical engineering. However, in most of the actuators used, the conversion of motion is realized by breaking the kinematic chain. This leads to high loads on the actuator links due to shocks occurring at the beginning or end of the movement phase. An urgent task is the development of compact and reliable mechanical converters where the required motion is carried out smoothly and without breaking the kinematic chain. The article presents the new kinematic schemes of the intermittent motion mechanisms based on planetary gears with elliptical gearwheels. For ease of balancing it is proposed to use elliptical wheels with a rotation axis in the center of symmetry. The kinematic analysis of the developed mechanisms is carried out, the rotation angle and the analog of the output shaft angular velocity are determined. Mechanisms can be widely used in machine tools, robotics, automatic machines, and conveyors.

Analysis on the Twinning of FCC Metals by EBSD

2012 ◽  
Vol 472-475 ◽  
pp. 700-706 ◽  
Author(s):  
Guang Sheng Song ◽  
Shi Hong Zhang ◽  
Ming Cheng ◽  
Bin Wang
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Face Centered Cubic ◽  
Practical Application ◽  
Fcc Metals ◽  
Twinning Plane ◽  
Twinning Direction ◽  
Face Centered

For the technology of EBSD, the twinning of metals was described in the form of rotation angle combined with rotation axis, while the twinning of metals was usually described in the form of twinning plane combined with twinning direction. In this report, the corresponding relationship between the two description forms of twinning of face-centered cubic (FCC) metals has been built, based on this relationship, the twinning plane and twinning direction of FCC metals can be determined by EBSD. As the practical application of this relationship above, the twinning variants of two kinds of Ni based superalloys were analyzed.

Quantitative Evaluation of Crystal Rotation Behavior Around the Fatigue Crack in Copper Films

2005 ◽  
Author(s):  
Kenichi Shimizu ◽  
Tasiyuki Torii
Keyword(s):  
Fatigue Crack ◽  
Quantitative Evaluation ◽  
Crystal Orientation ◽  
Rotation Angle ◽  
Fatigue Testing ◽  
Rotation Axis ◽  
Fatigue Cracks ◽  
Pure Copper ◽  
Copper Films ◽  
Crystal Rotation

Using a fatigue testing method by which fatigue cracks can be initiated and propagated in a film adhered to cover a circular through-hole in a base plate subjected to pull-pull cyclic loads, annealed rolled pure copper films of 100μm thickness were fatigued. In order to discuss about the correlation between fatigue crack propagation and the change of crystal orientation, crystal orientation on the surface of the film materials was measured before and after fatigue testing. The crystallo-graphic information of these films was analyzed using the EBSD (Electron Back-scatter Diffraction) system and the quantitative evaluation method for the crystal rotation angle, the rotation axis and the rotation direction with fatigue testing was developed based on the analysis of crystal orientation matrix. As a result, the crystal rotation angle near the fatigue crack is larger than that apart from the crack and the crystal rotation was quantitatively larger around the transgranular crack than the intergranular crack propagated along the annealing twin boundary.

Discussions about the One Time Rotation of a Body of Revolution to the Designated Position

2011 ◽  
Vol 130-134 ◽  
pp. 1762-1764
Author(s):  
Pei Bei Dong ◽  
Shuo Li
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Economic Benefits ◽  
Descriptive Geometry ◽  
Body Of Revolution ◽  
Production Processes ◽  
The One ◽  
Product Costs ◽  
Graphic Solution

In this article, the one time rotation of a body of revolution to the designated position under any conditions is studied, especially when the rotating front and rear of a body of revolution are on different planes, in which case, descriptive geometry graphic solution is employed to solve the rotation axis and rotation angle. Compared with other methods, this method is simpler and more visible. The solution of the problem will greatly contribute to the simplification of production processes, thus lowering product costs and improving economic benefits.

scholarly journals Mt. Wendelstein imaging of comet 41P/Tuttle-Giacobini-Kresak during the 2017 perihelion arc

2020 ◽  
Vol 638 ◽  
pp. A8
Author(s):  
Hermann Boehnhardt ◽  
Arno Riffeser ◽  
Christoph Ries ◽  
Michael Schmidt ◽  
Ulrich Hopp
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Active Regions ◽  
Brightness Distribution ◽  
Axis Ratio ◽  
Image Normalization ◽  
Equivalent Radius ◽  
Latitude Range ◽  
Time Period ◽  
Radial Profiles

Context. In April 2017 comet 41P/Tuttle-Giacobini-Kresak (41P) passed perihelion close to Earth. Aims. We present observations of the comet and aim to use these to characterize its dust environment in the coma and of physical properties of the nucleus. Methods. 41P was imaged in the visible at the Mt. Wendelstein observatory over 45 nights between 2017 March 11 and 2017 July 17. Photometry of the coma provides information on the dust activity of the comet. Nucleus dimensions are determined by comparing radial profiles of observed images with those from models of the brightness distribution in the near-nucleus coma. Coma and tail structures in the images are enhanced using adaptive Laplace filtering and image normalization techniques. The geometry of the coma structures found are compared to results from dust models of the comet in order to obtain information on the nucleus rotation axis and the location of active regions on the nucleus. Results. Analyzing the evolution of the dust activity of the comet versus solar distance gives different results depending on the assumed phase angle corrections for the dust. Contamination from light emissions of the coma gas modified the colors in the coma and the slopes of the radial profiles. The light distribution in the very inner coma indicates some dust fragmentation close to the nucleus. The appearance of a prominent coma fan during the time period from March to May 2017 constrains the rotation axis of the nucleus (52 deg obliquity, 50 deg rotation angle of the perihelion meridian). This fan is produced by an extended region beyond about 50 deg latitude on the nucleus. Two narrow regions at low northern to equatorial latitudes were responsible for the time-variable occurrence of linear dust jets in the coma. An outburst of about 0.6 mag on 2017 May 11 generated various dust structures in the coma. The outburst originated from a region extending over a latitude range from +30 deg to −5 deg and released about 3000 tons of dust into the coma. The brightness of the nucleus indicates an equivalent radius of about 600 m and a minimum axis ratio larger than two.

Angle Detection Methods for a CMOS Smart Rotary Encoder

2005 ◽  
Vol 17 (4) ◽  
pp. 469-474 ◽  
Author(s):  
Kazuhiro Nakano ◽  
◽  
Toru Takahashi ◽  
Shoji Kawahito ◽  
◽  
...  
Keyword(s):  
Rotation Angle ◽  
Detection Method ◽  
Rotation Axis ◽  
Digital Signal ◽  
Detection Algorithm ◽  
Detection Methods ◽  
Zero Crossing ◽  
Rotary Encoder ◽  
Personal Robots ◽  
Signal Profile

A CMOS smart magnetic rotary encoder is useful for compact equipment such as personal robots. In the proposed encoder, the geometrical angle of a magnet is detected with digital signal processing. The output signal of our proposed system is susceptible to sensor offsets caused by sensors mismatch and misalignment between sensor chip and magnet. The accuracy of angle detection method and the robustness to rotation axis misalignment depend greatly on the angle detection algorithm. This paper presents three types of angle detection and their performance in precision and compares rotation angle misalignment tolerance. Simulation results show that detecting two zero-crossing points in the signal profile, ZC method, is the most adequate for this system.

An Exploration of the Method for Repairing Knee Joint Trauma-Induced Skin Defects with a Saphenous Flap

2021 ◽  
Vol 11 (9) ◽  
pp. 1848-1852
Author(s):  
Li-Feng Guan ◽  
Chao Fang ◽  
Xiao-Ni Wang ◽  
Qi-Yun Luo ◽  
Yin-Sheng Wu
Keyword(s):  
Knee Joint ◽  
Rotation Angle ◽  
Rotation Axis ◽  
Subcutaneous Tissue ◽  
Donor Site ◽  
Medium Size ◽  
Soft Tissue Defects ◽  
Skin Defects ◽  
Joint Trauma ◽  
Tissue Defects

The present study aimed to explore the methods and clinical outcomes of repairing skin defects caused by knee joint trauma with a saphenous flap. Fifteen patients with skin and soft tissue defects after knee joint trauma received saphenous flap transposition, and the rotation axis point of the pedicle and transposition method were modified. The flap was transposed through a “semi-open tunnel” using a subcutaneous tissue pedicle and by parallel “bilobed” flap transposition through an “open tunnel.” The whole flap survived after the surgery in all patients. The pedicle was not bulging, had no “cat’s ear” deformity, and was flattened. The donor site was sutured directly, and the wound healed by primary intention without postoperative complications. When a saphenous flap is applied to repair the knee joint trauma skin defects with small and medium size, by modifying the pedicle transposition method, the flap rotation angle is more flexible, surgery is safer and more reliable, and flap appearance is more optimized.

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