Rotation Formula Research Articles

Vibration-based identification of rotating blades using Rodrigues' rotation formula from a 3-D measurement

In this study, the geometrical setup of a turbine blade is tracked. A research-scale rotating turbine blade system is setup with a single 3-axes accelerometer mounted on one of the blades. The turbine system is rotated by a controlled motor. The tilt and rolling angles of the rotating blade under operating conditions are determined from the response measurement of the single accelerometer. Data acquisition is achieved using a prototype wireless sensing system. First, the Rodrigues\' rotation formula and an optimization algorithm are used to track the blade rolling angle and pitching angles of the turbine blade system. In addition, the blade flapwise natural frequency is identified by removing the rotation-related response induced by gravity and centrifuge force. To verify the result of calculations, a covariance-driven stochastic subspace identification method (SSI-COV) is applied to the vibration measurements of the blades to determine the system natural frequencies. It is thus proven that by using a single sensor and through a series of coordinate transformations and the Rodrigues\' rotation formula, the geometrical setup of the blade can be tracked and the blade flapwise vibration frequency can be determined successfully.

Measuring strain and rotation fields at the dislocation core in graphene

Strain fields, dislocations and defects may be used to control electronic properties of graphene. By using advanced imaging techniques with high-resolution transmission electron microscopes, we have measured the strain and rotation fields about dislocations in monolayer graphene with single-atom sensitivity. These fields differ qualitatively from those given by conventional linear elasticity. However, atom positions calculated from two dimensional (2D) discrete elasticity and three dimensional discrete periodized F\"oppl-von K\'arm\'an equations (dpFvKEs) yield fields close to experiments when determined by geometric phase analysis. 2D theories produce symmetric fields whereas those from experiments exhibit asymmetries. Numerical solutions of dpFvKEs provide strain and rotation fields of dislocation dipoles and pairs that also exhibit asymmetries and, compared with experiments, may yield information on out-of-plane displacements of atoms. While discrete theories need to be solved numerically, analytical formulas for strains and rotation about dislocations can be obtained from 2D Mindlin's hyperstress theory. These formulas are very useful for fitting experimental data and provide a template to ascertain the importance of nonlinear and nonplanar effects. Measuring the parameters of this theory, we find two characteristic lengths between three and four times the lattice spacings that control dilatation and rotation about a dislocation. At larger distances from the dislocation core, the elastic fields decay to those of conventional elasticity. Our results may be relevant for strain engineering in graphene and other 2D materials of current interest.

On the position analysis of a new spherical parallel robot with orientation applications

In this paper, we propose a new spherical parallel robot for celestial orientation, and rehabilitation applications (TV satellite dish, tracking systems, solar panels, cameras, telescopes, table of the machine tools, ankle, shoulder, wrist and etc.). The proposed robot can completely rotate about an axis. After describing the robot and its inverse position analysis, using the genetic algorithm, the dimensional optimization to maximize the workspace of the robot is performed. The workspace analysis shows that the proposed robot has a relatively large workspace. Also, singularity analysis represents that the manipulator is a singularity-free workspace. It is a great advantage of the proposed robot. Next, an optimal approach is proposed for solving the direct position problem of the robot. According to the geometry of the robot, two coupled trigonometric equations are obtained through using a special form of Rodrigues' rotation formula. Next, the two coupled equations are transformed to a 8-degrees polynomial using the Sylvester's Dialytic elimination method. Finally, a numerical example for the robot with an asymmetric structure is given with eight real solutions. Therefore, the polynomial being minimal and the proposed approach is optimal. This greatly decreases computational time, which is necessary for dynamics, control and simulation. A new spherical parallel robot for orientation applications is proposed.The proposed robot can completely rotate about an axis.The dimensional optimization to maximize the workspace of the robot is performed.An innovative approach is proposed for solving the direct position problem.

A closed-form solution for the position analysis of a novel fully spherical parallel manipulator

SUMMARYIn this paper, a novel 3(RPSP)-S fully spherical parallel manipulator (SPM) is introduced. Also, an innovative method based on the geometry of the manipulator is presented for solving the forward position problem of the manipulator. The presented method provides a framework for the future research to solve the forward position problem of the other fully spherical PMs (for examples 3(UPS)-S and 3(RSS)-S). In the proposed method, two coupled trigonometric equations are obtained by utilizing the geometry of the manipulator and Rodrigues' rotation formula. Using Bezout's elimination technique, the two coupled equations lead to a polynomial of degree eight. We show that the polynomial is minimal and optimal. Furthermore, the other method is proposed for selecting an admissible solution of the forward position problem. This algorithm is required to control modeling and dynamic simulations.

On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R q about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.

Measure of segments which intersect a convex body from rotational formulae

Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in ℝ3. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.

The Basics of Flat Space Cosmology

We present a new model of cosmology which appears to show great promise. Our flat space cosmology model, using only four basic and reasonable assumptions, derives highly accurate Hubble parameter H0, Hubble radius R0 and total mass M0 values for our observable universe. Our model derives a current Hubble parameter of , in excellent agreement with the newly reported (lower limit) results of the 2015 Planck Survey. Remarkably, all of these derivations can be made with only these basic assumptions and the current CMB radiation temperature . The thermodynamic equations we have generated follow Hawking’s black hole temperature formula. We have also derived a variety of other useful cosmological formulae. These include angular velocity and other rotational formulae. A particularly useful hyperbolic equation, , has been derived, which appears to be an excellent fit for the Planck scale as well as the current observable universe scale. Using the flat space Minkowski relativistic formula for Doppler effect, and a formula for staging our cosmological model according to its average mass-energy density at every Hubble time (universal age) in its expansion, a persuasive argument can be made that the observable phenomena attributed to dark energy are actually manifestations of Doppler and gravitational redshift. Finally, a theory of cosmic inflation becomes completely unnecessary because our flat space cosmology model is always at critical density.

Damage in edge cracking of rolled metal slabs

Materials get damaged under shear deformations. Edge cracking is one of the most serious damage to the metal rolling industry, which is caused by the shear damage process and the evolution of anisotropy. To investigate the physics of the edge cracking process, simulations of a shear deformation for an orthotropic plastic material are performed. To perform the simulation, this paper proposes an elasto-aniso-plastic constitutive model that takes into account the evolution of the orthotropic axes by using a bases rotation formula, which is based upon the slip process in the plastic deformation. It is found through the shear simulation that the void can grow in shear deformations due to the evolution of anisotropy and that stress triaxiality in shear deformations of (induced) anisotropic metals can develop as high as in the uniaxial tension deformation of isotropic materials, which increases void volume. This echoes the same physics found through a crystal plasticity based damage model that porosity evolves due to the grain-to-grain interaction. The evolution of stress components, stress triaxiality and the direction of the orthotropic axes in shear deformations are discussed.

Towards kinematic modeling of a multi-DOF tendon driven robotic catheter.

This paper presents a recent study on kinematic modeling of a remote-controlled active catheter. The tip steering motion of the catheter is actuated in a tendon-driven manner. Two antagonistic groups of tendon actuation realize the distal tip deflecting with two-degree-of-freedom (2-DOF) allowing it to reach a considerable large spatial workspace without catheter shaft rotation. However, when modeling such bending deformation, the sequential rotation approach is easily misapplied. We address this problem and introduce a novel and unified modeling methodology based on the concept of simultaneous rotation and the use of Rodrigues' rotation formula. An accurate model is created for robotic catheters and also can be generalized to common multi-tendon-driven continuum manipulators. It is essential in achieving accurate control and improving autonomous execution of command tracking tasks.

The invariator principle in convex geometry

The invariator principle is a measure decomposition that was rediscovered in local stereology in 2005 and has since been used widely in the stereological literature. We give an exposition of invariator related results where existing formulae are generalized and new ones proposed. In particular, we look at rotational Crofton-type formulae that are obtained by combining the invariator principle and classical Crofton formulae. This results in geometrical quantities represented as averages over weighted Crofton-type integrals in linear sections. We refer to these weighted integrals as measurement functions and derive several, more explicit representations of these functions. Inspired by classical Morse theory we rewrite the measurement function in terms of critical values of the sectioned object. This is particularly useful for surface area estimation.

Improvements on the present theoretical understanding of octupole correlations

Some intriguing results, obtained in a recent survey of octupole properties for all even-even nuclei, are reanalyzed in order to understand the origin of the strong disagreement with experimental data and/or the strange behaviours observed. The limi- tations of the rotational formula to describe E1 and E3 transition strengths are discussed as well as the role played by octupole-quadrupole coupling in some specific nuclei.

Study and Application on Airport Clearance Based on Secant Method in Gauss Projection Algorithm for Inverse Solution

This article adopts WGS-84 Coordinate, applying the Secant Method successive approximation to solve geographic coordinates according to the Gauss-Krueger formula, realizing the transformation from plane rectangular coordinates to geographic coordinates, using the coordinate translation and rotation formula to achieve the transformation between the airport coordinate and Gauss plane rectangular coordinates, thus geographic coordinate, gauss plane rectangular coordinates and the airport coordinate can change into each other.

Positional description applied to the solution of geometrically non-linear plates and shells.

In this work the finite element method (FEM) is applied to simulate orthotropic laminated plates and shells presenting large deformations. Alternatively to co-rotational and strain or stress assumed enhanced solid-like elements, a total Lagrangian technique based on unconstrained vectors and position description is proposed. As a consequence, there is no necessity to apply the linearized finite rotation formulae, as for example the Euler–Rodrigues one. The proposed laminate orthotropic strategy can be used for any order isoparametric curved elements; in this study we only explored cubic elements. Equilibrium is achieved from the principle of minimum total potential energy. Examples are solved in order to demonstrate the precision and robustness of the proposed procedure comparing results to analytical and other numerical solutions.

Electromagnetic transition strengths in soft deformed nuclei

Spectroscopic observables such as electromagnetic transitions strengths can be related to the properties of the intrinsic mean-field wave function when the latter are strongly deformed, but the standard rotational formulas break down when the deformation decreases. Nevertheless there is a well-defined, non-zero, spherical limit that can be evaluated in terms of overlaps of mean-field intrinsic deformed wave functions. We examine the transition between the spherical limit and strongly deformed one for a range of nuclei comparing the two limiting formulas with exact projection results. We find a simple criterion for the validity of the rotational formula depending on $<\Delta \vec{J}^2>$, the mean square fluctuation in the angular momentum of the intrinsic state. We also propose an interpolation formula which describes the transition strengths over the entire range of deformations, reducing to the two simple expressions in the appropriate limits.

Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.

Calculation of the Chirality in Solutions

Abstract The chirality parameter of a chiral chemical solution is derived as a new function of the specific rotation analytically. The Maxwell’s equations are solved together for an incident linearly polarized wave to find this function. The specific rotation formula found by the measurement is used to find the chirality parameter of chiral solutions. The calculated chirality parameter can be used to simulate the chiral molecule structure in different wavelengths and also to determine the specific rotation and the concentration of a chiral chemical solution.

Rotational states and masses of heavy and superheavy nuclei

The macroscopic-microscopic model with the Lublin-Strasbourg drop, the Strutinsky shell-correction method, and the BCS approach for pairing correlations is used with the cranking model to describe nuclear masses and rotational bands in even-even Ra to Cn isotopes of actinide and transactinide nuclei. The single-particle levels and potential-energy surfaces are calculated with the Yukawa-folded single-particle potential using the “modified funny hills” (c, h) shape parametrization. A monopole pairing force is used in our calculations. At equilibrium deformation the pairing strength is adjusted for every nucleus so as to reproduce the experimentally known rotational E2+ state within the cranking model. The pairing strength obtained in this way is used to predict the masses and rotational states in superheavy No to Cn nuclei. It is also shown that the rotational states with L 10 in Ra to No nuclei evaluated using a simple rotational formula agree quite well with the data. We propose a simple mechanism which takes into account a dynamical coupling of rotation with the pairing field that then allows one to obtain an excellent agreement with the data up to the states with the largest angular momenta (L 30) measured in this mass region.

Closed form of the rotational Crofton formula

AbstractThe closed form of a rotational version of the famous Crofton formula is derived. In the case where the sectioned object is a compact d‐dimensional C2 manifold with boundary, the rotational average of intrinsic volumes (total mean curvatures) measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^d$\end{document} with positive reach, the rotational average also involves hypergeometric functions. For convex bodies, we show that the rotational average can be expressed as an integral with respect to a natural measure on supporting flats. It is an open question whether the rotational average of intrinsic volumes studied in the present paper can be expressed as a limit of polynomial rotation invariant valuations.

VARIANCES OF LENGTH AND SURFACE AREA ESTIMATES BY SPATIAL GRIDS: PRELIMINARY STUDY

Periodic spatial grids can be used for unbiased estimation of length and surface area of objects by counting or measuring intersections of the objects with the grids. The estimators are theoretically based on discrete approximation of well established integral geometric formulas. The variance of the estimates depends on properties of both the grid and the measured objects. Main results of the theory of variance of the isotropic uniform random (IUR) volume estimation by spatial grids, especially a formula relating the variance of the volume estimator with the object surface area and the grid constant, are recapitulated. To identify main features of length and surface area IUR estimates the variance due to rotation and simple asymptotic formulas for the residual variance of estimates of selected model objects is calculated. Surface area estimates by multiple grids of parallel lines in 3D and of the variance of length estimates by periodic grids of planes or spheres in ddimensional space are studied.

Flowers and wedges for the stereology of particles

Recently, a new decomposition has been found for the motion invariant density of straight lines in, with applications in stereology. The new principle, called the invariator, leads to new rotational formulae which express the surface area and the volume of a bounded subset (called a 'particle') in terms of an observable functional defined in an isotropically oriented section (called a pivotal section) through a fixed point (called the pivotal point). The results have been extended to intrinsic volumes of manifolds in general space forms. The purpose of this paper is to present new results and computational formulae for three-dimensional particles. Explicit estimators are obtained for a convex polyhedral particle with a pivotal point in its interior, in terms of the coordinates of the vertices of the pivotal section. The results are applied to a population of polyhedral grains from a cemented carbide which was studied earlier by alternative methods.