Arbitrary Axis Research Articles

О геометрической интерпретации кватернионов конусами

The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date.
 The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation.
 The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs.
 For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis.
 Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.

Geometric error identification of five-axis machine tools using dual quaternion

A universal identification method for position independent geometric errors (PIGEs) of dual rotary axes of five-axis machine tools with arbitrary rotary axis position structures is proposed based on the unit dual quaternion (UDQ). The PIGEs of translational/rotary axes are redefined based on UDQ transformation, which effectively reduces the transformation parameters and avoids the unitization process of UDQ compared with the existing UDQ kinematics model. Based on the redefined PIGEs, a kinematic error model of a five-axis machine tool is established for error identification. Taking the cradle-type five-axis machine too with non-intersecting rotary axes as the research objective, a novel method for identifying PIGEs through the synchronous motion of dual rotary axes using a double ball bar (DBB) is proposed. Compared with BK1 and BK2 in ISO10791-6 and other existing methods, the advantage of the proposed method is that it can greatly reduce the complexity of the experiment and the operation steps by directly constructing the relative kinematic relationship between the tool and the workpiece based on the error model. Only one installation and one measurement trajectory are needed to simultaneously identify the eight PIGEs of the dual rotary axes. In addition, a trajectory equalization algorithm is proposed to effectively solve the asynchronous problem of motion and data sampling in DBB trajectory operation. By combining the simplified error model with the sampling data of DBB, the pseudo-inverse matrix is used for decoupling. The effectiveness of the proposed method is verified by DBB compensation comparison experiment. The proposed error identification method can be applied to other types of five-axis machine tools with a few changes in the configural parameters, which is ideal for periodic checks and calibration to ensure the machining accuracy.

Mechanical Mode Engineering with Orthotropic Metamaterial Membranes

AbstractMetamaterials are structures engineered at a small scale with respect to the wavelength of the excitations they interact with. These structures behave as artificial materials whose properties can be chosen by design, mocking and even outperforming natural materials, and making them the quintessential tool for manipulation of wave systems. In this paper it is shown how the acoustic properties of a silicon nitride membrane can be affected by nanopatterning. The degree of asymmetry in the pattern geometry induces an artificial anisotropic elasticity, resulting in the splitting of otherwise degenerate mechanical modes. The artificial material introduced has a maximum Ledbetter–Migliori anisotropy of 1.568, favorably comparing to most bulk natural crystals. With an additional freedom in defining arbitrary asymmetry axes by pattern rotation, the described approach can be useful for fundamental investigation of material properties as well as for devising improved sensors of light, mass, or acceleration based on micromechanical resonators.

Electromagnetic beam transmission in a uniaxial anisotropic slab with arbitrary optical axis

Based on the electromagnetic fields expansions over rectangular vector wave functions or eigen plane waves, the reflection and refraction of an electromagnetic beam by a uniaxial anisotropic slab are analyzed. The unknown expansion coefficients are determined from a system of linear equations derived by using the boundary conditions. In the uniaxial anisotropic slab, the optical axis is arbitrarily oriented with respect to the interface. As two typical examples, the normalized field intensity distributions are evaluated for a Gaussian beam and Hermite-Gaussian beam, and the propagation features are discussed briefly.

A template method to measure the t {bar{t}} polarisation

We develop a template method for the measurement of the polarisation of t {bar{t}} pairs produced in hadron collisions. The method would allow to extract the individual fractions of t_L bar{t}_L, t_R {bar{t}}_R, t_L {bar{t}}_R and t_R {bar{t}}_L pairs with a fit to data, where L, R refer to the polarisation along any axis. These polarisation fractions have not been independently measured at present. Secondarily, the method also provides the net polarisation of t and {bar{t}}, as well as their spin correlation for arbitrary axes.

Flutter stability analysis of an elastically supported flexible foil. Application to the energy harvesting of a fully-passive flexible flapping-foil of small amplitude

The aerodynamic forces on an oscillating flexible foil are used to study the flutter instability when the flexible foil is elastically mounted to translational and torsional springs and dampers at an arbitrary pivot axis. The present linear theory, valid for small amplitudes of the heaving, pitching and flexural deflection motions, and therefore valid for sufficiently large stiffness ratios, characterizes analytically the onset of the flutter instability and the corresponding leading frequency in terms of the flow velocity and all the structural parameters of the system. The analysis may serve to guide the search for the parametric ranges of energy extraction by a fully-passive flexible flapping-foil hydrokinetic turbine, including the effect of some relevant nondimensional parameters which have not been considered before. The results for the rigid-foil case are validated with recent numerical simulations for a fully-passive flapping-foil turbine. As the stiffness of the foil decreases, the coupled-mode flutter instability of the elastically supported rigid foil may weaken and disappear, or become enhanced, depending on the remaining parameters, most particularly on the location of the centre of mass in relation to the pivot point, whose dependence is investigated for specific values of the rest of the nondimensional parameters.

Development of dynamic motion models of SPACE code for ocean nuclear reactor analysis

Lately, ocean nuclear power plants have attracted attention as one of diverse uses of nuclear power plants. Because ocean nuclear power plants are movable or transportable, it is necessary to analyze the thermal hydraulics in a moving frame of reference, and computer codes have been developed to predict thermal hydraulics in large moving systems. The purpose of this study is to incorporate a three dimensional dynamic motion model into the SPACE code (Safety and Performance Analysis CodE) so that the code is able to analyze thermal hydraulics in an ocean nuclear power plant. A rotation system that describes three-dimensional rotations about an arbitrary axis was implemented, and modifications were made to the one-dimensional momentum equations to reflect the rectilinear and rotational acceleration effects. To demonstrate the code's ability to solve a problem utilizing a rotational frame of reference, code calculations were conducted on various conceptual problems in the two-dimensional and three-dimensional pipeline loops. In particular, the code results for the three-dimensional pipeline loop with a tilted rotation axis agreed well with the multi-dimensional CFD results.

Determinação do eixo intercondilar

This study is for the arbitrary intercondilar to the real intercondilar hinge axis, with the help of a kinematic face bow. In the 21 patients we determined the real and the arbitrary hinge axis, registering the results in a melimetred paper. The results obtained were compared confirming that 58% the real intercondilar hinge axis was localized into a radius of 4 mm from the arbitrary intercondilar hinge axis.

Energy Harvesting and Propulsion of Pitching Airfoils with Passive Heave and Deformation

A general formulation for the fluid–structure interaction of an airfoil undergoing prescribed pitching and passive heave and deformation about an arbitrary axis is given in the linearized potential flow limit. The resulting three algebraic equations provide the heave and deformation amplitudes linearly from two of the equations, as well as the torque that generates the pitching motion (and hence the input power) from the third. Both, the energy harvesting and the propulsion problems are analyzed. In the first case, energy is harvested at the pivot point through dampers so that the efficiency is obtained once the heaving motion and the input power are computed. In the second problem, the prescribed pitch and the computed heave and deformation yield the thrust as well as the propulsive efficiency with the obtained input power. The present analytical results allow for a general survey of these two physical problems as the multiple governing parameters are varied. It is generally found that the best performance is achieved around the first natural frequency, which is obtained here from a simple algebraic expression. The formulation is validated with previous experimental and numerical results. The effect of flexibility maximizing the energy harvesting efficiency is particularly analyzed.

The University M\xfcnster model surgery system for orthognathic surgery - The digital update

AimThe aim of this work is to present a digital methodology of a conventional articulator based planning protocol.MethodsThe digital counterpart consists of intra-oral scans (3Shape) and a free available 3D mesh software (Meshmixer, Autodesk). The maxillary position in relation to the reference plane used and the arbitrary hinge axis were determined mathematically from landmarks on cephalometric x-rays and frontal photographs. Distances and angles were calculated to virtually mount the digital jaws in Meshmixer’s wold frame. Virtual planning is done by cloning and moving the jaws according to the preliminary surgery plan. The spatial movements of the jaws are measured by attached reference points.ResultsThis digital approach eliminate the need for articulator hardware and laboratory plaster work. It enables all planning scenarios as they are also possible with conventional plaster-based procedures. The method is time-saving, practical and cost-effective. Standard dimensions of articulators and face-bows have been incorporated in the implementation. This reduction of individual patient values puts the accuracy of the presented method within the range of conventional model surgery.ConclusionArbitrary planning will continue to have its place in orthognathic surgery, especially when digital methods can improve the overall process. The method presented can be seen as a cost-effective alternative for patients who do not require technically complex planning.

A Novel Technique to Align the Intraoral Scans to the Virtual Articulator and Set the Patient-Specific Sagittal Condylar Inclination.

Customized cast orientations and parameter settings of the virtual articulator according to the patient's condyles are indispensable parts of today's digital workflows in prosthodontics. This article describes a digital technique to align the intraoral scans to a virtual articulator by using a facial scanner to locate the patient's cutaneous landmarks of the arbitrary hinge axis and the reference plane, and to customize the sagittal condylar inclination of the virtual articulator through a digital protrusive interocclusal record and a dental computer-aided design software program. It enables individual cast orientations and virtual articulator parameter settings without conventional facebow transferring and bite registration procedures and can be easily integrated with most virtual articulator systems on the market to allow clinicians and technicians to work in a complete digital workflow and facilitate customized treatment planning and dental prosthesis fabrication.

Optimization of Magnetic Flux Ropes Modeled with the Regularized Biot–Savart Law Method

The so-called regularized Biot-Savart laws (RBSLs) provide an efficient and flexible method for modeling pre-eruptive magnetic configurations of coronal mass ejections (CMEs) whose characteristics are constrained by observational images and magnetic-field data. This method allows one to calculate the field of magnetic flux ropes (MFRs) with small circular cross-sections and an arbitrary axis shape. The field of the whole configuration is constructed as a superposition of (1) such a flux-rope field and (2) an ambient potential field derived, for example, from an observed magnetogram. The RBSL kernels are determined from the requirement that the MFR field for a straight cylinder must be exactly force-free. For a curved MFR, however, the magnetic forces are generally unbalanced over the whole path of the MFR. To minimize these forces, we apply a modified Gauss-Newton method to find optimal MFR parameters. This is done by iteratively adjusting the MFR axis path and axial current. We then try to relax the resulting optimized configuration in a subsequent line-tied zero-beta magnetohydrodynamic simulation toward a force-free equilibrium. By considering two models of the sigmoidal pre-eruption configuration for the 2009 February 13 CME, we demonstrate how this approach works and what it is capable of. We show, in particular, that the building blocks of the core magnetic structure described by these models match to morphological features typically observed in such type of configurations. Our method will be useful for both the modeling of particular eruptive events and theoretical studies of idealized pre-eruptive MFR configurations.

An Analysis of the Effect of a Change in the Support Point Location on the Vibration of Thin-Walled Beams

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

Rapid computational cell-rotation around arbitrary axes in 3D with multi-core fiber

Optical trapping is a vital tool in biology, allowing precise optical manipulation of nanoparticles, micro-robots, and cells. Due to the low risk of photodamage and high trap stiffness, fiber-based dual-beam traps are widely used for optical manipulation of large cells. Besides trapping, advanced applications like 3D refractive index tomography need a rotation of cells, which requires precise control of the forces, for example, the acting-point of the forces and the intensities in the region of interest (ROI). A precise rotation of large cells in 3D about arbitrary axes has not been reported yet in dual-beam traps. We introduce a novel dual-beam optical trap in which a multi-core fiber (MCF) is transformed to a phased array, using wavefront shaping and computationally programmable light. The light-field distribution in the trapping region is holographically controlled within 0.1 s, which determines the orientation and the rotation axis of the cell with small retardation. We demonstrate real-time controlled rotation of HL60 cells about all 3D axes with a very high degree of freedom by holographic controlled light through an MCF with a resolution close to the diffraction limit. For the first time, the orientation of the cell can be precisely controlled about all 3D axes in a dual-beam trap. MCFs provide much higher flexibility beyond the bulky optics, enabling lab-on-a-chip applications and can be easily integrated for applications like contactless cell surgery, refractive index tomography, cell-elasticity measurement, which require precise 3D manipulation of cells.

Microrheology With an Anisotropic Optical Trap

Microrheology with optical tweezers (MOT) measurements are usually performed using optical traps that are close to isotropic across the plane being imaged, but little is known about what happens when this is not the case. In this work, we investigate the effect of anisotropic optical traps on microrheology measurements. This is an interesting problem from a fundamental physics perspective, but it also has practical ramifications because in 3D all optical traps are anisotropic due to the difference in the intensity distribution of the trapping laser along axes parallel and perpendicular to the direction of beam propagation. We find that attempting viscosity measurements with highly anisotropic optical traps will return spurious results, unless the axis with maximum variance in bead position is identified. However, for anisotropic traps with two axes of symmetry such as traps with an elliptical cross section, the analytical approach introduced in this work allows us to explore a wider range of time scales than those accessible with symmetric traps. We have also identified a threshold level of anisotropy in optical trap strength of ~30%, below which conventional methods using a single arbitrary axis can still be used to extract valuable microrheological results. We envisage that the outcomes of this study will have important practical ramifications on how all MOT measurements should be conducted and analyzed in future applications.

Light refraction from a uniaxial crystal with arbitrary optic axis orientation using matrix representation

We have studied theoretically the refraction of a light beam in a uniaxial crystal plate by observing the electric field vector components, for an arbitrary orientation of optic axis with respect to plane of incidence and plane of interface. The electric field components for ordinary and extraordinary light beams are studied with varying the angle of optic axis with respect to plane of incidence and crystal surface and also with angle of incidence, in order to find the optimized position of optic axis orientation for various application prospects. We have used the matrix formulation to transform and simplify the electric field components from principal coordinate of crystal to the laboratory coordinate frame of crystal plate. It was observed that when optic axis is in the plane of incidence and optic axis is in the surface of crystal plate, ordinary and extraordinary beams have nonzero x and y components. While in other cases when optic axis is perpendicular to the plane of incidence and perpendicular to the surface of crystal plate, ordinary beam and extraordinary beams have only y and x components, respectively.

Experimental demonstration of novel beam characterization using a polarizable X-band transverse deflection structure

The PolariX TDS (Polarizable X-Band Transverse Deflection Structure) is an innovative TDS-design operating in the X-band frequency-range. The design gives full control of the streaking plane, which can be tuned in order to characterize the projections of the beam distribution onto arbitrary transverse axes. This novel feature opens up new opportunities for detailed characterization of the electron beam. In this paper we present first measurements of the Polarix TDS at the FLASHForward beamline at DESY, including three-dimensional reconstruction of the charge-density distribution of the bunch and slice emittance measurements in both transverse directions. The experimental results open the path toward novel and more extensive beam characterization in the direction of multi-dimensional-beam-phase-space reconstruction.

General Stiffness Matrix for a Thin-Walled, Open-Section Beam Structure

This paper is to review the theory of thin-walled beam structures of the open cross-section. There is scant information on the performance of structures made from thin-walled beam elements, particularly those of open sections, where the behavior is considerably complicated by the coupling of tensile, bending and torsional loading modes. In the combined loading theory of thin-walled structures, it is useful to mention that for a thin-walled beam, the value of direct stress at a point on the cross-section depends on its position, the geometrical properties of the cross-section and the applied loading. This applies whether the thin-walled section is closed or open but this study will be directed primarily at the latter. Theoretical analyses of structures are fairly well established, considered in multi-various applications by many scientists. However, due to the present interest in lightweight structures, it is necessary to specify where the present theory lies. It does not, for example, deal with compression and the consequent failure modes under global and local buckling. Indeed, with the inclusion of strut buckling failure and any other unforeseen collapse modes, the need was perceived for further research into the subject. Presently, a survey of the published works has shown in the following: 1) The assumptions used in deriving the underlying theory of thin-walled beams are not clearly stated or easily understood; 2) The transformations of a load system from arbitrary axis to those at the relevant centre of rotation are incomplete. Thus, an incorrect stress distribution may result in; 3) Several methods are found in the recent literature for analyzing the behaviour of thin-walled open section beams under combined loading. These reveal the need appears for further study upon their torsion/flexural behaviour when referred to any arbitrary axis, a common case found in practice. This review covers the following areas: 1) Refinement to existing theory to clarify those observations made in 1 - 3 above; 2) Derivation of a general elastic stiffness matrix for combined loading; 3) Calculation of the stress distribution on the cross-section of a thin-walled beam. A general transformation matrix that accounts for a load system applied at an arbitrary point on the cross-section will be published in a future paper.

Thermal Lens Astigmatism Induced by the Photoelastic Effect in m3m, 432, and 43m Symmetry Cubic Crystals

The expressions for the thermal lens of linearly polarized radiation arising in m3m, 432 and $\bar {4}3\text{m}$ cubic crystals with arbitrary crystallographic axes orientation in the presence of circular birefringence $\delta _{\mathbf {c}}$ are analyzed. It is shown that the thermal lens may have astigmatism the magnitude of which is proportional to the power of heat generation. The Euler angles determining the crystallographic axes orientation [P] are found and the conditions for the material constant – optical anisotropy parameter $\xi $ – under which the thermal lens astigmatism in active and magneto-active optical elements can be eliminated, are formulated. It is demonstrated that the orientation [P] depends only on optical anisotropy parameter $\xi $ of the crystal and on circular birefringence $\delta _{\mathbf {c}}$ .

Modeling and analysis of solar Doppler difference bias with arbitrary rotation axis

The solar rotation causes the solar Doppler difference bias, which leads to the decline of the velocity measurement accuracy. Modeling and compensation are an effective solution. The limited model with specific geometric direction, where the solar rotation axis is perpendicular to the plane through the Sun, the Earth and Mars, was established. However, in fact, the geometric relationship among the Sun, Mars and the spacecraft is not fully in line with the hypothesis of the model due to the spacecraft orbital angle and the solar rotation axis drift. Thus, this model is not consistent with the fact. In order to solve this problem, a universal solar Doppler difference bias model, which provides the expression with arbitrary rotation axis, is established in this paper. In this method, for any point at the solar surface, four variables including the direction of the solar rotation linear velocity at this point, the distance from this point to the rotation axis, the vector from this point to Mars, and the vector from this point to the spacecraft are calculated. Based on these four variables, the solar Doppler difference bias corresponding to this point is obtained. The theoretical analysis and simulation results demonstrate that the solar Doppler difference bias model with the actual rotation axis is different from that with one of the specific rotation axes. Therefore, it is indispensable to build the proposed model for compensation. Besides, the direction of the solar rotation axis, the spacecraft-Mars-Sun angle and the spacecraft-to-Mars distance are important impact factors for the proposed model.