Rotation Formula Research Articles

Solar differential rotation coefficients fitted from synoptic magnetic maps

ABSTRACT Based on the consecutive synoptic magnetic maps, we devise a new method to calculate the solar differential rotation coefficients. This method is very easy to implement and has a high accuracy. Firstly, based on the two-term or three-term differential rotation formula, we simulate a synoptic map CR$_{n}$ evolves one Carrington Rotation (CR) time only under the effect of the differential rotation, and thereby a stretched synoptic map CR$_{n*}$ is obtained. Then, through searching the maximum covariance between the maps CR$_{n*}$ and CR$_{n+1}$ by the grid search method, the rotation coefficients can be determined. Based on the synoptic maps of CRs 1625 to 2278 (during the years 1975–2023), the two-term coefficients A and B for latitude region between $\pm 40^{\circ }$ are calculated. The rotation coefficient B shows an obvious 11-yr period. From the time series of B, we find that the Sun usually rotates more differentially in the rising phases of the sunspot cycles than in the falling phases. Moreover, the strong magnetic field corresponds to an increasing of B (note that B has a negative sign) or decreasing of differential. The evolutionary trend of B also indicates that there are several years until the maximum value of B will be reached in solar cycle 25, and the coefficient B will be still in the rising phase in the few coming years. The two-term rotation coefficients for the two hemispheres are also calculated separately, and in the studied time-scale, the largest N–S asymmetry of the rotation rate appeared in October 2007.

Geometric error compensation method using the Laser R-test

Traditional methods for measuring geometric errors in machine tools, including interferometry and Double Ball Bar (DBB), are known to be expensive and time-intensive. Consequently, a non-contact calibration system called the “Laser R-test” has been developed. This innovative system is designed to measure both position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) efficiently. Since its development in 2000, this tool has been instrumental in analyzing eccentricity errors, angular position errors, and simultaneous trajectory errors. Through extensive research, it has been determined that the total error in a five-axis machine tool can be controlled to below 40 µm after compensating for eccentricity parameters and angular position errors. However, reducing this error to below ± 10 µm is challenging, primarily due to wobble errors in the orientation of the rotary axis without compensating. In this study, a new methodology based on Laser R-test and Rodrigues’ rotation formula has been developed to establish a PIGE model of rotary axis. Based on the methodology, the 8 PIGEs can be analyzed by measuring 5 coordinate positions. The compensation of 8 PIGEs in the rotary axis is completed within 30 min using the inspection path. Compatibility with ISO-10791–6 standards for BK1, BK2, and BK4 path tests is confirmed, validating the compensation effects. A precision of below ± 10 µm is achieved, with inspection time reduced by over 50%. This system can complete multiple errors by simply using the different paths. This greatly reduces the setup time for future users, enhancing its commercial applicability.

A generalized and comprehensive slant channel modeling method for underwater wireless optical communication and its system performance analysis

The underwater wireless slant optical link is a more generalized type of communication links compared with vertical and horizontal links. Since several oceanic factors including temperature and salinity variations with the seawater depth, the performance of the slant link for the underwater wireless optical communication (UWOC) will be overestimated under the assumptions of constant temperature and salinity. To this end, this paper proposes a model of slant link considering the depth-dependent and inclination-angle-dependent turbulence and scattering that are the main causes of multipath fading for UWOC. Specifically, to model the turbulence-induced fluctuations of light fields, the random phase screen method is exploited, where the power spectrum of refractive index is calculated using the oceanic turbulence optical power spectrum (OTOPS) model and the light fields at the receivers are derived by the angular spectrum method. To emulate the scattering-induced divergence, the scattering phase function is applied, where the motion directions and coordinates of light spots propagating among phase screens are deduced using Rodrigues’ Rotation Formula. Simulation results demonstrate the effects of the depths and inclination angles of transmitters on the performance of UWOC by analyzing the scintillation index, channel impulse response, and BER (bit error rate).

XLPE cable joint defects measurement method based on point cloud remapping

For the problems of low detection accuracy caused by weak texture and strong reflection on the surface of Cross-linked polyethylene (XLPE) high-voltage cable joint, this paper proposed a joint point cloud remapping and image segmentation method for the cable joint defects measurement, which improved the defects measurement accuracy from the correlation of the disordered point cloud and the salience of defects. Firstly, the discrete noise in the joint point cloud is removed using a modified radius filtering algorithm. Secondly, for the problems of disorderliness and weak association of the cable joint point cloud, we utilized Rodrigues' rotation formula (RRF) to correct the position of the XLPE cable joint point cloud. Then, for the demand of improving feature expression of non-significant defects, a point cloud remapping method is proposed, which remapping insignificant defects point cloud into saliency image for more effective detection. Lastly, an image segmentation method based on 8-neighborhood growth is applied for explicit defects detection, and the detected defects are quantified by the proposed measurement approach. Extensive experiments are conducted on the collected cable joints' point cloud and simulated cable joints' point cloud. With the help of the proposed method, we can achieve the best effects in DE, FR quantitative comparison, and the measurement inaccuracies are less than 4.5%. Experimental results demonstrate that the proposed method enjoys state-of-the-art performance in cable joint defects measurement.

Rotational Crofton formulae with a fixed subspace

The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. Motivated by stereological applications, we present variants of Crofton's formula, where the flats are constrained to contain a fixed linear subspace L0, but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case dim⁡L0=0. The proof combines a suitable Blaschke–Petkantschin formula with the classical Crofton formula. We also argue that our main result is best possible, in the sense that one cannot estimate intrinsic volumes of a set, based on lower-dimensional sections, other than those given by our result. Finally, we provide a proof for a well-established variant: an integral relation for vertical sections. Our formula is stated for intrinsic volumes of a given set, complementing the classical approach for Hausdorff measures.

Initial Structure Design and Optimization of an Automotive Remote Head-Up Display

Aiming at the problems of difficult construction and occlusion of the initial structure of an automotive augmented reality head-up display (AR-HUD), a method to quickly build the initial structure of an automotive AR-HUD is proposed. Firstly, the position of the mirrors in the initial structure is calculated based on the Rodrigues rotation formula. Secondly, the position of the mirrors is restricted by constraints during optimization to prevent the problem of structural occlusion. Finally, a virtual image display system with a visual distance of 7.5 m and a field of view angle of 10° × 5° is designed. The image quality analysis of the optimized system shows that the light spots in each field of view are all within Airy spots. At the spatial cutoff frequency of the virtual image plane of the optical system, the modulation transfer function (MTF) value of the full field of view is basically greater than 0.5, and the distortion is less than 1%. Finally, using the image for simulation, the results of the simulation image are satisfying, which proves the validity and feasibility of the structural design. It provides a useful reference for the structural design of the remote head-up display system.

Systematic Measurement of Temperature Errors of Positive and Negative FOG Scale Factors Using a Low-Precision Turntable

Measurement of temperature errors of fiber optic gyro (FOG) scale factor is important in FOG technology. In traditional methods, the measurement errors of FOG scale factors are greatly affected by the turntable’s orientation control errors. This article proposes a systematic method to measure the positive and negative FOG scale factor errors using a low-precision turntable. The relationship between the inertial navigation errors and the complete inertial measurement unit errors is revealed. A simplified tracking algorithm is devised to obtain the change rates of the velocity errors with lower computation. A measurement procedure is designed, with three groups of π and -π rotations around the east axis of the turntable at each temperature. Observation equations are derived to estimate the positive and negative FOG scale factor errors. Simulation and experimental results show that a low-precision turntable can accurately measure the FOG scale factor errors at each temperature. Furthermore, temperature errors of accelerometer biases, scale factors, and nonlinear scale factors can also be measured at the same time using the rotation sequences and formulas in this article without additional rotations.

Dispersive formalism for the nuclear structure correction δNS to the β decay rate

We analyze the axial $\gamma W$-box diagram for $I(J^P)=1(0^+)$ nuclei and provide a dispersion representation of the nuclear-structure correction $\delta_\text{NS}$ including its energy-dependent part. We also summarize useful isospin rotation formula and representations in nuclear theory that could facilitate the calculation of the parity-odd nuclear structure function $F_3(\nu,Q^2)$. They provide a rigorous theory framework for the future, high-precision calculation of the nuclear structure correction $\delta_\text{NS}$ necessary for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ud}|$ from superallowed nuclear $\beta$ decays.

Input-Constrained Hybrid Control of a Hyper-Redundant Mobile Medical Manipulator.

To reduce the risk of infection in medical personnel working in infectious-disease areas, we proposed a hyper-redundant mobile medical manipulator (HRMMM) to perform contact tasks in place of healthcare workers. A kinematics-based tracking algorithm was designed to obtain highly accurate pose tracking. A kinematic model of the HRMMM was established and its global Jacobian matrix was deduced. An expression of the tracking error based on the Rodrigues rotation formula was designed, and the relationship between tracking errors and gripper velocities was derived to ensure accurate object tracking. Considering the input constraints of the physical system, a joint-constraint model of the HRMMM was established, and the variable-substitution method was used to transform asymmetric constraints to symmetric constraints. All constraints were normalized by dividing by their maximum values. A hybrid controller based on pseudo-inverse (PI) and quadratic programming (QP) was designed to satisfy the real-time motion-control requirements in medical events. The PI method was used when there was no input saturation, and the QP method was used when saturation occurred. A quadratic performance index was designed to ensure smooth switching between PI and QP. The simulation results showed that the HRMMM could approach the target pose with a smooth motion trajectory, while meeting different types of input constraints.

Finite Element Numerical Simulation of Local Scour of a Three-Dimensional Cylinder under Steady Flow

Pile safety has received increasing attention in marine engineering, especially in the field of local scour. In this paper, a finite element numerical model is established for local scour around a cylinder in steady currents. The flow is described by unsteady Reynolds–averaged Navier–Stokes equations with a traditional turbulent closure model. The proposed scour model takes bed load into account. The Exner equation is solved to determine the bed variation and the moving mesh approach is used to capture the evolution of the bed. When the resulting slope exceeds the angle of repose, a novel sand-slide model based on Rodrigues' rotation formula is used to prevent simulation distortion. All the equations are discretized by the two-step Taylor–Galerkin algorithm, and the resulting approach is fast to implement with second-order accuracy in space. The numerical results are found to be in good agreement with the experimental data.

Indirect measurement of solar array rotation angle for satellite emergency rescue

In the emergency rescue mission, the primary task is to achieve the satellite’s orientation to the sun and ensure the energy supply. For this purpose, it is necessary to determine the rotation angle of solar array relative to satellite body. However, the rotation angle cannot be measured directly by the angle sensor in case of satellite emergency. To handle this issue, an indirect measurement method of solar array rotation angle is proposed in this paper. First, the indirect point measurement value of solar array rotation angle is given via the relationship between satellite body coordinate frame and solar array coordinate frame and the Rodrigues’ rotation formula. Then, the interval analysis technique is used to determine the upper and lower bounds of rotation angle and evaluate the influence of measurement noises. Based on the determined point measurement value and the interval analysis results, a zonotopic Kalman filter-based strategy is further proposed to improve the measurement accuracy. Finally, the effectiveness and validity of the proposed approach are demonstrated by numerical simulations, which illustrate its engineering application value.

Seismic performance of a narrow flange H-beam-to-column full-bolt end-plate weak-axis connection

This study proposes a full-bolted end-plate weak-axis connection T-shape joint based on a narrow flange H-beam-to-column section. A quasi-static test was conducted to solve the problems of large steel cross-section and high steel utilization, as well as the decline in space utilization and aesthetics caused by convex beam-to-column components in low-rise and multi-story steel frame residential buildings. The failure mode, hysteresis curve, and bearing capacity of the specimens were discussed and investigated. The test results show that the stiffness of the joints of the specimens are relatively high, the failure mode is the beam hinge failure mechanism, the hysteresis curves of the joints are full fusiform, and the seismic performance is good. An experimental numerical model was established using the ABAQUS software to determine the factors that affect the performance of the joint. The effects of skin thickness, bolt diameter, beam end-plate thickness, stiffener thickness and rid slope on the seismic performance of the T-shape joint were analyzed. The numerical simulation results indicate that when the thickness of the skin plate increases, the energy dissipation capacity of the joint significantly increases, whereas the ductility decreases. a moment–rotation formula that can be used as a reference in the control design of the key parameters of the joint was obtained, which is useful in practical engineering applications.

Needle Tip Pose Estimation for Ultrasound-Guided Steerable Flexible Needle With a Complicated Trajectory in Soft Tissue

Visualization of the surgical needle is critical for an image guided insertion. However, it is difficult to obtain a 3D tip pose of a steerable flexible needle under 2D US images. In this letter, an image processing method is used to extract the needle shaft radial cross-sectional centroid coordinates (NSRCCCs). Based on the NSRCCCs, two methods are proposed to estimate the needle tip pose in real-time, which are based on Rodrigues’ rotation formula, and Fourier expansion with three-terms and linear interpolation algorithm (FEWT&LIA), respectively. In order to validate the methods proposed in this letter, the insertion experiments with different depths are conducted in phantom tissues, and the results show that the mean errors are 0.68 mm and 0.65 mm with using the Rodrigues’ rotation formula based method and the FEWT&LIA based method, respectively, showing clinically acceptable accuracy.

Rigid Body Approximation for the Anharmonic Description of Molecule-Surface Vibrations.

We present an anharmonic approach for molecule-surface vibrations that employs rigid body coordinates, based on the Rodrigues rotation formula, to describe curvilinear displacements (rotations and translations) of the molecule along normal modes. These displacements are used to calculate energy data points from which one-dimensional polynomial potentials are fitted using cubic splines. In these potentials, for each of the six rigid body modes separately, one-dimensional Schrödinger equations are solved with harmonic oscillator or Fourier functions (for pure rotations) as basis sets. The anharmonic vibrational energies obtained are used to calculate partition functions and from them enthalpies, entropies, and Gibbs free energies of adsorption. Our numerical implementation has been successfully tested for Morse and cosine potentials with known analytical solutions. The methods have been applied to adsorption of CH4 on the hydroxyl group of the proton form of the chabazite zeolite (H-CHA), as well as to adsorption of CH4 and CO on the Mg2+ ions of the metal-organic framework (MOF) Mg2(dobdc). To obtain the best estimates for thermodynamic functions, we include the coupling between molecule-surface vibrations and intrasystem vibrations at the harmonic level. The calculated Gibbs free energies show that the coupling is small for CH4/H-CHA and CO/MOF (between -0.7 and +0.1 kJ/mol) but substantial for CH4/MOF (-3.4 kJ/mol). The predicted anharmonic effect on the Gibbs free energy of adsorption for CH4/H-CHA, CH4/MOF, and CO/MOF is -4.7, 0.3 ± 0.7, and -2.4 ± 0.6 kJ/mol, respectively, which results in +4.2, +0.9 ± 0.7, and -0.4 ± 0.6 kJ/mol, respectively, for the deviation from experiment. This is well within chemical accuracy limits (±4.2 kJ/mol) for the adsorption of CH4 and CO in the MOF. The larger deviation for CH4/H-CHA, at the edge of the chemical accuracy range, is most likely due to contributions from soft zeolite modes which are neglected in our approach.

Correction of full-field rigid body rotation and strain using a hybrid peridynamics and digital image correlation approach

Rigid body rotation is difficult to be removed independently while using digital image correlation (DIC) technique for monitoring the displacement and deformation of an object. Furthermore, strain measurement can also become inaccurate due to the presence of displacement caused by rigid body rotation. Common solutions to this problem, such as tracking a single-point's rotation, require additional reference information and only obtain one rigid body rotation angle of the object. In this study, a hybrid computational and experimental mechanics method based on peridynamics (PD) and DIC technique is proposed to measure the full-field rigid body rotation and strain. A DIC-based PD model can be established based on the full-field position data obtained by DIC. The calculation formulas for rigid body rotation and strain are derived in the framework of PD and the calculations of rotation and strain are mutually insensitive and independent. Therefore, without any additional reference information, the rigid body rotation and strain information can be obtained in the presence of simultaneous rigid body rotation and deformation. Different rigid body rotations in various regions of an object, whether large or small rotations, can be calculated by this method. Correspondingly, this method is effective for calculating the local rigid body rotations of objects undergoing local fracture. Numerical simulations and experimental tests were performed to verify the effectiveness of the proposed method. The calculation results show that the proposed method owns the ability to accurately calculate the rotation and strain in the above scenarios.

Subassemblage test and gusset rotation response of buckling-restrained braced steel frames

Gusset rotation would cause adverse influence on the buckling-restrained brace (BRB) end zone and even lead to premature buckling of the BRB end zone. However, limited studies to date referred to the gusset rotation response. To evaluate the cyclic behavior and gusset rotation response, five 1/2-scaled buckling-restrained braced frame (BRBF) subassemblages with various gusset connections, i.e., pinned gusset connection, bolted gusset connection, welded gusset connection, pinned-welded gusset connection, and pinned-bolted gusset connection, were conducted under horizontal cyclic loading. The effects of gusset connection type on the failure modes, horizontal initial stiffness, energy dissipation and rotation response of specimens were analyzed in detail. It showed that all specimens performed plump and stable hysteresis behavior. The upper gusset rotation of the specimen with pinned gusset connection was larger than that of bolted gusset connection, followed by welded connection. Meanwhile, the finite element (FE) models were established and validated by test results. Furthermore, the standard FE model of a full-scale one-story one-bay steel moment BRBF with a single diagonal BRB was developed and then extended to a large number of models to investigate the contribution of various parameters on the upper gusset rotation. Finally, the gusset rotational formulae for predicting its response, as the steel moment BRBF reached to either 2.0% or 3.0% drift, were proposed on the basis of the results of the extensive parametric analysis. It could be used to determine the rotation of the BRB end zone, and the flexural moment acting on the BRB end zone and corresponding practical design procedure can be finally derived to avoid the potential premature failure of the BRB end zone.

Spectral element formulation for damped transversely isotropic Micropolar-Cosserat layered composite panels

The present paper aims to develop governing equation of motion for in-plane dynamics of Micropolar-Cosserat composite models with damping. Constitutive model of linear elastic damping system is formulated for an anisotropic domain fiber-reinforced composite panels (FRCP); undergoing large macro as well as micro geometric deformations. The air damping and Kelvin–Voigt strain linear rate damping have been considered into the governing equations of model, while mathematical modelling and simulation of composite panel is restricted to the free-vibration and in-plane static response. The composite panel has been modeled as a Micropolar-Cosserat continuum assuming second-order micro-length of the fiber deformation; by embedding an additional equation of kinematics through the micro-rotation degree of freedom in the classical continuum model. This account for the in-plane curvature bending effects of composite panels during the loss of ellipticity of the governing equations. A transformation matrix based on Rodrigues’ rotational formula for transversely isotropic Micropolar-Cosserat lamina has been introduced; which reduces it to the well-known non-classical (classical and couple-stress) elastic formulation. The equivalent single layer (ESL) resultant stresses of FRCP in global coordinates is introduced to calculate in-plane damped and undamped response. The geometric and material linear elastic model for FRCP is derived using the spectral element method within state–space approach, and the corresponding plane-stress finite element model is validated with the undamped responses. Analytical response of damped composite panel is proposed based on available undamped simulation results.

The Behavior of Moments of Inertia and Energy Staggering in Superdeformed Nuclei.

Three pairs of signature partners' transition energies of Thallium (Tl) odd mass superdeformed (SD) nuclei's (A ~ 191-195), were fitted with the experimental one using the Bohr-Mottelson four-parameter collective rotational model. We chose the Bohr-Mottelson four-parameter rotational energy formula because it has been reported that it has excellent compatibility with the γ-ray transition energies. The four model parameters were extracted using a suitable search program. Harris's method was used to calculate the superdeformed rotational bands' (SDRBs) bandhead spins. The values of the adopted parameters, which were obtained using a simulated fitting search software, were used to measure the rotational frequency, dynamic J(2), and kinematic J(1) moments of inertia to the transition energies. When compared to the experimental values, there is a great deal of agreement J(2) and J(1) have been studied as a function of increasing rotational frequency. The suggested staggering function was used to investigate the ΔI= 1 energy staggering in Tl odd mass SD nuclei.

Bimodulus constitutive relation and mesoscopic model of braided composites

Braided composites are widely used in engineering, but there is little research on the bimodulus constitutive relation of tension and compression. In this paper, a bimodulus constitutive model suitable for two-dimensional (2D) 2 × 2 braided carbon fibre composites is proposed. The modulus prediction method and stress rotation formula were optimised and verified. According to the tension and compression tests and two stacking sequences that were conducted, the degrees of deviation of the moduli were found. To identify the underlying cause of the modulus deviation, two unit cell models were built with different stacking angles. By changing the fibre volume fraction and braiding angle of the unit cell models, the tensile and compressive moduli under the influence of these two parameters were obtained. As the fibre volume fraction and braiding angle increased, the deviation degree of the two moduli increased. When the braiding angle was small, the tensile modulus was greater than the compressive modulus, while when the braiding angle was relatively large, the compressive modulus was greater than the tensile modulus.

Effects of manufacturing micro-structure on vibration of FFF 3D printing plates: Material characterisation, numerical analysis and experimental study

This paper presents the effects of manufacturing micro-structure on the elastic properties of FFF (fused filament fabrication) 3D printing material and the vibration characteristics of FFF 3D printing plates. According to the results of scanning electron microscopy, three different material directions are defined to describe the orthotropy of FFF 3D printing material including fibre direction, intra-layer direction, and inter-layer direction. Then, the constitutive model of the orthotropic elastic 3D printing material is established based on static experimental results and plane stress rotation formula. Static experimental results (totally 27 kinds of specimens) show that the largest difference of Young’s moduli in different material directions is 1104.684 MPa which is even larger than the Young’s modulus in the intra-layer direction. Finite element models of FFF 3D printing plates with six different printing directions (α-0°, α- 90°, β-0°, β-90°, λ-0°, λ-90°) and three different layer thicknesses (0.2 mm, 0.25 mm, 0.3 mm) are built and their natural frequencies are determined. Simulation results show that the largest difference of the natural frequencies among plates with different printing directions are 10.645 Hz, 39.588 Hz, 66.710 Hz, 123.780 Hz, and 185.720 Hz when the mode of the plates changes from one to five. Meanwhile, these differences are relative large when compared with the first five out-of-plane natural frequencies of these plates. Additionally, vibration experiments of FFF 3D printing plates are carried out to verify the validity of the simulation process and all the relative errors of the natural frequencies between simulation and the experimental results are found to be very small. Therefore, the accuracy of the finite element models is good enough and one can confirm that the effects of manufacturing micro-structure on the vibration characteristics are significant.