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This letter introduces a novel approach that combines the use of exponential barrier functions and satisfiability modulo theories solving to the formal safety analysis of flight controllers. Specifically, exponential barrier functions are used to construct candidate invariant regions and proofs of invariance are constructed using satisfiability modulo theories solving. This approach is demonstrated by analyzing safety properties related to command tracking and operation in cases of rotor failures on an octorotor platform.

ВВ криптографических системах шифрования информации используются гиперкомплексные числа: кватернионы и октонионы. В качестве ключа применяется кватернион, который производит вращения группы выборок информации. Кватернионы и бикватернионы являются частными случаями геометрической алгебры Клиффорда. Использование векторов и мультивекторов геометрической алгебры для шифрования информации позволяет расширить разнообразие этих векторов. Для шифрования информации, представленной совокупностью векторов геометрической алгебры, эти векторы умножаются на мультивектора, которые осуществляют операцию ротор (rotor). В качестве ключа используется мультивектор (ротор). Для дешифрования информации применяется операция, которая соответствует обратному ротору. Алгоритмы геометрической алгебры повышают безопасность шифрования информации за счет повышения размерности алгебры. Для повышения производительности шифрования предлагается коэффициенты информационного вектора и мультивектора вращения выбирать из поля Z256. Предлагается вектор информации с коэффициентами из Z256 складывать со случайным вектором с коэффициентами из Z256 и считать эти коэффициенты ключами шифрования. Приведены базисные векторы применяемых геометрических алгебр и таблицы геометрических произведений базисных векторов.

The authors solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. The authors show its efficiency in two applications: the as-rigid-as-possible (ARAP) surface modelling and the more smooth rotation enhanced ARAP mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. The ARAP technique along with its improved variants, although have been extensively studied, can still not be achieved efficiently. Linear geometric algebra based rotor solution proposed in this study gives another perspective of the kernel problem. This, however, not only improves the real performance of the three-dimensional mesh deformation, but also provides a brand new computationally efficient solution to the Wahba's problem and point cloud registration, which has been closely related to the automation science and engineering.

The problem of rolling the balanced dynamically nonsymmetric ball with a rotor on a rough horizontal plane is considered. Topological types of isoenergy surfaces of this integrable Hamiltonian system are found. Curves are constructed on the plane of the parameters $$\mathbb{R}^{2}(h,c)$$ separating it into regions so that all points of the same region correspond to isoenergy surfaces with identical Fomenko–Zieschang invariants.

We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal non-invariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI $\rightarrow$ TRNI transition depend on $\alpha^2/N$, where $\alpha$ is the chaos parameter of the QKR and $N$ is the dimensionality of the evolution operator matrix. For $\alpha^2/N \gtrsim N$, the transition coincides with that in random matrix theory. For $\alpha^2/N < N$, the transition shows a marked deviation from random matrix theory. Further, the speed of this transition as a function of the magnetic field is significantly enhanced as $\alpha^2/N$ decreases.

In the semiclassical theory of rotational transitions, S-matrix elements are expressed as integrals over initial and final angles of probability amplitudes calculated along the classical paths joining these angles, before final passage to an initial value representation [W. H. Miller, J. Phys. Chem. A 105, 2942 (2001)]. These angles can be either natural angles fixing the orientation of the rotor or angles shifted with respect to the previous ones so as to vary only within the interaction region causing the transitions. The two approaches, however, were recently shown to lead to different predictions. While the theory in natural angles lacks precision and exhibits unphysical behavior, the theory in shifted angles is much more accurate and physically well behaved [L. Bonnet, J. Chem. Phys. 153, 174102 (2020)]. The present work is devoted to the analysis of this unexpected finding.

In complex-valued Hopfield neural networks (CHNNs), the neuron states are complex numbers whose amplitudes are: 1) they can also be described in special orthogonal matrices of order and 2) here, we propose a new Hopfield model, the $O(2)$ -valued Hopfield neural network [ $O(2)$ -HNN], whose neuron states are extended to orthogonal matrices. Its neuron states are embedded in 4-D space, while those of CHNNs are embedded in 2-D space. Computer simulations were conducted to compare the noise tolerance (NT) and storage capacity (SC) of CHNNs, $O(2)$ -HNNs, and rotor Hopfield neural networks. In terms of SC, $O(2)$ -HNNs outperformed the others, while in NT, they outdid CHNNs.

This paper proposes a nonlinear finite-element analysis for slender composite structures with anisotropic material properties using the mixed geometrically exact beam formulation and exponential map finite rotation. Rodrigues finite rotation expression used in the existing mixed-variational formulation of the geometrically exact beam is further improved by replacing it with the exponential map finite rotation, which is free of singularities. As a result, the existing mixed-variational formulation is further simplified by the matrix arithmetic, so that numerical implementation of the resulting equations may become more compact. Using the present beam formulation, nonlinear static displacements of a straight beam, twisted straight beam, and initially curved beam are precisely predicted within smaller number of iterations compared to the existing formulations. In addition, the present eigenmode analysis is capable of providing the rotating natural frequencies for a full-scale realistic rotor blade accurately. As a result, the present analysis is applicable for large deflection behavior of a composite structure without the singularities in a better convergence.

This work is interested in studying the motion of a rigid body carrying a rotor that rotates with a constant angular velocity about an axis parallel to the axis of dynamical symmetry. This motion is assumed to take place due to the effect of a combination of both uniform fields of gravity and magnetism that do not possess an axis of common symmetry. The equations of motion are constructed, and they are rewritten by means of the Hamiltonian function in the framework of the Lie–Poisson system. The equilibrium positions are inserted. The necessary conditions for the stability are introduced by applying the linear approximation method, while the sufficient conditions for stability are determined by utilizing the energy-Casimir method.

In this letter, starting from recent experiments about the circular motion of a rotor with an absorber of photons emitted by a Mossbauer source, we want to underline some mathematical aspects in General Relativity framework. We do not want to discuss in detail the different physical interpretations of the experimental results proposed during the recent years and we do not want to propose a new one. Indeed, starting from a paper awarded to Gravity Research Competition 2018, the aim of our work is to analyze three different types of time involved in this experiment linking a term introduced in the above mentioned paper, to the difference between coordinate and physical velocity of light.

The quantum kicked rotor is well-known to display dynamical localization in the non-interacting limit. In the interacting case, while the mean-field (Gross-Pitaevskii) approximation displays a destruction of dynamical localization, its fate remains debated beyond mean-field. Here we study the kicked Lieb-Liniger model in the few-body limit. We show that for any interaction strength, two kicked interacting bosons always dynamically localize, in the sense that the energy of the system saturates at long time. However, contrary to the non-interacting limit, the momentum distribution $\Pi(k)$ of the bosons is not exponentially localized, but decays as $\mathcal C/k^4$, as expected for interacting quantum particles, with Tan's contact $\mathcal C$ which remains finite at long time. We discuss how our results will impact the experimental study of kicked interacting bosons., Comment: v1) 10 pages, 10 figures; v2) minor typos corrected, published version

We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus., Reformatted for DMTCS

The paper focuses on the loads applied to the helicopter blade cross-sections in the multibody dynamics system. The main objective is to simplify the blade aerodynamics calculation and avoid time-consuming CFD methods. For this reason, the way of computing blade aerodynamics is proposed by using multibody dynamics methods with a linear-elastic blade model. As the primary tool for further research, the MCS Adams software package is selected. Splitting the main rotor blade into a finite number of sections, each having its own average value of installation and coning angles, simplifies the calculation. Afterward, expressions for the total flow velocity around the blade section and its angle of attack are obtained through vector operations. This provides a measure of aerodynamic forces acting on each section in its cross-sectional coordinate system. In conclusion, the article provides the formalized method of aerodynamic force distribution between blade sections in the multibody model as well as the correlation between the flow coordinate system and the blade chord coordinate system.

We apply the semi-classical limit of the generalized $SO(3)$ map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on $T^{\ast }\mathcal{S}_{2}$. Using the asymptotic form of the star-product, we manage to "quantize" one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared., Comment: 19 pages, 3 figures. https://doi.org/10.3390/e23060684

One of the most successful applications of geometric calculus to engineering refers to robotics and computer vision. In this line, this chapter presents an overview of the main classical problems in robot kinematics and motion planning and explains how geometric calculus has been used to solve them by exploiting their algebraic and geometric properties (such as, for instance, that every isometry can be compactly represented, the geometric covariance, the properties of the rotor group and the bivector algebra). Besides, it also introduces recent open problems in robotics and explains how geometric calculus can be used to contribute to their solutions.

Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar.

The wobbling spectrum of $^{163}$Lu is described through a novel approach, starting from a triaxial rotor model within a semi-classical picture, and obtaining a new set of equations for all four rotational bands that have wobbling character. Redefining the band structure in the present model is done by adopting the concepts of Signature Partner Bands and Parity Partner Bands. Indeed, describing a wobbling spectrum in an even-odd nucleus through signature and parity quantum numbers is an inedited interpretation of the triaxial super-deformed bands., Comment: 8 pages, 4 figures

The coupled aerodynamic performance influence of multiple deviations in the blade profile is a complex phenomena. In this paper, a high-pressure compressor rotor blade mid-span profile is studied with the design of experiments (DOE), numerical simulation and surrogate model to analyze the influence of the deviations on the blade aerodynamic performance. The purpose of this work is to provide a new rapid evaluation approach for the blade aerodynamic performance under multiple geometric deviations influence. First, the Hicks-Henne function was used to model the local geometric deviations of the blade profile, and the blade aerodynamic performance was calculated by using the computational fluid dynamics tools. By analyzing the calculation results, the momentum thickness of the boundary layer, the deviations height and the distance between the deviations are combined into a coupled effect model. Then, the coupled effect model was used to rapidly evaluate the blade aerodynamic performance when the two-position local geometric deviations exist on the blade surface. Finally, the evaluated performance were compared with the results predicted by a high-precision surrogate model, which verifies the high accuracy of the coupled effect model in evaluating the positive incidence range of the blade profile.

In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects.

We analyze the out-of-time-order correlation functions of a solvable model of a large number, $N$, of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,~M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticle-like excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and point out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions., Comment: 5+6 pages, 3+2 figures

The positive-parity doublet bands with configuration $\ensuremath{\pi}{h}_{11/2}\ensuremath{\bigotimes}\ensuremath{\nu}{h}_{11/2}$ and negative-parity bands with $\ensuremath{\pi}{h}_{11/2}\ensuremath{\bigotimes}\ensuremath{\nu}({g}_{7/2}{d}_{5/2})$ observed in $^{124}\mathrm{Cs}$, and the possibility of multiple chiral doublets ($\mathrm{M}\ensuremath{\chi}\mathrm{D}$) with octupole correlations are investigated using the reflection-asymmetric triaxial particle rotor model. The calculated excitation energies, energy staggering parameters, and the electromagnetic transitional probabilities are in good agreement with the data available and satisfy the expected features of chiral doublet bands. The angular momentum geometry and its evolution with spin are studied with the angular momentum components and the azimuthal plot. In comparison with a typical chiral vibration pattern for the positive-parity doublet bands, a transient static chirality around $I=12\ensuremath{\hbar}$ is shown for the calculated negative-parity doublet bands.

A reflection-asymmetric triaxial particle rotor model with three quasiparticles and a reflection-asymmetric triaxial rotor is developed, and applied to investigate the observed multiple chiral doublets (M\c{hi}D) candidates with octupole correlations in 131Ba, i.e., two pairs of positive-parity bands D3-D4 and D5-D6, as well as one pair of negative-parity bands D7-D8. The energy spectra, the energy staggering parameters, the B(M1)/B(E2) ratios, and the B(E1)/B(E2) ratios are reproduced well. The chiral geometries for these M\c{hi}D candidates are examined by the azimuthal plots, and the evolution of chiral geometry with spin is clearly demonstrated. The intrinsic structure for the positive-parity bands is analyzed and the possible pseudospin-chiral quartet bands are suggested., 16 pages, 8 figures, PHYSICAL REVIEW C (in press)

Computational research is carried out of the effect of geometric scatter within the manufacturing tolerances on the natural frequency values of a low pressure compressor blade. The first stage blade natural frequencies are estimated and compared considering various distribution laws of dimensions within the manufacturing tolerances. The joint approach to the blade natural frequency optimization and robust design is presented. The obtained blade design is less sensitive to geometric dimension scattering compared to the original design.

A rotor Hopfield neural network (RHNN) is an extension of a complex-valued Hopfield neural network (CHNN) and has excellent noise tolerance. The RHNN decomposition theorem says that an RHNN decomposes into a CHNN and a symmetric CHNN. For a large number of training patterns, the projection rule for RHNNs generates large self-feedbacks, which deteriorates the noise tolerance. To remove self-feedbacks, we propose a projection rule using quaternions based on the decomposition theorem. Using computer simulations, we show that the quaternion projection rule improves noise tolerance.

Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate momentof inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect,quickly averages to zero in classical ensembles since the flipping period varies significantly uponapproaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermalasymmetric nanorotors and show that classically forbidden tunnelling gives rise to persistent tennisracket dynamics, in stark contrast to the classical expectation. We characterise this effect, demon-strating that quantum coherent flipping dynamics can persist even in the regime where millions ofangular momentum states are occupied. This persistent flipping offers a promising route for observ-ing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles., Comment: 12 pages, 8 figures

In this paper, an analytical method for predicting the rotor overhang (OH) effect is presented using two types of conformal mappings (CMs), which are the ${r}$ – ${\theta }$ and zr plane CMs. First, the no-load air-gap flux density (AFD) of a slotted air gap without OH is determined using the ${r} {\theta }$ plane CM. Second, the complex relative air-gap permeance (CRAP) in the zr plane is obtained by the zr plane CM. This helps in taking the OH effect into account. Finally, the no-load AFD of the slotted and OH air gaps in cylindrical coordinates is derived from the no-load AFD in the ${r} {\theta }$ plane with CRAP along the ${z}$ -axis. The accuracy of the presented method is verified by comparing the results of no-load analysis with those of 3-D finite-element analysis.

In order to overcome the problem of “singular points”, a method has been developed for the kinematically accurate separation of a large rotation into an axial Euler vector and a transverse Euler vector. The proposal is based on the fact that in the problems of the rotor dynamics of machines consisting of shafts, gears, bearings, etc., the transverse rotation never reaches a value of 2π (a critical value for the Euler vector). The axial rotation is not limited in any way. A numerical dynamics example illustrating the method is presented. The result of the dynamics problem is checked by observing the law of conservation of total energy.

We show that a heavy, magnetized and electrically charged asymmetric rigid body moving about a fixed point while carrying a rotor and acted upon by three uniform fields can perform a regular precession about a nonvertical axis, of the type described for the case of a single field by Grioli in 1947. This is the first, and the only known by now, non-equilibrium solution of the problem of motion of a body in the presence of three classical fields, which are irreducible to a less number of fields.

An extraction method is proposed to investigate blade vortex interaction noise emitted during helicopter transient maneuvering flight. The extraction method allows for the investigation of blade vortex interactions, independent of other sound sources. It is based on filtering the spectral representation of experimentally acquired full-scale helicopter acoustic data. The data is first transformed into time-frequency space through the wavelet transformation, with blade vortex interactions identified and filtered by their high amplitude, high frequency impulsive content. The filtered wavelet coefficients are then inverse transformed to create a pressure signature solely related to blade vortex interactions. Analysis on a synthetic data set is conducted, and it is shown that blade vortex interactions can be accurately extracted so long as the blade vortex interaction peak energy signal is greater or equal to the energy in the main rotor harmonic. A brief analysis shows that the extraction method performs admirably throughout a fast advancing side roll maneuver. Using this method, it was shown that peak blade vortex interaction noise levels are linked directly to the roll rate of the vehicle, and are directed towards the retreating side during the transient portion of the maneuver.

The chirality suggested for the doublet bands B2 and B2a in $^{195}$Tl in $A\sim190$ region is reexamined. The potential-energy curves and the configurations together with the deformation parameters are obtained by the constrained covariant density functional theory. The corresponding experimental energy spectra, energy differences between doublet bands, and the available $B(M1)/B(E2)$ values are investigated by the fully quantal particle rotor model. Analysis on the basis of the angular momentum components, the $K$-plots, and the azimuthal plots suggest a planar rotation interpretation for the bands B2 and B2a. Hence, it coexists with the aplanar rotation in the other doublet bands B4 and B4a in $^{195}$Tl., Comment: 7 pages, 5 figures. arXiv admin note: text overlap with arXiv:1904.12100

In this paper, the dynamics of a tailored composite rotating blade with curved tips are investigated, with a view to improving the dynamic behaviour of the blade in flight. The blade tip is curved either in the out-of-plane or in the in-plane directions. The composite blade is modelled by using the exact beam formulation, and the cross-sectional properties of the blade are obtained using the variational asymptotic method. The resulting nonlinear partial differential equations are discretised using a time-space scheme, and the stationary and rotating frequencies of the blade are obtained from the eigenvalues of the linearised system. Three case studies are considered here each of them representing one of the main elastic couplings that might happen in a composite blade. These three elastic couplings are the flap-twist, lag-twist, and extension-twist couplings. All these couplings are very important in the blade design as they can affect the twist and hence the dynamics of the blade. The blade tip length and curvature value are two main parameters that this paper is focused on. It is shown that the curved tip of the blade affects the blade frequencies by adding extra couplings, and therefore could be used as a potential morphing concept for tuning the frequencies, enhancing the aeroelastic stability or performance of the blade in flight.

We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.

The generalized triaxial rotor model with independent variable moments of inertia is applied to describe the transitional behavior of staggering patterns observed in the $\ensuremath{\gamma}$ bands of $^{108,110,112}\mathrm{Ru}$. An oscillating behavior is introduced in the variations of the moments of inertia ${\mathcal{I}}_{1}$ and ${\mathcal{I}}_{2}$ to describe the $\ensuremath{\gamma}$-soft pattern in $^{108}\mathrm{Ru}$. Then the amplitude of the oscillating term is exponentially reduced as a function of spin in $^{110}\mathrm{Ru}$. The oscillating term is taken away completely in $^{112}\mathrm{Ru}$. A good agreement is obtained between the model calculation and the experimental data both in the phase and amplitude of the staggering pattern. Using the obtained eigenfunctions, effective $\ensuremath{\gamma}$ values are so determined to reproduce best the branching ratios of $\ensuremath{\gamma}$ decays from the $\ensuremath{\gamma}$ bands. The stable values of ${\ensuremath{\gamma}}_{\mathrm{eff}}\ensuremath{\approx}{20}^{\ensuremath{\circ}}$ are obtained for the lower-spin states in any of $^{108,110,112}\mathrm{Ru}$.

Macrocycles Within proteins, individual peptide bonds have free rotation but typically keep to a narrow conformational space. Macrocycles are already conformationally hindered and provide an interesting venue for pushing peptide bonds into uncharted territory. Diaz et al. prepared a series of small macrocycles using so-called dominant rotors, peptide-like units that are kinetically locked in a particular conformation. Structural analysis revealed unusual conformations of the peptide backbone beside the dominant rotor, perhaps indicative of the kinds of fleeting intermediates expected during peptide folding. Nat. Chem. 13 , 218 (2021).

Quantum-chemical INDO/S calculations of Thiofl avin T have been carried out taking account of variation of the angle φ between the planes of the benzothiazole (BTZ) and dimethylaniline (DMA) rings. It was found that when the angle φ changes from 40° to 90° the energy of the triplet state increases by 4000 cm−1, whereas the energy of the singlet state S 1 decreases by 1900 cm−1 and reaches a minimum. The function $$ {E}_{T_1}\left(\upvarphi \right) $$ has a minimum at φ = 30°, which is 300 cm−1 less than at φ = 40°. The calculated T 1 S 0 interval at φ = 30–40° amounts to 15,600–16,000 cm−1, which agrees well with the phosphorescence data (17,100–17,400 cm−1). For φ = 80–90° the T 1, T 2 , and T 3 levels (T 1 and T 2 are lower than S 1 ) are close to the S 1 level. The S 1 and T 3 levels intersect at φ ~ 85°, and at φ = 90° the interval $$ \Delta {E}_{S_1{T}_3} $$ amounts to only 100 cm−1 due to the small value of the exchange integral corresponding to electron transfer DMA → BTZ.

We show that a new attempt by Corda (2016), just like his previous attempt (Corda, 2015) that we had answered before (A.L. Kholmetskii et al., 2015), to reinterpret Mossbauer experiments in a rotating system as a “new, strong and independent proof of the correctness of Einstein’s vision of gravity” is erroneous. In addition, we demonstrate that Corda’s criticism of Yarman–Arik–Kholmetskii gravitation theory (in short YARK), is based on the application of ill-posed logic; thus rendering his claims against YARK as unfounded.

Purpose This paper aims to compare the comprehensive rotorcraft analyses using the two different blade section property data sets for the blade natural frequencies, airloads, elastic deformations, the trimmed rotor pitch control angles and the blade structural loads of a small-scale model rotor in a blade vortex interaction (BVI) phenomenon. Design/methodology/approach The two different blade section property data sets for the first Higher-harmonic control Aeroacoustic Rotor Test (HART-I) are considered for the present rotor aeromechanics analyses. One is the blade property data set using the predicted values which is one of the estimated data sets used for the previous validation works. The other data set uses the measured values for an uninstrumented blade. A comprehensive rotorcraft analysis code, CAMRAD II (comprehensive analytical model of rotorcraft aerodynamics and dynamics II), is used to predict the rotor aeromechanics such as the blade natural frequencies, airloads, elastic deformations, the trimmed rotor pitch control angles and the blade structural loads for the three test cases with and without higher-harmonic control pitch inputs. In CAMRAD II modelling with the two different blade property data sets, the blade is represented as a geometrically nonlinear elastic beam, and the multiple-trailer wake with consolidation model is used to consider more elaborately the BVI effect in low-speed descending flight. The aeromechanics analysis result sets using the two different blade section property data sets are compared with each other as well as are correlated with the wind-tunnel test data. Findings The predicted blade natural frequencies using the two different blade section property data sets at non-rotating condition are quite similar to each other except for the natural frequency in the fourth flap mode. However, the natural frequencies using the predicted blade properties at nominal rotating condition are lower than those with the measured blade properties except for the second lead-lag frequency. The trimmed collective pitch control angle with the predicted blade properties is higher than both the wind-tunnel test data and the result using the measured blade properties in all the three test cases. The two different blade property data sets both give reasonable predictions on the blade section normal forces with BVI in the three test cases, and the two analysis results are reasonably similar to each other. The blade elastic deformations at the tip using the measured blade properties are correlated more closely with the wind-tunnel test data than those using the predicted blade properties in most correlation examples. In addition, the predictions of blade structural loads can be slightly or moderately improved by using the measured blade properties particularly for the oscillatory flap bending moments. Finally, the movement of the sectional centre of gravity location of the uninstrumented blade has a moderate influence on the blade elastic twist at the tip in the baseline case and the oscillatory flap bending moment in the minimum noise case. Practical implications The present comparison study on rotor aeromechanics analyses using the two different blade property data sets will show the influence of blade section properties on rotor aeromechanics analysis. Originality/value This paper is the first attempt to compare the aeromechanics analysis results using the two different blade section property data sets for all three test cases (baseline, minimum noise and minimum vibration) of HART-I in low-speed descending flight.

Positive-parity doublet bands were reported in 120I. Based on these, we discuss the corresponding experimental characteristics, including rotational alignment, and re-examine the corresponding configuration assignment. The self-consistent tilted axis cranking relativistic mean-field calculations indicate that the doublet bands are built on the configuration . By adopting the two quasiparticles coupled with a triaxial rotor model, the excitation energies, energy staggering parameter S(I), , effective angles, and K plots are discussed and compared with available data. The obtained results support the interpretation of chiral doublet bands for the positive-parity doublet bands in 120I, and hence identify this nucleus as the border of the A ≈ 130 island of chiral candidates.

In this paper, the effect of various parameters of a specific rotor blade cross-section on the effectiveness of a twist morphing concept is investigated. Then, by considering different constraints, a cross-section consistent with this morphing concept with high lag-twist coupling and low extension-twist, is developed. This lag bending-torsion coupling is used to change the twist of the blade during the flight, while the high values of extension-twist coupling is avoided. To this end, a concentrated mass is added to the blade, where its chordwise location varies in flight. When the mass moves in the chordwise direction, a local lag bending is introduced into the blade. This in-plane bending moment then changes the blade twist distribution through lag-twist coupling induced through stiffness tailoring in the blade cross-section. Therefore, this coupling plays an important role in this morphing concept. The one-dimensional dynamics of the blade is modelled by using the geometrically exact fully intrinsic bean equations while the 2D cross-sectional stiffness values are determined by using the VABS software. First, a blade which resembles the BO-105 main rotor blade in the fundamental frequencies is designed. Then, the effect of various parameters of the cross-section on the fundamental frequencies, the lag-twist coupling, and the extension-twist coupling are determined. It is found that the skin of the spar has the highest contribution to both the extension-twist and the lag-twist coupling. Finally, a cross-section compatible with the proposed morphing concept is designed and it is demonstrated that the twist of the blade may be changed significantly.

Dynamical kicking systems possess rich topological structures. In this work, we study Floquet states of matter in a non-Hermitian extension of double kicked rotor model. Under the on-resonance condition, we find various non-Hermitian Floquet topological phases, with each being characterized by a pair of topological winding numbers. A generalized mean chiral displacement is introduced to detect these winding numbers dynamically in two symmetric time frames. Furthermore, by mapping the system to a periodically quenched lattice model, we obtain the topological edge states and unravel the bulk-edge correspondence of the non-Hermitian double kicked rotor. These results uncover the richness of Floquet topological states in non-Hermitian dynamical kicking systems., Comment: 13 pages, 9 figures

The behavior of the collective rotor in the chiral motion of triaxially deformed nuclei is investigated using the particle rotor model by transforming the wave functions from the $K$-representation to the $R$-representation. After examining the energy spectra of the doublet bands and their energy differences as functions of the triaxial deformation, the angular momentum components of the rotor, proton, neutron, and the total system are investigated. Moreover, the probability distributions of the rotor angular momentum ($R$-plots) and their projections onto the three principal axes ($K_R$-plots) are analyzed. The evolution of the chiral mode from a chiral vibration at the low spins to a chiral rotation at high spins is illustrated at triaxial deformations $\gamma=20^\circ$ and $30^\circ$., Comment: 21 pages, 6 figures

The aim of this paper is to evaluate the theoretical efficiency of optimal propellers by means of a variational approach; non classical propellers are included in the analysis. A solution to the optimum rotor problem, in the context of propeller vortex theory, was given by Goldstein in 1929; in this paper, a variational formulation of the optimum rotor problem is proposed. The formulation consists into finding a class of functions (the circulation along the blade axis) for which the thrust and drag momentum functionals are well defined; in this class, the functional is proved to be strictly convex and, thus, the global minimum exists and is unique. Some configurations are analysed using the Ritz method: classical straight blade, parabolic blade, elliptical blade; single and multiple blade propellers are compared.

In this paper, the 3D model of the centrifugal fan impeller will be obtained using CAD software (CATIA). The blade will be obtained with 4 design methods: constant blade width and variable blade angle; hyperbolic blade width and variable blade angle; hyperbolic blade width and constant blade angle; linear blade width and variable blade angle. A comparative analysis of the weight of the blade, as well as of the weight of the rotor composed by cascade of blades, front and rear shrouds will be presented. Two materials have been used for all the elements of the centrifugal fan impeller: steel and aluminium. For computational purposes the MATLAB programming language will be used. The centrifugal fan blade will be designed as untwisted geometry with =2 reference curves (one on the rear shroud and the other one on the front shroud) with different numbers of reference points ={5, 10, 20, 30, 40, 50}. After running the MATLAB script file, the coordinates of reference points of the blade will be obtained for all 4 blade design methods. The transfer of the reference points coordinates from MATLAB workspace to CATIA has been performed using the automatic surface generation method, via an EXCEL macro file. This is the recommended method especially for a big number of reference points.

We report the derivation and experimental evaluation of a stable adaptive identifier to estimate rigid body rotations using rotors in Geometric Algebra GA. This work is motivated by the need for in situ estimation of the alignment between sensors commonly used in underwater vehicle navigation. Here we derive an adaptive identifier using a geometric interpretation of the error to drive first-order rotor kinematics. We prove that it is Lyapunov stable, and we show that it is asymptotically stable in the presence of persistent excitation. We use the identifier to estimate the alignment between the Doppler velocity log sonar and the fiber optic gyrocompass used by underwater vehicles for dead reckoning DR. We evaluate this method in the laboratory with a remotely operated vehicle ROV, and then with an autonomous underwater vehicle AUV operating in the field at 1,200i¾?m depth. Our results show that this technique reduces dead reckoning navigation errors on these platforms and provides comparable performance to previously reported SO3 constrained Linear Algebra LA approaches. The rotor identifier has a number of advantages over these previously reported methods, including a more straightforward derivation, simpler gain tuning, increased computational efficiency, and reduced data manipulation.

A rare conditional integrable case of rigid body dynamics, recently introduced in J. Phys. A: Math. Theor. 2013, is shown to remain integrable when one of the two singular terms of the potential is removed and a symmetric rotor is attached along the axis of dynamical symmetry of the body. Physical interpretation of the new case is possible as a motion of a heavy magnetized gyrostat carrying electric charges in an axially symmetric combination of the three classical fields.

The dominant, outstanding characteristic of formation vector-rotor, which provide adequate description of images was considered. It was received the algebraical expressions for determination of step vector as under direct action of rotor. It is shown as a consequence of this approach it is possible to differentiate the growth of the color vector by the individual components and establish the direction of its growth without the use of vector gradient. The inverse representation of the color vector at an arbitrary point of the environment thruogh value of vector-rotor has been written. The ratio of minimum volume or maximum compression of information can be determined for processing by first and second oder vector-rotor.

An asymmetric mechanism for correlated motion occurring in noninteracting pairs of adjacent orthogonal 1,4-bis(carboxyethynyl)bicyclo[1.1.1]pentane (BCP) rotators 1 in the solid state is unraveled and shown to play an important role in understanding the dynamics in the crystalline rotor, Bu4N+[1–]·H2O. Single crystal X-ray diffraction and calculation of rotor–rotor interaction energies combined with variable-temperature, variable-field 1H spin–lattice relaxation experiments led to the identification and microscopic rationalization of two distinct relaxation processes., Work in Bellaterra and Tarragona was supported by the Spanish Ministerio de Economía y Competitividad (projects FIS2015-64886-C5-4-P and CTQ2017-87269-P) and Generalitat de Catalunya (2014GR301 and 2014GR199). E.C. acknowledges the support of the Spanish MINECO through the Severo Ochoa Centers of Excellence Program under grant SEV-2015-0496. J.K. and J.M. gratefully acknowledge financial support from the European Research Council under the European Community’s Framework Programme (FP7/2007-2013) ERC grant agreement no. 227756 and the Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic (RVO: 61388963). This material is based upon work supported by the National Science Foundation under grant no. CHE-1566435. Work at Orsay was supported by the CNRS. Work at Angers was supported by the CNRS and the Région des Pays de la Loire grant MOVAMOL. Joint work at Angers, Orsay, and Prague was supported by the Ministère de l’Europe et des Affaires Etrangères under the grant PHC Barrande no. 864563J.

This chapter is devoted to a closer study of the rotor group \(\mathcal {R}=\mathcal {R}_{r,s}\), and as a byproduct of its primacy (in the sense given to this expression in the last chapter), also of the other spinorial and orthogonal groups.

In this work we present the extended two-dimensional torsion (E2DT) method and use it to analyze the performance of several methods that incorporate torsional anharmonicity more approximately for calculating rotational–vibrational partition functions. Twenty molecules having two hindered rotors were studied for temperatures between 100 and 2500 K. These molecules present several kinds of situations; they include molecules with nearly separable rotors, molecules in which the reduced moments of inertia change substantially with the internal rotation, and molecules presenting compound rotation. Partition functions obtained by the rigid-rotor harmonic oscillator approximation, a method involving global separability of torsions and the multistructural methods without explicit potential coupling [MS-T(U)] and with explicit potential coupling [MS-T(C)] of torsions, are compared to those obtained with a quantized version – called the extended two-dimensional torsion (E2DT) method – of the extended hindered rotor approximation of Vansteenkiste et al. (Vansteenkiste et al. J. Chem. Phys. 2006, 124, 044314). In the E2DT method, quantum effects due to the torsional modes were incorporated by the two-dimensional nonseparable method, which is a method that is based on the solution of the torsional Schrödinger equation and that includes full coupling in both the kinetic and potential energy. By comparing other methods to the E2DT method and to experimental thermochemical data, this study concludes that the harmonic approximation yields very poor results at high temperatures; the global separation of torsions from the rest of the degrees of freedom is not justified even when an accurate method to treat the torsions is employed; it is confirmed that methods based on less complete potential energy coupling of torsions, such as MS-T(U), are not accurate when dealing with rotors with different barrier heights, and more complete inclusion of torsional coupling to the method in MS-T(C) improves substantially the results in such a way that it could be used in cases where the E2DT method is unaffordable Financial support from the Ministerio de Economia y Competitividad of Spain (Research Grant No CTQ2014-58617-R), the Conselleria de Cultura, Educación e Ordenación Universitaria (Centro singular de investigación de Galicia acreditación 2016-2019, ED431G/09), and the European Regional Development Fund (ERDF) is gratefully acknowledged SI

In the present paper methods are described for how a variety of new ‘Twisting Towers’ can be derived on the basis of three special Archimedean polyhedrons. Twisting Towers are mechanisms that in motion deform roto-symmetrically. Three basic Twisting Towers are obtained directly by ‘mobilizing’ these special Archimedean polyhedrons, i.e. by removing square side faces and mounting double rotor joints at the vertices. The basic mechanisms encompass in their most extended position the Archimedean polyhedrons from which they are derived. While moving they enclose several other Archimedean polyhedrons. A number of kinematic chains connect the body at the top of a Twisting Tower with the body at the bottom. A careful investigation of these kinematic chains reveals that only two types of body-connections secure finite mobility of the tower. With these two body-connections a great variety of different new Twisting Towers is obtained.