Superposition Of Rotation Research Articles

Superposition of rotations in mechanisms

Superposition of rotations in mechanisms

Exact ground state of the Lieb-Mattis Hamiltonian as a superposition of Néel states

We show that the exact ground state of the Lieb-Mattis Hamiltonian is an equal-weight superposition of all possible classical N\'{e}el states, and provide an exact formulation of this superposition in the $z$-spin basis for both $S=1/2$ and general $S$ using Schwinger bosons. In general, a superposition of possible rotations on a general initial state is symmetric if and only if the initial state has a nonzero overlap with a singlet state and is otherwise made up of states that vanish due to the symmetrization. Most notably, $|s, m=0 \rangle$ states will vanish if symmetrized, which explains how a superposition of N\'{e}el states projects onto its singlet component.

The Milky Way bar/bulge in proper motions: a 3D view from VIRAC and Gaia

Abstract We have derived absolute proper motions of the entire Galactic bulge region from VVV Infrared Astrometric Catalogue (VIRAC) and Gaia. We present these both as integrated on-sky maps and, after isolating standard candle red clump (RC) stars, as a function of distance using RC magnitude as a proxy. These data provide a new global, 3D view of the Milky Way barred bulge kinematics. We find a gradient in the mean longitudinal proper motion, $\langle \mu _ l^\star \rangle $, between the different sides of the bar, which is sensitive to the bar pattern speed. The split RC has distinct proper motions and is colder than other stars at similar distance. The proper motion correlation map has a quadrupole pattern in all magnitude slices showing no evidence for a separate, more axisymmetric inner bulge component. The line-of-sight integrated kinematic maps show a high central velocity dispersion surrounded by a more asymmetric dispersion profile. $\sigma _{\mu _l} / \sigma _{\mu _b}$ is smallest, ≈1.1, near the minor axis and reaches ≈1.4 near the disc plane. The integrated $\langle \mu_b\rangle$ pattern signals a superposition of bar rotation and internal streaming motion, with the near part shrinking in latitude and the far part expanding. To understand and interpret these remarkable data, we compare to a made-to-measure barred dynamical model, folding in the VIRAC selection function to construct mock maps. We find that our model of the barred bulge, with a pattern speed of 37.5 $\mathrm{ \mathrm{ km \, s^{-1}} \, kpc^{-1} }$, is able to reproduce all observed features impressively well. Dynamical models like this will be key to unlocking the full potential of these data.

Various angle periods of parabolic coincidence fringes in violation of the Bell inequality with high-dimensional two-photon entanglement

Two quantum states of two half-charge optical vortexes with relative azimuthal angle {\pi} are orthogonal, by which two half-charge spiral phase plates with intersection angle {\pi} can be used to demonstrate a parabolic coincident fringe in the Bell inequality experiment with high-dimensional two-photon orbital angular momentum entanglement and to thereby obtain a strong Bell parameter of 3.2. I theoretically demonstrate various orthogonal relations between two quantum states, each of which is a state with a rotational symmetry superposition made up of n fractional orbital angular momentum states, where n is N. I propose a Bell inequality experiment with two n-section spiral phase plates that have these two quantum states, whereby various angle periods with parabolic coincident fringes 2{\pi}/n and Bell's parameter 3.2 in each period can be obtained.

Towards real-time simulation of deformable structures by means of co-rotational finite element formulation

Whereas typical Finite Element (FE) computations are performed off-line, many virtual-reality (VR) applications put a demand for interactive simulations involving deformable objects. Interactive simulation implies real-time or nearly real-time computation and graphical representation of modeled deformable objects. The growing computational power of modern conventional hardware calls for FEM developments in this direction. Depending on specific VR applications, the developments need to account for different aspects of physical behavior, with geometrically nonlinear deformations emerging as one of most important and frequent. This paper proposes a simplified co-rotational FE formulation that considers the overall finite element motion as a superposition of rigid-body rotation and deformation described by a linear model with respect to the co-rotated reference frame. By neglecting the stress stiffening effect and the dependence of the element stiffness matrix on the deformational displacements, the formulation aims at meeting the objectives of highly efficient simulation, under certain conditions even real-time simulation, and with acceptable deviation in accuracy compared to the rigorous nonlinear FE formulation. Computations with both solid and shell elements are addressed. A set of examples is provided to illustrate and discuss the aspects of accuracy and achievable simulation speed.

Torsional balloon flight line oscillations: Comparison of modelling to flight data

During the EBEX2013 long duration flight the payload was free to rotate in azimuth. The observed azimuth motion consisted of a superposition of full rotations with a period of 10—30min and oscillatory motion with an amplitude of tens of degrees, average period of 79s, and period dispersion of 12s. We interpret the full rotations as induced by slow rotations of the balloon and the shorter period oscillatory motion as due to torsional oscillations of the flight line. We derive the torsional stiffness of the flight line using the bifilar pendulum model and apply it to the flight line of the EBEX2013 payload. We find a torsional spring constant of 36kgm2/s2 corresponding to a period of 58s. We conclude that the bifilar model, which accounts for the geometry of the flight line but neglects all material properties, predicts a stiffness and period that are 45% larger and 25% shorter than those observed. It is useful to have a simple, easy to use, coarse approximation for the torsional constant of the flight line.

Control-Oriented Modeling of Coupled Electromechanical-Aeroelastic Dynamics for Flapping-Wing Vehicles

A computationally tractable nonlinear aeroelastic model of a bioinspired flapping-wing micro air vehicle, which includes motor dynamics and is suitable for control evaluation, is developed in this paper. The structural model accounts for geometrically nonlinear deformations using an implicit condensation technique. The aerodynamic loads are computed as a superposition of quasi-steady translation and rotation as well as added mass effects with an unsteady correction. The aeroelastic model is compared to experiments and higher-fidelity computations and shows acceptable agreement. The model is subsequently implemented in open-loop control simulations and demonstrates that motor dynamics and wing aeroelasticity have a significant impact on the loads generated by the wings, potentially introducing challenges in vehicle control.

An Analysis of Form Changes of a Toroidal Body with a Meridional Arrangement of Fibers on the Basis of the Two-Level Carcass Theory and of a Homogeneous Body Congruent to It

Results of an analysis of form changes of a toroidal body highly filled with fibers at large torsional deformations and rotational motions are presented. The body is reinforced in the meridional direction. The investigation was carried out by using the two-level carcass theory of fibrous media at large deformations, according to which the macroscopic fields of a reinforced body are determined by its internal fields. These fields are represented by the material configurations of nodal material blocks of the body, for which, on the basis of the model of a piecewise homogeneous medium, boundary-values problems of the micromechanical level of the theory are solved. The results obtained are compared with those for a homogeneous body. The congruent deformation of the homogeneous body at which its initial form and dimensions are practically restored upon superposition of torsion and rotation is determined.

Design of multi-bolted joints to prevent self-loosening failure

The rotational self-loosening of bolted joints under cyclic loading is a process, which means a screw rotation against the assembly direction by periodic load cycles. Thus, the preload balance is released and the clamping function is no longer maintained. The effect is well known, but prevention is usually performed experimentally only after occurrence of self-loosening events. The paper gives a systematic presentation of the dimensioning and the possibility to consider self-loosening in the development process. The procedure for self-loosening dimensioning and the influences are demonstrated with safety-relevant multi-bolted connections from the automotive industry. The aim of this paper is to provide a numerical design method with finite element analysis for detecting and understanding of the self-loosening process at bolted joints. Another focus of the work is to investigate the self-loosening behaviour under combined loading with a superposition of translation and rotation. Combined loading can lead to self-loosening even before reaching the limit of self-loosening for transverse loading. The computational results of the numerical simulation (FEA) are compared with experimental investigations.

Single rotational state preparation via coherent control of laser pulse

We present a scheme to prepare a single rotational state with high probability from an arbitrary superposition state. The NO molecule is used as a computational example. By using a series of linearly and circularly polarized laser pulses and adjusting their parameters, a ground rotational state with probability of 99.982% can be obtained. Starting from the ground rotational state prepared, we can prepare any desired single rotational state with high probability and some rotational superposition states. On this basis, the molecular orientation with initial thermal distribution can be considerably enhanced and well controlled.

Terahertz-Induced Field-Free Orientation of Rotationally Excited Molecules

We have used picosecond THz pulses to induce transient field-free orientation of OCS molecules. Coherent optical Raman excitation prepares the molecules in rotational superposition states prior to THz irradiation, substantially enhancing the degree of orientation. The time-dependent alignment and orientation are characterized via Coulomb explosion in an intense probe laser. The degree of OCS orientation is an order of magnitude larger than previously observed following THz irradiation and is achieved with a significantly smaller THz field.The field-free orientation level is comparable to that generated using pulsed, two-color laser fields but is obtained with negligible target ionization.

Shear resonance and torsional oscillator measurements of solid4He under dc rotation

Nonclassical response and acoustic resonance of a torsional oscillator (TO) containing solid helium is investigated under dc rotation. Acoustic resonance measurements can capture the elastic property of solid helium in the annular channel where we simultaneously measure the torsional oscillator response. Superposition of dc rotation on top of two resonance measurements can clarify whether or not the reduction in the resonant period of the TO is originated from the elastic property of solid helium more clearly. We find that the acoustic resonance is independent of dc rotation. On the contrary, the TO response measured simultaneously shows strong suppression which was accompanied by additional dissipation of the TO. This study supports the existence of a macroscopically phase coherent superfluid state in solid helium.

Friction-induced whirl vibration: Root cause of squeaking in total hip arthroplasty

Squeaking is reported for ceramic-on-ceramic hip arthroplasty, and risk factors leading to this phenomenon have been investigated empirically in the past, this way giving hints to when this phenomenon occurs. The aim of this study is to present an experimentally validated explanation for the dynamical mechanism underlying the squeak, i.e. a description of what happens when noise is generated.First the kinematics of the ceramic bearing couple in relative motion are reconsidered. The relative motion at the contact zone can be understood as superposition of relative rotation and translation. The relative weight of both components depends substantially on the instantaneous load vector, which primarily determines the position of the contact area, and the instantaneous relative rotation vector. For the investigated gait scenarios, both load vector and rotation axis vary strongly during the gait cycle.Second, experimental vibration analysis during squeak is performed. A pronounced micrometer scale elliptical motion of the ball inside the liner is found. It is shown that the rotational component of the relative kinematics during gait indeed leads to friction induced vibrations. We show that a generic whirl type friction induced flutter instability, also known from similar (non bio-) mechanical systems, is the root cause of the emitted squeaking noise.Based on the identified mechanism, the role of THA system parameters (materials, design), patient risk factors, as well as the role of the gait cycle, will have to be reconsidered and linked in the future to develop effective measures against squeaking.

Similarity recognition of molecular structures by optimal atomic matching and rotational superposition

An algorithm for similarity recognition of molecules and molecular clusters is presented which also establishes the optimum matching among atoms of different structures. In the first step of the algorithm, a set of molecules are coarsely superimposed by transforming them into a common reference coordinate system. The optimum atomic matching among structures is then found with the help of the Hungarian algorithm. For this, pairs of structures are represented as complete bipartite graphs with a weight function that uses intermolecular atomic distances. In the final step, a rotational superposition method is applied using the optimum atomic matching found. This yields the minimum root mean square deviation of intermolecular atomic distances with respect to arbitrary rotation and translation of the molecules. Combined with an effective similarity prescreening method, our algorithm shows robustness and an effective quadratic scaling of computational time with the number of atoms.

Least constraint approach to the extraction of internal motions from molecular dynamics trajectories of flexible macromolecules

We propose a rigorous method for removing rigid-body motions from a given molecular dynamics trajectory of a flexible macromolecule. The method becomes exact in the limit of an infinitesimally small sampling step for the input trajectory. In a recent paper [G. Kneller, J. Chem. Phys. 128, 194101 (2008)], one of us showed that virtual internal atomic displacements for small time increments can be derived from Gauss' principle of least constraint, which leads to a rotational superposition problem for the atomic coordinates in two consecutive time frames of the input trajectory. Here, we demonstrate that the accumulation of these displacements in a molecular-fixed frame, which evolves in time according to the virtual rigid-body motions, leads to the desired trajectory for internal motions. The atomic coordinates in the input and output trajectory are related by a roto-translation, which guarantees that the internal energy of the molecule is left invariant. We present a convenient implementation of our method, in which the accumulation of the internal displacements is performed implicitly. Two numerical examples illustrate the difference to the classical approach for removing macromolecular rigid-body motions, which consists of aligning its configurations in the input trajectory with a fixed reference structure.

Comment on “Fast determination of the optimal rotational matrix for macromolecular superpositions” [J. Comp. Chem. 31, 1561 (2010)

Recently Liu et al. published a fast algorithm to solve the eigenvector problem arising in the quaternion-based method for the rotational superposition of molecular structures (J Comput Chem 2010, 31, 1561.). In this Comment, it is shown that the construction of the 4 × 4 matrix to be diagonalized—and not the diagonalization itself—represents the dominating part of the computational effort for the quaternion-based solution of the rotational superposition problem if molecules with more than about 100 atoms are considered.

Eckart axis conditions, Gauss’ principle of least constraint, and the optimal superposition of molecular structures

The relation of the Eckart axis conditions for polyatomic vibrating molecules to the problem of optimal superposition of molecular structures has been pointed out recently [J. Chem. Phys. 122, 224105 (2005)]. Here, it is shown that both problems are intimately related to Gauss' principle of least constraint, for which a concise derivation is presented. In the context of this article, Gauss' principle leads to a rotational superposition problem of the unconstrained atomic displacements and the corresponding displacements due to a molecular rigid-body motion. The Eckart axis conditions appear here as necessary conditions for a minimum of the constraint function. The importance of Eckart's problem for extracting the internal motions of macromolecules from simulated molecular dynamics trajectories is pointed out, and it is shown how the case of coarse-grained sampled trajectories can be treated.

Helical flow during rotational pipe extrusion

The helical flow resulting from the superposition of mandrel rotation with tubular polymer extrusion is studied. The experiments, carried out under conditions relevant to industrial practice, confirmed a pressure drop reduction upon increasing mandrel rotation and a decrease in torque when increasing the flowrate. An available analytical model was extended to consider the case when the inner wall of the channel is adiabatic and modified to predict torque and pressure drop. The results show a good agreement with experimental data. Information on velocity and viscosity profiles, melt temperature at the outlet, and first and second normal stress differences is also reported. Theoretical and experimental data confirmed unfavourable energy consumption upon mandrel rotation, but at high flowrates the energy consumption is similar to that for conventional static mandrel operation. The model is also used to estimate processing windows favouring the mechanical performance of the extrudate.

Rotational superposition and least squares: The SVD and quaternions approaches yield identical results. Reply to the preceding comment by G. Kneller

We clarify once again that Kabsch's method and the Quaternion method are mathematically equivalent methods, that is, that they contain identical information and, when properly understood and applied, lead to identical answers to any questions regarding least-squares rotational superposition, either by proper rotations or by rotation-reflections. We also provide the correct bounds for the eigenvalue spectrum.

Comment on “Using quaternions to calculate RMSD” [J. Comp. Chem. 25, 1849 (2004)

Coutsias et al. have recently published a method to find the optimal rotational superposition of two molecular structures, which is based on a representation of rotations by quaternions (J. Comp. Chem. 25(15), 1849 (2004)). The method, which has been suggested by other authors before, is compared to the one by Kabsch, where the elements of the rotation matrix are directly used as variables of the optimization problem. The statement that the two methods are equivalent is misleading in the sense that the Kabsch method may yield an improper optimal rotation, which must be explicitly checked for, whereas the quaternion method does not mix proper and improper rotations. Nevertheless, both types of solutions can be considered by solving the same eigenvector problem. The relation between the two types of solutions is briefly discussed and bounds for the eigenvalues are given.