Bearing Rigidity Research Articles

Nonspreading relativistic electron wavepacket in a strong laser field

A solution of the Dirac equation in a strong laser field presenting a nonspreading wave packet of the electron is derived. It consists of a generalization of the self-accelerating free electron wavepacket [I. Kaminer , ] to the case with the background of a strong laser field. Built upon the notion of nonspreading for an extended relativistic wavepacket, the concept of Born rigidity for accelerated motion in relativity is the key ingredient of the solution. At its core, the solution comes from the connection between the self-accelerated free electron wavepacket and the eigenstate of a Dirac electron in a constant and homogeneous gravitational field via the equivalence principle. The solution is an essential step towards the realization of the laser-driven relativistic collider [S. Meuren , ], where the large spreading of a common Gaussian wavepacket during the excursion in a strong laser field strongly limits the expectable yields. Published by the American Physical Society 2024

Comparative study on static and dynamic characteristics of pumping-out and pumping-in herringbone grooved journal bearings

The aim of this paper is to compare the static and dynamic performances of two types of herringbone grooved journal bearings (HGJBs), that is, pumping-out HGJB and pumping-in HGJB. Firstly, based on finite control volume method and the principle of flow balance, an equivalent Reynolds equation considering the turbulence effect is derived. Then, the journal motion considering angular displacement is analyzed with linear perturbation method. The equivalent Reynolds equation is linearized into static-form one and perturbed-form ones. These equations are solved by finite difference method and their solutions are integrated on the bearing surface to determinate the static characteristics and dynamic coefficients of HGJB. Additionally, the critical mass and critical whirl frequency are defined for evaluating the stability and rigidity of bearing by solving a motion equation with four degree of freedoms (DOFs). Finally, the static and dynamic characteristics of two types of HGJBS are analyzed and compared symmetrically in consideration of effects of eccentricity ratio, length to diameter ratio, groove angle, groove depth, the width of sealing damp, tilting degree of journal, and rotation speed. The results show that, within the interested range of investigated parameters, static performances of pumping-out HGJB are superior to that of pumping-in HGJB generally. Stability of pumping-in HGJB is better than pumping-out HGJB but bearing rigidity of pumping-in HGJB is inferior to pumping-out HGJB.

Some properties of time-varying bearing formation

This paper explores the properties of Bearing Persistently Exciting (BPE) formations defined in d(d≥2)-dimensional space, a class of time-varying bearing formations whose configuration can be uniquely determined up to a translational Euclidean vector using only bearing and velocity measurements. By analyzing the bearing Laplacian matrix in relation to PE conditions on the bearings, we provide results on necessary conditions, and necessary & sufficient conditions to guarantee that a formation is BPE. We also explore the concept of Relaxed Bearing Rigidity (RBR), a particular case of BPE formation, which relaxes the classical conditions on interaction topologies required by Bearing Rigidity to maintain a rigid shape. Distributed estimators are also proposed to estimate the configuration of the BPE formation using bearing and velocity measurements. Simulation results are provided to validate the proposed estimators.

A Unified Dissertation on Bearing Rigidity Theory

This work focuses on bearing rigidity theory, namely the branch of knowledge investigating the structural properties necessary for multi-element systems to preserve the inter-unit bearings under deformations. The contributions of this work are twofold. The first one consists in the development of a general framework for the statement of the principal definitions and properties of bearing rigidity. We show that this approach encompasses results existing in the literature, and also provides a systematic approach for studying bearing rigidity on any differential manifold in SE(3)^n, where n is the number of agents. The second contribution is the derivation of a general form of the rigidity matrix, a central construct in the study of rigidity theory. We provide a necessary and sufficient condition for the infinitesimal rigidity of a bearing framework as a property of the rank of the rigidity matrix. Finally, we present two examples of multi-agent systems not encountered in the literature and we study their rigidity properties using the developed methods.

The Frenet-Serret description of Born rigidity and its application to the Dirac equation

The role played by non-inertial frames in physics is one of the most interesting subjects that we can study when dealing with a physical theory. It does not matter whether we are studying classical theories such as special relativity or quantum theory, the idea of an accelerated frame is always one of the first ideas to come to our minds. In the case of special relativity, a problem with the concept of rigidity emerged as soon as Max Born gave a reasonable definition of rigid motion: the Herglotz-Noether theorem imposes a strong restriction on the possible rigid motions. In this paper, the equivalence of this theorem with another one that is formulated with the help of Frenet-Serret formalism is proved, showing the connection between the rigid motion and the curvatures of the observer's trajectory in spacetime. In addition, the Dirac equation in the Frenet-Serret frame for an arbitrary observer is obtained and applied to the rotating observers. The solution in the rotating frame is given in terms of that of an inertial one.

Minimal and Redundant Bearing Rigidity: Conditions and Applications

This article studies the notions of minimal and 1-redundant bearing rigidity. A necessary and sufficient condition for the numbers of edges in a graph of $n\,(n \geq 3)$ vertices to be minimally bearing rigid (MBR) in $\mathbb {R}^d\,(d \geq 2)$ is proposed. If $3 \leq n \leq d+1$ , a graph is MBR if and only if it is the cycle graph. In case $n > d+1$ , a generically bearing rigid graph is minimal if it has precisely $1 + \lfloor \frac{n-2}{d-1}\rfloor \times d + \text{mod}(n-2, d-1) + \text{sgn}(\text{mod}(n-2, d-1))$ edges. Then, several conditions for 1-redundant bearing rigidity are derived. Based on the mathematical conditions, some algorithms for generating generically, minimally, and 1-redundantly bearing rigid graphs are given. Furthermore, two applications of the new notions to optimal network design and formation merging are also reported.

Frequency spectrum of an optical resonator in a curved spacetime

The effect of gravity and proper acceleration on the frequency spectrum of an optical resonator—both rigid or deformable—is considered in the framework of general relativity. The optical resonator is modeled either as a rod of matter connecting two mirrors or as a dielectric rod whose ends function as mirrors. Explicit expressions for the frequency spectrum are derived for the case that it is only perturbed slightly and variations are slow enough to avoid any elastic resonances of the rod. For a deformable resonator, the perturbation of the frequency spectrum depends on the speed of sound in the rod supporting the mirrors. A connection is found to a relativistic concept of rigidity when the speed of sound approaches the speed of light. In contrast, the corresponding result for the assumption of Born rigidity is recovered when the speed of sound becomes infinite. The results presented in this article can be used as the basis for the description of optical and opto-mechanical systems in a curved spacetime. We apply our results to the examples of a uniformly accelerating resonator and an optical resonator in the gravitational field of a small moving sphere. To exemplify the applicability of our approach beyond the framework of linearized gravity, we consider the fictitious situation of an optical resonator falling into a black hole.

Corrigendum: A New Class of Amphiphiles Bearing Rigid Hydrophobic Groups for Solubilization and Stabilization of Membrane Proteins

Corrigendum: A New Class of Amphiphiles Bearing Rigid Hydrophobic Groups for Solubilization and Stabilization of Membrane Proteins

Evaluation of the Dynamic Characteristics of Coupled Double-Rotor Spindle System in High Speed Grinder

In recent years, large numbers of patents have been devoted to evaluating the dynamic characteristics of coupled double-rotor spindle system of high speed grinder for significant improvement in both the removal rate and quality of grinding. This study was focused on the theoretical modeling and numerical simulation about the dynamic characteristics of coupled double-rotor spindle system of high speed grinder. The influences of the grinder spindle’s major structural parameters on its critical speed and vibration mode were investigated. The results showed that the critical speeds of coupled double-rotor system were arranged in a similar increasing order as to those of each single rotor. It was thus indicated that the critical speeds of each order for coupled double-rotor system corresponded to those of single rotor. Furthermore, we analyzed the effect of preload of bearing, overhang length, span of bearing, rotor structure and mass of spindle system on critical speeds of various orders for the coupled spindle system. The results showed that increased bearing rigidity could increase critical speeds of each order and the effect of overhang length of coupled rotors was the superimposition of effect of single rotor on critical speeds of each order of the coupled double order system. The critical speeds of spindle system of coupled double-rotor system could be decreased by increasing the mass of its spindle system.

Inside Cover: A New Class of Amphiphiles Bearing Rigid Hydrophobic Groups for Solubilization and Stabilization of Membrane Proteins (Chem. Eur. J. 31/2012)

A class of structurally new amphiphile based on steroidal lipophilic groups have been prepared by P. S. Chae et al. on page 9485 ff. The results show that the new amphiphiles confer enhanced stability to a variety of membrane proteins in solution relative to popular conventional detergent dodecylmaltoside (DDM). The potential of the new amphiphiles as tools for solubilization, isolation, and stabilization is evident from their success with a variety of membrane proteins.

A new class of amphiphiles bearing rigid hydrophobic groups for solubilization and stabilization of membrane proteins.

Integral membrane proteins (IMPs) are crucial cellular components, mediating the transfer of material and signals between the environment and the cytoplasm, or between different cellular compartments. Structural and functional analysis of IMPs is important; more than half of current pharmaceutical agents target proteins in this class. [1] IMP characterization is often challenging, and sometimes impossible, because of difficulties associated with handling these macromolecules.[2] IMPs in the native state display large hydrophobic surfaces, which are not compatible with an aqueous environment; therefore, detergents are required to extract IMPs from the lipid bilayer and to maintain the native state of the protein in solution.[3] Nonionic detergents, such as dodecyl-β-D-maltoside (DDM) and octyl-β-D-glucoside (OG), are generally preferred for these applications. Despite the comparatively mild nature of DDM, OG and related detergents, many membrane proteins denature and/or aggregate upon solubilization with these agents.[4]

About deformation and rigidity in relativity

The notion of deformation involves that of rigidity. In relativity, starting from Born's early definition of rigidity, some other ones have been proposed, offering more or less interesting aspects but also accompanied of undesired or even pathological properties. In order to clarify the origin of these difficulties presented by the notion of rigidity in relativity, we analyze with some detail significant aspects of the unambiguous classical, Newtonian, notion. In particular, the relative character of its kinetic definition is pointed out, allowing to predict and to understand the limitations imposed by Herglotz-Noether theorem. Also, its equivalent dynamic definition is obtained and, in contrast, its absolute character is shown. But in spite of this absolute character, the dynamic definition is shown to be not extensible to relativity. The metric deformation of Minkowski space by the presence of a gravitational field is interpreted as a universal deformation, and it is shown that, under natural conditions, only a simple deformation law is possible, relating locally, but in an one-to-one way, gravitational fields and gauge classes of two-forms. We argue that fields of unit vectors associated to the internal gauge class of two-forms of every space-time (and, in particular, of Minkowski space-time) are the relativistic analogues of the classical accelerated observers, i.e. of the classical rigid motions. Some other consequences of the universal law of gravitational deformation are commented.

ChemInform Abstract: Design and Synthesis of Porphyrins Bearing Rigid Hydrogen Bonding Motifs: Highly Versatile Building Blocks for Self‐Assembly of Polymers and Discrete Arrays.

AbstractChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 100 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a “Full Text” option. The original article is trackable via the “References” option.

Spatially affine motions in relativity

Aiming to avoid the limitations that the Herglotz--Noether theorem imposes on the existence of rigid motions, as defined by Born's rigidity condition, we propose an extension of the latter, that we call spatial affinities. We then analyse these motions in Minkowski spacetime and obtain a sort of extension of the Herglotz--Noether theorem: although the space affinity condition is weaker than Born's rigidity, it does not allow for congruences of worldlines with arbitrary translational and rotational motion on one of these worldlines.

Kinematics of relativistic deformations and Thomas precession

Relativistic transformation laws of angular velocity and deformation rate for a continuous arbitrary medium are given, with respect to any change of Galileian frame of reference, as well as in the intrinsic form. In particular, in the context of Born rigidity, the relativistic effect of a spinning electron is shown: Thomas precession, as well as anew deformation effect of the same order (1/c2).

Weak rigidity in almost-thermodynamic material schemes

To avoid the restrictions that the Born rigidity supposes for the motions in relativity, the definition of a weakly rigid almost-thermodynamic material scheme is proposed. From it the relativistic incompressibility condition given by Ferrando and Olivert is obtained. Moreover, it is proved that, for the weakly rigid irrotational and geodesic almost-thermodynamic material schemes, the scalar curvature of the Landau manifolds is constant along the streamlines.

Compatibility of rigid motions with the relativistic incompressibility condition

The compatibility of the hypoelastic-Synge (isotropic case) and hypoelastic-Carter and Quintana (fully isotropic case) almost-thermodynamic material schemes with the relativistic incompressibility condition, given by Ferrando and Olivert, is analyzed. The behavior of those schemes in Born-rigid motion is also studied and an additional stipulation is joined to the Born rigidity concept. This requisite leads to the vanishing of the relativistic stress tensor spatial variation and, in hypoelastic-Carter and Quintana and hypoelastic-Maugin almost-thermodynamic material schemes, leads to the absence of mechanical power originated by internal rotation. The rigidity definition that is proposed remains valid for more general almost-thermodynamic material schemes, as far as the compatibility with the incompressibility condition quoted above is concerned.

Rigid-motion conditions in special relativity

A rigid-motion condition is proposed as a generalization of the well-known Born rigidity in order to include rotational degrees of freedom. The explicit solution is obtained in the general case and comparison is made with Born and other conditions. Classical rigidity is recovered in full in the nonrelativistic limit.

Some results on pseudorigid motions

Ehlers' and Rudolph's generalization of Born rigidity is analyzed and some properties of the corresponding flow field are derived. We investigate if at the center of motion pseudorigidity reduces to Born rigidity and give some weak criteria for determining if a given motion is pseudo rigid.

A class of coupled spaces

The algebraic structure and analytic properties of a class of Einstein spaces whose metrics are coupled by a particular type of algebraic relation are analyzed. Several invariant relations among the geometric properties of these spaces are obtained; in particular, a detailed analysis is made of the transformation of time-like congruences. The particular case in which one of the spaces is planar is used to establish the consequences of the Herglotz-Noether theorem.